Research Article A Global Multilevel Thresholding Using ...

Research ArticleA Global Multilevel Thresholding Using DifferentialEvolution Approach

Kanjana Charansiriphaisan Sirapat Chiewchanwattana and Khamron Sunat

Department of Computer Science Faculty of Science Khon Kaen University Khon Kaen 40002 Thailand

Correspondence should be addressed to Khamron Sunat khamron sunatyahoocom

Received 3 October 2013 Revised 23 January 2014 Accepted 3 February 2014 Published 20 March 2014

Academic Editor Yi-Hung Liu

Copyright copy 2014 Kanjana Charansiriphaisan et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Otsursquos function measures the properness of threshold values in multilevel image thresholding Optimal threshold values arenecessary for some applications and a global search algorithm is required Differential evolution (DE) is an algorithm that has beenused successfully for solving this problem Because the difficulty of a problem grows exponentially when the number of thresholdsincreases the ordinaryDE fails when the number of thresholds is greater than 12 An improvedDE using a newmutation strategy isproposed to overcome this problem Experiments were conducted on 20 real images and the number of thresholds varied from 2 to16 Existing global optimization algorithms were compared with the proposed algorithms that is DE rank-DE artificial bee colony(ABC) particle swarm optimization (PSO) DPSO and FODPSO The experimental results show that the proposed algorithm notonly achieves a more successful rate but also yields a lower threshold value distortion than its competitors in the search for optimalthreshold values especially when the number of thresholds is large

1 Introduction

Thresholding is the simplest and most commonly usedmethod of image segmentation It can be bilevel or multilevel[1] Both of these types can be classified into parametricand nonparametric approaches [1] Surveys of thresholdingtechniques for image segmentation can be found in [2ndash7]The surveys revealed that Otsursquos method is a commonly usedtechnique [4 8] This method finds the optimal thresholdsby maximizing the weighted sum of between-class variances(BCV) [9] The BCV function is also called Otsursquos functionHowever the solution finding process is an exhaustive searchand it is a very time-consuming process because the complex-ity grows exponentially with the number of thresholds

Multilevel image thresholding based on Otsursquos functionhas been used as a benchmark for comparing the capability ofevolutionary algorithms (EA)The EA is a nongradient basedoptimization algorithm Several algorithms have been widelyapplied to solvemultilevel thresholdingA group of successfulworks were based on a combination of Otsursquos function withsome state-of-the-art algorithms PSO [10] DE [11] ABC[12] and FOSPSO [13] Kulkarni and Venayagamoorthy [14]

showed that PSO was faster than Otsursquos method in searchingthe optimal thresholds of multilevel image thresholdingAkay [15] presented a comprehensive comparative study ofthe ABC and PSO algorithms The results showed that theABC algorithm with both the between-class variance andthe entropy criterion can be efficiently used in multilevelthresholding Hammouche et al [16] focused on solving theimage thresholding problem by combining Otsursquos functionwith metaheuristic techniques that is genetic algorithm(GA) PSO DE ant colony simulated annealing and Tabusearch Their results revealed that DE was the most efficientwith respect to the quality of solution Osuna-Enciso et al[17] presented an empirical comparative study of the ABCPSO andDE algorithms to perform image thresholding usinga mixture of Gaussian functions The results showed that theDE algorithmwas superior in performance inminimizing theHellinger distance and used less evaluations of the Hellingerdistance Ghamisia et al [18] showed that a global optimalsearch for optimal threshold values of Otsursquos function wasessential for the multilevel segmentation of multispectral andhyperspectral images

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 974024 23 pageshttpdxdoiorg1011552014974024

2 Mathematical Problems in Engineering

The DE algorithm was selected for multilevel imagethresholding It is simple to implement and produces goodresults However based on our experiments DE couldnot reach an optimal solution when it was applied to avery difficult problem Therefore a better DE algorithm isrequired We noticed that the mechanism of vector selectionand the size of the higher ranked population are an importantcriterion for success

The contribution of this paper is as followsDE with the onlooker and ranking-based mutation oper-

ation named 119874(120573)119877-DE is proposed to overcome the draw-back of the DE algorithm for multilevel image thresholdingespecially when the number of thresholds is large Theproposed algorithm homogenizes the onlooker phase of theABC algorithm and the ranking-based mutation operatorof the rank-DE [19] The main advantage of the proposedalgorithm is that a user can adjust the balancing of theexploitation and exploration capabilities of the algorithm

To verify the capabilities of the proposed 119874(120573)119877-DEalgorithm experiments to find the optimal solutions inthe multilevel image thresholding when the number ofthresholds ranged from two to 16 were set up It was foundthat the optimal solutions could be effectively reached usingthe proposed 119874(120573)119877-DE algorithm

The remainder of the paper is organized as followsSection 2 describes the multilevel thresholding problemSection 3 presents a brief review of the differential evolutionalgorithm (DE) In Section 4 the proposed new versionof the DE algorithm with the onlooker and ranking-basedmutation operator algorithm 119874(120573)119877-DE is described indetail Section 5 shows the experimental results of applyingthe proposed method to multilevel segmentation in differentimages Finally the conclusion of the paper is discussed inSection 6

2 Multilevel ThresholdingProblem Formulation

Otsursquos method [9] is based on the maximization of thebetween-class variance Consider a digital image having thesize 119867 times 119882 where 119882 is the width and 119867 is the height Thepixels of a given picture are represented in 119871 gray levels andthey are in 0 1 2 119871 minus 1 The number of pixels at level119894 is denoted by 119899

119894and the total number of pixels by 119873 =

1198991+ 1198992+ sdot sdot sdot + 119899

119871 The gray-level histogram is normalized

and regarded as a probability distribution and is written asfollows

119901119894=

119899119894

119873 119901119894ge 0

119871

sum

119894=1

119901119894= 1 (1)

The total mean of the image can be defined as

120583119879=

119871

sum

119894=1

119894 times 119901119894 (2)

The multilevel thresholding with respect to the given 119899 minus 1

threshold values 119905119895 119895 = 1 119899 minus 1 can be performed as

follows

119865 (119909 119910) =

0 119891 (119909 119910) le 1199051

1

2(1199051+ 1199052) 119905

1lt 119891 (119909 119910) le 119905

2

1

2(119905119899minus2

+ 119905119899minus1

) 119905119899minus2

lt 119891 (119909 119910) le 119905119899minus1

119871 119891 (119909 119910) gt 119905119899minus1

(3)

where (119909 119910) is the coordinate of a pixel and 119891(119909 119910) denotesthe intensity level of a pixel The pixels of a given image willbe divided into 119899 classes 119863

1 119863

119899in this regard

The optimal threshold can be determined by maximizingthe between-class variance function (BCV) 1205902

119861 which can be

defined by

1205902

119861=

119899

sum

119895=1

119908119895(120583119895minus 120583119879)2

(4)

where 119895 represents a specific class in such a way that 119908119895and

120583119895are the probability of occurrence and the mean of class

119895 respectively Equation (4) is also called Otsursquos functionThe probabilities of occurrence 119908

119895of classes 119863

1 119863

119899are

defined by

119908119895=

119905119895

sum

119894=1

119901119894 119895 = 1

119905119895

sum

119894=119905119895minus1+1

119901119894 1 lt 119895 lt 119899

119871

sum

119894=119905119895minus1+1

119901119894 119895 = 119899

(5)

The mean of each class 120583119895can be given by

120583119895=

119905119895

sum

119894=1

119894 times 119901119894

119908119895

119895 = 1

119905119895

sum

119894=119905119895minus1+1

119894 times 119901119894

119908119895

1 lt 119895 lt 119899

119871

sum

119894=119905119895minus1+1

119894 times 119901119894

119908119895

119895 = 119899

(6)

Thus the 119899-level thresholding problem is transformed to anoptimization problem The process is to search for 119899 minus 1

thresholds 119905119895that maximize the value 120593 which is generally

defined as

120593 = max1lt1199051ltsdotsdotsdotlt119905119899minus1lt119871

1205902

119861(119905119895) (7)

3 Differential Evolution Algorithm

The DE algorithm is an evolutionary optimization techniqueproposed by Storn and Price [11]Themain procedures of DEare briefly described as follows

Mathematical Problems in Engineering 3

31 Initialization The DE algorithm starts with a popu-lation of initial solutions each of dimension 119863 119883

119894119892=

(1199091198941 1199091198942 119909

119894119863) 119894 = 1 NP where the index 119894 denotes

the 119894th solution or vector of the population 119892 is thegeneration and NP is the population size The initial pop-ulation (at 119892 = 0) is randomly generated to be within thesearch space constrained by the minimum and maximumbounds 119883min = 119909

1min 1199092min 119909119863min and 119883max =

1199091max 1199092max 119909119863max The 119894th vector 119909

119894is initialized as

follows

1199091198951198940

= 119909119895min + rndreal

119894119895 [0 1) sdot (119909119895max minus 119909119895min) (8)

where rndreal119894119895[0 1) is a uniformly distributed random real

number between 0 and 1 (0 le rndreal119894119895[0 1) lt 1)

32 Mutation Operators The differential mutation operatoris one of the three operators of DE The mutation operator isapplied to generate the mutant vector V

119894for each target vector

119909119894in the current population A mutant vector is generated

according to

V119894119892+1

= 1199091199031 119892

+ 119865 sdot (1199091199032 119892

minus 1199091199033 119892

) (9)

where the randomly chosen indexes random indexes1199031 1199032 1199033

isin 1 2 NP are mutually different randominteger indices and they are also different from the runningindex 119894 Further 119894 119903

1 1199032 and 119903

3are different so that NP ge 4

119865 is a real and constant factor 119865 isin [ 0 2] which controls theamplification of the differential variation 119909

1199031 119892is called the

base vector1199091199032119892

is called the terminal vector1199091199033119892

is called theother vector and (119909

1199032 119892minus 1199091199033 119892

) is called the difference vectorThere have been many proposed mutation strategies for

DE [20 21] Each different strategy has different character-istics and is suitable for a set of problems However thechoice of the best mutation operators for DE is difficultfor a specific problem [22ndash24] The ldquoDErand1binrdquo strategyhas been widely used in DE literature [25ndash28] It is morereliable than the strategies based on the best-so-far solutionsuch as ldquoDEbest1rdquo and ldquoDEcurrent-to-best1rdquo HoweverldquoDErand1binrdquo has slower convergence Simply put it hashigh exploration but low exploitation abilities

33 Crossover DE utilizes the crossover operation to gen-erate new solutions by shuffling competing vectors and toincrease the diversity of the population The classical versionof the DE (DErand1bin) uses the binary crossover Itdefines the following trial vector

119906119894119892+1

= (1199061119894119892+1

1199062119894119892+1

119906119863119894119892+1

) (10)

where 119895 = 1 119863 (119863 = problem dimension) and

119906119895119894119892+1

=

V119895119894119892+1

if (randb (119895) le CR) and 119895 = rnbr (119894)

119909119895119894119892

if (randb (119895) gt CR) and 119895 = rnbr (119894) (11)

CR is the crossover rate isin [0 1] randb(119895) is the 119895th evaluationof a uniform random number generator with outcome isin

[0 1] and rnbr(119894) is a randomly chosen index isin 1 2 119863

that ensures 119906119894119892+1

will get at least one parameter from V119894119892+1

34 Selection Selection determines whether the target or thetrial vector survives to the next generation The selectionoperation is described as

119909119894119892+1

=

119906119894119892

if 119891 (119906119894119892) le 119891 (119909

119894119892)

119909119894119892

if 119891 (119906119894119892) gt 119891 (119909

119894119892)

(12)

where 119891(119909) is the objective function to be minimizedTherefore if the objective of the new trial vector 119891(119906

119894119892)

is equal to or less than the objective of the old trial vector119891(119909119894119892) then 119909

119894119892+1is set to 119906

119894119892 otherwise the old value 119909

119894119892

is retainedThe pseudocode of basic DE with ldquoDErand1binrdquo strat-

egy is shown in Algorithm 1The function rndint[1 119863] returns a uniformly distributed

random integer number between 1 and D rndreal119895[0 1) is a

uniformly distributed random real value of [0 1) The wordldquobetterrdquo in line 17 means ldquoless thanrdquo if the problem requiresminimization see (12) and its explanation and it meansldquogreater thanrdquo if the problem requiresmaximizationThe best119883119894119866 where 119866 is the maximum number of generations is the

solution of the algorithm The word ldquobestrdquo also depends onthe type of problem

4 The Proposed DE with Onlooker Ranking-Based Mutation Operator

In 2013 Gong and Cai [19] proposed a rank-DE algorithmThey claimed that probabilistically selecting the vectors 119909

1199031

and 1199091199032in the mutation operator from the better population

can improve the exploitation ability of basic DE To thebest of the authorsrsquo knowledge rank-DE may however alsolead to premature convergence (this will be shown in theexperiments) That means that the rank-DE has too muchexploitation ability Furthermore it cannot balance betweenthe exploration and the exploitation abilities In order tobalance between the two abilities we propose DE withthe onlooker and ranking-based mutation operator named119874(120573)119877-DEThe proposed algorithm is an improvement of therank-DE by homogenizing the rank-DE with the onlookerphase of ABC algorithm The detail of the 119874(120573)119877-DE algo-rithm is described as follows

41 Ranking Assignment To perform the maximization thefitness of each vector is sorted in ascending order (ie fromworst to best) Then the rank of the 119894th vector 119877

119894 is assigned

based on its sorted ordering as follows

119877order = order order = 1 2 NP (13)

As a result the best vector in the current population willobtain the highest ranking that is NP


(1) Generate the initial population randomly(2) Evaluate the fitness for each individual in the population(3) while the maximum generation G is not reached do(4) for 119894 = 1 to NP do(5) Select uniform randomly 119903

1= 1199032

= 1199033

= 119894

(6) 119895rand = rndint [1 119863]

(7) for 119895 = 1 to119863 do(8) if rndreal

119895[0 1) le CR or 119895 is equal to 119895rand then

(9) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(10) else(11) 119906

119894119895= 119909119894119895

(12) end if(13) end for(14) end for(15) for 119894 = 1 to NP do(16) Evaluate the offspring 119906

119894

(17) if 119891(119906119894) is better than or equal to 119891(119909

119894) then

(18) Replace 119909119894with 119906

119894

(19) end if(20) end for(21) end while

Algorithm 1 The DE algorithm with ldquoDErand1binrdquo strategy

42 Probabilistic Selection After assigning the ranking foreach vector the selection probability 119901

119894of the 119894th vector 119909

119894

is calculated as

119901119894=

119877119894

NP 119894 = 1 2 NP (14)

43 A New Strategy for Base Vector Terminal Pointand the Other Vector Selections

Definition 1 (a worse population and a better population)Let 120577 be a real value and 0 le 120577 lt 1 A population havingprobability less than 120577 is called a worse population and apopulation having probability greater than or equal to 120577 iscalled a better population

In the rank-DE the base vector 1199091199031and the terminal point

1199091199032

were based on their selection probabilities The othervector in themutation operator119909

1199033 is selected randomly as in

the original DE algorithm The vectors with higher rankings(higher selection probabilities) aremore likely to be chosen asthe base vector or the terminal point in themutation operator

Our investigation revealed that if both 1199091199031and 119909

1199032vec-

tors of rank-DE were chosen from better vectors then thedistribution of the target vector may collapse quickly andpossibly lead to premature convergence Accordingly whenthe rank-DE was applied to a very difficult problem it couldnot reach the optimal solution

If the steps of the DE algorithm are compared with theABC algorithm the population in the current generation canbe considered as the employed bees and the population inthe next generation can be considered as the onlooker beesTo follow the concept of ABC a new vector 119909

1199031 which is

called the base vector chooses a food source with respect to

the probability that is computed from the fitness values ofthe current population The probability value 119901

119905 of which

119909119905is chosen by a base vector 119909

1199031can be calculated by using

the expression given in (14) After a base source 1199091199031

fora new vector is probabilistically chosen both 119909

1199032and 119909

1199033

are also chosen in the same manner as the terminal pointand the other vector selections in the rank-DE The targetvector is created by a mutation formula of DE The mutantvector 119906

119894is created after the target vector is crossed with

a randomly selected vector and then the fitness value iscomputedAs in the ordinaryDE a greedy selection is appliedbetween 119906

119894and 119909

119894 Hence the new population contains better

sources and positive feedback behavior appears This ideacan be expressed as pseudocode as in Algorithm 2 Since theselection of 119909

1199031is the onlooker selection and the selections of

1199091199032and 119909

1199033are brought from the rank-DE then the algorithm

is called onlooker and ranking-based vector selectionThe pseudocode of onlooker and ranking-based vector

selection is shown in Algorithm 2 The differences betweenthe original ranking-based and onlooker and ranking basedselection are highlighted by ldquolArrrdquo

The function minprop(120573) is added to generalize thealgorithm Its output depends on the parameter 120573 Theoutcome can be either a constant value of [0 1) or a valueof the uniform random function rndreal[0 1) The balanceof the exploration and exploitation ability can be set by theparameter 120573 And the function is defined by

minprop (120573)

= 120573 if120573 is a constant and 0 le 120573 lt 1

rndreal [0 1) otherwise(15)


(1) Input The target vector index 119894 the last index of onlooker 1199031 and 120573 lArr

(2) Output The selected vector indexes 1199031 1199032 1199033

(3) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr

(4) while minprop(120573) gt 1199011199031

onlooker-like selection lArr

(5) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr

(6) end while(7) Randomly select 119903

2isin 1NP terminal vector index

(8) while rndreal[0 1) gt 1199011199032or 1199032== 119894 or 119903

2== 1199031do

(9) Randomly select 1199032isin 1NP


3isin 1NP the other vector index

(12) while 1199033== 1199032or 1199033== 1199031or 1199033== 119894 do


(14) end while

Algorithm 2 Onlooker and ranking-based vector selection for DE

(1) Randomly generate the initial population(2) Evaluate the fitness for each individual in the population(3) while the maximum generation G is not reached do(4) Sort and rank the fitness values of population according to (13)(5) Calculate the selection probability for each individual according to (14)(6) 119903

1= 0 lArr

(7) for 119894 = 1 to NP do(8) Select 119903

1 1199032 1199033as shown in Algorithm 2 based on the current 119903

1and 120573 lArr

(9) 119895rand = rndint[1 119863]



(12) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(13) else(14) 119906

119894119895= 119909119894119895


119894


119894) then

(21) Replace 119909119894with 119906

119894


Algorithm 3 DE with onlooker and ranking-based mutation

44TheDEwithOnlooker-Ranking-BasedMutationOperatorThe procedures in Sections 41 42 and 43 are combinedtogether to create a better DE algorithm The parameter 0 le

120573 lt 1 determines the fraction of the worse population tobe eliminated When 120573 = 0 there is no worse populationeach single vector in the current population will act as thebase vector If 0 lt 120573 lt 1 then each single vector havinga probability less than 120573 is a worse vector and will not beselected as the base vector If 120573 is not a constant or is outside[0 1) each single base vector is an onlooker bee Accordinglythe name of the algorithm is Onlooker(120573) Ranking-BaseDifferential Evolution (119874(120573)119877-DE) To achieve the globalsolution a user can set a proper value for 120573 to control

the balance of the exploration and exploitation abilities ofthe algorithm The pseudocode of 119874(120573)119877-DE is shown inAlgorithm 3 and the differences between the rank-DE and119874(120573)119877-DE are highlighted by ldquolArrrdquo

5 Experiments and Results

51 Experimental Setup The global multilevel thresholdingproblem deals with finding optimal thresholds within therange [0 119871 minus 1] that maximize the BCV function Thedimension of the optimization problem is the number ofthresholds 119899 and the search space is [0 119871minus1]119899Theparameter


120573 of 119874(120573)119877-DE is rndreal[0 1) or is set to be one of 0001 09 The variation of the proposed 119874(120573)119877-DE wasimplemented and compared with the existing metaheuristicsthat performed image thresholding that is PSO DPSOFODPSO ABC and several variations of DE algorithmsAll the methods were programmed in Matlab R2013a andwere run on a personal computer with a 34GHz CPU 8GBRAM with Microsoft Windows 7 64-bit operating systemThe experiments were conducted on 20 real images The 19images namely starfish mountain cactus butterfly circussnow palace flower wherry waterfall bird police ostrichviaduct fish houses mushroom snowmountain and snakewere taken from the Berkeley Segmentation Dataset andBenchmark [29] The last image namely Riosanpablo is asatellite image ldquoNew ISS Eyes see Rio San Pablordquo March 12013 (httpvisibleearthnasagovviewphpid=80561) Eachimage has a unique gray level histogram These originalimages and their histograms are depictedin Figure 1 Anexperiment of an image with a specific number of thresholdsis called a ldquosubproblemrdquo The number of thresholds inves-tigated in the experiments was 2 3 16 Thus there are20 times 15 subproblems per algorithm Each subproblem wasrepeated 50 times and each time is called a run

To compare with PSO ABC and DEs algorithms theobjective function evaluation is computed for NPtimes119873

119894 where

NP is population size and 119873119894is the number of generations

A population of PSO and the DEs calls Otsursquos function onetime per generation The population size in the PSO andDEs algorithms was set to 50 A bee in the ABC calls Otsursquosfunction two times per generation therefore their number offood sources were set to a half of the PSOrsquos size that is 25The stopping criteria were set by the maximum amount ofgenerations 119866 In this experiment 119866 was set to 50 100 150200 300 400 600 800 1000 1500 2000 3000 4000 5000and 6000 when 119899 was 2 3 4 5 6 7 8 9 10 11 12 13 1415 and 16 respectively For the PSO DPSO and FODPSOalgorithms the parameters were set as per the suggestion in[30] and is shown in Table 1 The other control parameterof the ABC algorithm limit was set to 50 [15] The controlparameters 119865 and CR of the DE algorithms were set to 05and 09 respectively [31 32]

52 Comparison Strategies and Metrics To compare theperformance of different algorithms there are three metrics(1) the convergence rate of algorithms was compared by theaverage of generations (NG) a lower NG means a fasterconvergence rate (2) the stability of algorithmswas comparedby the average of the success rate (SRHM) a higher SRHMmeans higher stability (3) the reliability was compared bythe threshold value distortion measure (TVD) a lower TVDmeans higher reliability The details of the three metrics aredescribed as follows

When all 50 runs of an algorithm performing on animage with a specific number of thresholds are terminatedthe outcomes will be analyzed Run 119903rsquoth is called a successfulrun if there is a generation of 119905 le G such that BCV

119903(119905) ge VTR

Table 1 Essential parameters of the PSO DPSO and FODPSOtaken from [30]

Parameter PSO DPSO FODPSOPopulation 50 50 501205881

15 15 151205882

15 15 15119882 12 12 12119881max 2 2 2119881min minus2 minus2 minus2119909max 255 255 255119909min 0 0 0Min population mdash 10 10Max population mdash 50 50No of swarms mdash 4 4Min swarms mdash 2 2Max swarms mdash 6 6Stagnancy mdash 10 10Fractional coefficient mdash mdash 075

and the number of generations (NG) of the successful run isrecorded Thus the number can be defined by

NG119903= ArgMin

119905

(BCV119903 (119905) ge VTR)

if 119903 is a successful run and otherwise undefined(16)

The average of NG119903from those successful runs is represented

by NG as follows

NG =1

number of successful runssum

All successful runsNG119903

(17)

The ratio of success rate (SR) for which the algorithmsucceeds to reach the VTR for each subproblem is computedas

SR =number of successful runs

total number of runs (18)

The experiments were conducted on 20 images The arith-meticmean (AM) ofNG (NGAM) over the entire set of imageswith a specific number of thresholds is calculated as

NGAM =1

119873sum

All imagesNG (19)

where 119873 is the total number of images NGAM is shown inTable 3 The worst-case scenario is that there is no successfulrun for a subproblem this subproblem is called an ldquounsuc-cessful subproblemrdquo If an algorithm encounters this scenariothe subproblem will be grouped by its number of thresholdsand the number of images in the group will be counted andassigned to 119909 These scenarios will be represented by NA(119909)as shown in Tables 3 and 4


0

400

800

1200

1600

0

400

800

1200

1600

0

400

800

1200

1600

0

1000

2000

3000

4000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

0 50 100 150 200 2500 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250 0 50 100 150 200 250

0 50 100 150 200 2500

400

800

1200

1600

0 50 100 150 200 250

0 50 100 150 200 250

0

400

800

1200

1600

(1) Starfish (481 times 321) (6) Snow (481 times 321)

(2) Mountain (481 times 321)

(3) Cactus (481 times 321)

(4) Butterfly (481 times 321)

(5) Circus (481 times 321)

(8) Flower (481 times 321)

(9) Wherry (321 times 481)

(7) Palace (321 times 481)

(10) Waterfall (321 times 481)

(a)Figure 1 Continued


0500

1000150020002500

0 50 100 150 200 2500

500

1000

1500

2000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 2500

400

800

1200

1600

2000

0 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0500

1000150020002500

0 50 100 150 200 250

0500

10001500200025003000

0

2000

4000

6000

8000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 250

(11) Bird (321 times 481)

(12) Police (321 times 481)

(13) Ostrich (321 times 481)

(14) Viaduct (481 times 321)

(15) Fish (481 times 321)

(16) Houses (481 times 321)

(17) Mushroom (321 times 481)

(19) Snake (481 times 321)

(20) Riosanpablo (720 times 944)

(18) Snow mountain (321 times 481)

(b)

Figure 1 The test images and corresponding histograms


The average of the success rate over the entire dataset witha specific number of thresholds (SRHM) is averaged by theHarmonic mean HM as follows

SRHM =119873

sumAll images (1SR) (20)

The SRHM is very important in measuring the stability ofan algorithm and it means the ratio of runs that are achievingthe target solution Because the evolutionary methods arebased on stochastic searching algorithms the solutions arenot the same in each run of the algorithm and depend on thesearch ability of the algorithmTherefore the SRHM is vital inevaluating the stability of the algorithms The comparison ofthe stability gives us valuable information in terms of the ratiorepresenting the success rates (SRHM) A higher SRHM meansbetter stability of the algorithm

An algorithm producing SRHM lt 05 means that morethan 50 percent of the independent runs of the algorithmcannot reach the global solution Thus the algorithm thatyields SRHM lt 05 should not be selected to solve theproblemThe experiments were conducted for the number ofthresholds varying from 2 to 16These experiments containedthe maximum number of thresholds such that the algorithmyields SRHM ge 05 which is represented by 119899

05in Table 4

Furthermore the experiments also contained the maximumnumber of thresholds that the algorithm can solve and abovethis value there was the case such that all 50 runs of somesubproblems missed the VTRThis number is represented by119899max In this case the success rate was zero and the associatedSRHM was zero too And the definitions of the two values arepresented in (21)

11989905

= max (119899 | 119899 = number of thresholds that

has SRHM ge 05)

119899max = min (119899 | 119899 = number of thresholds that

has SRHM = 0) minus 1

(21)

Let 119899 be the number of thresholdsThe reliability of a solutionis measured by threshold value distortion measure (TVD)and is computed as

TVD =

sumAll images sumrun119903=1

sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816

1 + sumAll images sumrun119903=1

sum119899

119894=11119879lowast119903119894= 119879119898119903119894

times (1 minus SR) times 100

(22)

where119879lowast is the threshold value producing the VTR119879119898 is thethreshold value obtained from the algorithm and 1

119879lowast119903119894= 119879119898119903119894is

the indicator function which is equal to 1 when 119879lowast

119903119894= 119879119898

119903119894and

is zero otherwise TVD is zero if the algorithm can reach theVTR in every run The lower the TVD the more reliable thealgorithm is

521 The Value to Reach (VTR) Following the completionof all of the experiments the best values of the between-classvariance and the corresponding thresholds were collected

2 4 6 8 10 12 14 160

200

400

600

800

1000

1200

Number of thresholds

Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =

Mutlithresh 120575 = 8686

Figure 2The threshold value distortion (TVD) of algorithms versusnumber of thresholds

and are shown in Table 6 The results are shown image byimage and the numbers of thresholds vary from 2 to 16 Thebetween-class variance values in column 3 are used as theVTR values

522 Results Produced by Local Search Method The multi-thresh function of the Matlab toolbox was conducted on thesame images and number of thresholds as the other searchmethods The capabilities of solving the optimal solutionbetween a local search and a global search will be discussedhere This is the reason we focused on the global searchthat is the proposed 119874(00)119877-DE algorithm Table 2 showsthe between-class variances and threshold values of theldquomountainrdquo image These values were the best outcomes of50 runs produced by the multithresh function in the MatlabR2013a toolbox and by the proposed 119874(00)119877-DE algorithmThe terminated condition of the multithresh function was setby ldquoMaxFunEvalsrdquo = 500000That is themultithresh functionperforms more function calls than that of the 119874(00)119877-DEalgorithm It can be seen from columns 3 and 5 that allthe BCVs produced by the 119874(00)119877-DE algorithm are betterthan the BCVs produced by the multithresh function thedifference of the BCVs is shown in column 7 The differencesin the thresholds from the two algorithms shown in column8 tended to be large if the number of thresholds increased

Figure 2 shows the graph of the TVDof all the images andthresholdsThese results are in the same pattern of the resultsof the ldquomountainrdquo image in Table 2 That means the ability tosearch for the optimal solution of the proposed global searchalgorithm is higher than that of the multithresh functionespecially when the number of thresholds is large This goesto illustrate the difficulty of the problem The problem withthis kind is that it can be multimodal [33] or can be anearly flat top surface [34] The multithresh function solvesthe problem by performing the Nelder-Mead Simplex


Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21


Table 6 The between-class variance criterion and the best thresholds for test images

Image 119899 Between-class variance (VTR) Thresholds

Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227

method [35] which is a local search method that cannotguarantee an optimal solutionThus its solutions are inferiorto the solution produced by the algorithm using a globalsearch

523 Convergence Rate Comparison The number of gen-erations (NG) is a measure used for the convergence ratecomparisons If the target value VTR is achieved in a lessernumber of generations (NG) it means a faster convergencerate for the algorithm Table 3 shows the average of NG(NGAM) for each specific number of thresholdsThe results ofeach algorithmare represented in the corresponding columnrsquosname In each column the cell containing NGAM starts fromthe row associated with 119899 = 2 until the row associated with119899 = 119899max The cells associated with 119899 = 119899max + 1 to the row

associated with 119899 = 16 are filled by NA(119909) The second lastrow of the table is filled by the triple

(119899maxNA (number of unsuccessful subproblem)

AM (NGAM of 119899 = 2 until 119899 = 119899max)) (23)

The algorithm with the highest 119899max that is the lowestnumber of unsuccessful subproblems and the lowest averageof generation is the winner The ranking of the algorithmsdepends on the ordering of (a

1 b1 c1) and (a

2 b2 c2) as

followsFirst rank on a

1and a

2 Since both a

1and a

2are

numeric the higher value has the higher rankSecond rank on b

1and b

2 If a1is 16 then b

1must

be NA(0) If a1is less than 16 then b

1must be NA


(number of unsuccessful subproblems) and b2has the

same characteristics as b1 Perform the order of the

two numeric values in reverse the lower value has thehigher rankThird rank on c

1and c

2 They are numeric but the

lower is better perform the order of the two numericvalues in reverseThe lower value has the higher rank

When the ordering is finished assign the numeric value ofldquo1rdquo to the object having the highest rank assign the numericvalue of ldquo2rdquo to the first runner up and so onThese values arerepresented in the last row of the table

If the row having 119899 = 119898 must be ranked it can be donein the samemanner as above with someminormodificationsIf 119898 is less than 119899max the values of the triple pair will be 119898NA(0) NGAM If119898 is greater than 119899max the values of the triplepair will be (119899max NA(119909) infin) Thus the ranking can now beperformed

From the ranking results see the second last row ofTable 3 the convergence rate can be ranked from best toworst in the following order 119874(02)119877-DE 119874(01)119877-DE119874(00)119877-DE 119874(rand)119877-DE 119874(03)119877-DE 119874(04)119877-DE119874(05)119877-DE 119874(06)119877-DE DE 119874(07)119877-DE 119874(08)119877-DErank-DE 119874(09)119877-DE ABC PSO DPSO FODPSO

As can be seen from Table 3 the DE algorithm cannotcomplete the task when 119899 gt 12 and the rank-DE algorithmcannot complete the task when 119899 gt 9 Thus rank-DEcannot compete with DE on searching for global multilevelthresholding

In order for119874(120573)119877-DE to outperform DE then 120573must bein the range of [01 06] or set to rndreal[0 1)

524 Stability Analysis The harmonic mean of the successrate (SRHM) for each specific number of thresholds wascomputed and is presented in Table 4 The results of eachalgorithm are represented in the corresponding columnrsquosname The second row from the bottom shows the harmonicmean of the success rate of each algorithm for all thresholdlevels

In each column the cells containing SRHM start from therow associated with 119899 = 2 to the row associated with 119899 = 119899

05

The cells from the row associated with 119899 = 11989905

+ 1 to the rowassociatedwith 119899 = 119899max are filled by SRHMThe cells from therow associatedwith 119899 = 119899max+1 to the row associatedwith 119899 =

16 are the cells that have SRHM these cells will be excludedfrom the comparisonThe second last row of the table is filledwith the triple

(11989905

119899maxHM (SRHM of 119899 = 2 until 119899 = 119899max)) (24)

The algorithm with the highest 11989905 the highest 119899max and

the highest average success rate is the winner The ranking ofthe algorithms depends on the ordering of (a

1 b1 c1) and (a

2

b2 c2) as follows

First rank on a1and a2

Second rank on b1and b2

Third rank on c1and c2

Because they are numeric the higher value has the higherrank When the ordering is finished the numeric value of ldquo1rdquois assigned to the object having the highest rank the numericvalue of ldquo2rdquo is assigned to the first runner up and so on

If the row having 119899 = 119898must be ranked it can be done inthe samemanner as above with someminor modifications If119898 is less than or equal to 119899

05 the values of the triple pair will

be (119898 119898 SRHM) If 11989905 lt 119898 le 119899max then the values of thetriple pair will be (119899

05 119898 SRHM) If 119898 is greater than 119899max

the values of the triple pair will be (11989905 119899max SRHM) Thus

the ranking can now be performedFrom the ranking results see Table 4 the success rate

can be ranked from best to worst in the following order119874(00)119877-DE 119874(01)119877-DE 119874(02)119877-DE 119874(rand)119877-DE119874(03)119877-DE 119874(05)119877-DE 119874(04)119877-DE DE 119874(06)119877-DE119874(07)119877-DE 119874(08)119877-DE rank-DE 119874(09)119877-DE ABCPSO DPSO FODPSO multithresh

As can be seen from Table 4 the DE algorithm has anSRHM ge 05 until 119899 = 9 and the rank-DE algorithmhas an SRHM ge 05 until 119899 = 7 This result confirmsthat rank-DE cannot compete with DE on searching forglobal multilevel thresholding If the correct 120573 is selectedthe proposed algorithm can work very well For 120573 le 05119874(120573)119877-DE has a higher rank than DE The multithreshfunction cannot compete with any of the other algorithmsIt can also be seen that the proposed 119874(00)119877-DE algorithmhas the best stability because its SRHM is greater than 05when119899 = 2 to 16

525 Reliability Comparison The threshold value distortionaka TVD for each specific threshold is computed shownin Table 5 and depicted in Figure 2 The results of eachalgorithmare illustrated in the corresponding columnrsquos nameThe second last row of Table 5 shows the slope or theapproximated growth rate 120575 of the TVD of each algorithmfor all threshold levels The 120575 is the slope of the robust linearregression computed by the Matlab function ldquorobustfitrdquo Thelower slope exhibits the better reliability The 120575 of eachalgorithm is sorted in descending order From the results thereliability can be ranked from best to worst in the followingorder 119874(00)119877-DE 119874(01)119877-DE 119874(02)119877-DE 119874(03)119877-DE119874(04)119877-DE 119874(05)119877-DE 119874(rand)119877-DE DE 119874(06)119877-DEABC rank-DE 119874(07)119877-DE 119874(08)119877-DE 119874(09)119877-DEDPSO FODPSO PSO multithresh

We can see from these results that rank-DE has a higherapproximatedTVDgrowth rate thanDE119874(120573)119877-DEwith120573 le

05 and 119874(rand)119877-DE are still better than DE 119874(00)119877-DEproduced the best result with a very flat slope and a verylow 119910-intercept The multithresh function yielded a highergrowth rate of solution distortion and a higher 119910-interceptthan the other algorithmsThe higher growth rate of solutiondistortion means the quality of solution drops very fast ifthe number of thresholds increases The higher 119910-interceptmeans that the solution distortion at the lowest number ofthresholds is highThus the global optimization algorithm isrequired for solving the multilevel thresholding


6 Conclusions

The differential evolution with onlooker (120573) ranking-basedmutation operator 119874(120573)119877-DE algorithm was proposed andapplied to the multilevel image thresholding problem Theobjective of this proposed algorithm was to increase theability for adjusting the balance of the exploitation andexploration abilities Its concept is a combination of theranking-difference evolution and onlooker selection of theABC algorithm The experiments compared the proposed119874(120573)119877-DE algorithm with six existing algorithms PSODPSO FODPSO ABC DE and rank-DE on 20 real imagesof the Berkeley Segmentation Dataset and Benchmark anda satellite image The stability analysis convergence speedand the reliability were measured The results signified thatthe proposed 119874(120573)119877-DE algorithm is more efficient than thesix tested algorithms The onlooker ranking-based mutationoperator is able to enhance the performance of the proposedalgorithm The 119874(120573)119877-DE not only obtained more stabilityanalysis but it also achieved faster convergence rates to reachthe target BCV if a proper value of 120573 is set

For future work based on this paper the proposed119874(120573)R-DE algorithm has one parameter to be set by a userthe mechanism to automatically adapt this parameter is notpresented but is required

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Higher Education ResearchPromotion andNational Research University Project ofThai-land Office of the Higher Education Commission throughthe Cluster of Research to Enhance the Quality of BasicEducation

References

[1] W K Pratt Digital Image Processing John Wiley amp Sons NewYork NY USA 1978

[2] J S Weszka ldquoA survey of threshold selection techniquesrdquo Com-puter Graphics and Image Processing vol 7 no 2 pp 259ndash2651978

[3] K S Fu and J K Mui ldquoA survey on image segmentationrdquo Pat-tern Recognition vol 13 no 1 pp 3ndash16 1981

[4] P K Sahoo S Soltani and A K C Wong ldquoA survey ofthresholding techniquesrdquoComputer Vision Graphics and ImageProcessing vol 41 no 2 pp 233ndash260 1988

[5] N R Pal and S K Pal ldquoA review on image segmentation tech-niquesrdquo Pattern Recognition vol 26 no 9 pp 1277ndash1294 1993

[6] A T Abak U Baris and B Sankur ldquoThe performanceevaluation of thresholding algorithms for optical characterrecognitionrdquo in Proceedings of the 4th International Conferenceon Document Analysis and Recognition pp 697ndash700 UlmGermany August 1997

[7] M Sezgin and B Sankur ldquoSurvey over image thresholdingtechniques and quantitative performance evaluationrdquo Journal ofElectronic Imaging vol 13 no 1 pp 146ndash168 2004

[8] SU Lee and S YoonChung ldquoA comparative performance studyof several global thresholding techniques for segmentationrdquoComputer Vision Graphics and Image Processing vol 52 no 2pp 171ndash190 1990

[9] N Otsu ldquoA threshold selection method from gray level his-togramsrdquo IEEE Transactions on Systems Man and Cyberneticsvol 9 no 1 pp 62ndash66 1979

[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) vol 4 pp 1942ndash1948 Perth AustraliaDecember 1995

[11] R Storn and K Price ldquoDifferential evolution-a simple andefficient heuristic for global optimization over continuousspacesrdquo Tech Rep TR-95-012 International Computer SciencesInstitute Berkeley Calif USA 1995

[12] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] M S Couceiro R P Rocha N M F Ferreira and J A TMachado ldquoIntroducing the fractional-order Darwinian PSOrdquoSignal Image and Video Processing vol 6 no 3 pp 343ndash3502012

[14] R V Kulkarni and G K Venayagamoorthy ldquoBio-inspired algo-rithms for autonomous deployment and localization of sensornodesrdquo IEEE Transactions on Systems Man and Cybernetics Cvol 40 no 6 pp 663ndash675 2010

[15] B Akay ldquoA study on particle swarm optimization and artificialbee colony algorithms for multilevel thresholdingrdquo Applied SoftComputing Journal vol 3 no 6 pp 3066ndash3091 2013

[16] K Hammouche M Diaf and P Siarry ldquoA comparative studyof various meta-heuristic techniques applied to the multilevelthresholding problemrdquo Engineering Applications of ArtificialIntelligence vol 23 no 5 pp 676ndash688 2010

[17] V Osuna-Enciso E Cuevas and H Sossa ldquoA comparison ofnature inspired algorithms for multi-threshold image segmen-tationrdquo Expert Systems with Applications vol 40 no 4 pp 1213ndash1219 2013

[18] P Ghamisia M S Couceiro F Martins and J A BenediktssonldquoMultilevel image segmentation based on Fractional-OrderDarwinian particle swarm optimizationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 52 no 1 pp 1ndash44 2013

[19] W Gong and Z Cai ldquoDifferential evolution with ranking-basedmutation operatorsrdquo IEEE Transactions on Cybernetics vol 43no 6 pp 2066ndash2081 2013

[20] K Price R Storn and J Lampinen Differential Evolution APractical Approach to Global Optimization Springer BerlinGermany 2005

[21] R Storn and K Price Home Page of Differential EvolutionInternational Computer Science Institute Berkeley Calif USA2010

[22] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (CEC rsquo05)pp 1785ndash1791 September 2005

[23] EMezura-Montes J Velazquez-Reyes andC A Coello CoelloldquoA comparative study of differential evolution variants for globaloptimizationrdquo in Proceedings of the 8th Annual Genetic andEvolutionary Computation Conference pp 485ndash492 July 2006


[24] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[25] S Das A Abraham and A Konar ldquoAutomatic clustering usingan improved differential evolution algorithmrdquo IEEE Transac-tions on Systems Man and Cybernetics A vol 38 no 1 pp 218ndash237 2008

[26] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[27] N Noman andH Iba ldquoAccelerating differential evolution usingan adaptive local searchrdquo IEEE Transactions on EvolutionaryComputation vol 12 no 1 pp 107ndash125 2008

[28] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[29] D Martin C Fowlkes D Tal and J Malik ldquoA databaseof human segmented natural images and its application toevaluating segmentation algorithms and measuring ecologicalstatisticsrdquo in Proceedings of the 8th International Conference onComputer Vision vol 2 pp 416ndash423 July 2001

[30] P Ghamisi M S Couceiro J A Benediktsson and N M FFerreira ldquoAn efficient method for segmentation of image basedon fractional calculus and natural selectionrdquo Expert Systemswith Applications vol 39 pp 12407ndash12417 2012

[31] R Gamperle S D Muller and A Koumoutsakos ldquoA parameterstudy for differential evolutionrdquo in Proceedings of the Advancesin Intelligent Systems Fuzzy Systems Evolutionary Computationvol 10 pp 293ndash298 2002

[32] M Saraswat K V Arya and H Sharma ldquoLeukocyte segmen-tation in tissue images using differential evolution algorithmrdquoSwarm and Evolutionary Computation vol 11 pp 46ndash54 2013

[33] J Kittler and J Illingworth ldquoOn threshold selection usingclustering criteriardquo IEEE Transactions on Systems Man andCybernetics vol 15 no 5 pp 652ndash655 1985

[34] R Guo and S M Pandit ldquoAutomatic threshold selection basedon histogram modes and a discriminant criterionrdquo MachineVision and Applications vol 10 no 5-6 pp 331ndash338 1998

[35] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the Nelder-Mead simplex methodin low dimensionsrdquo SIAM Journal on Optimization vol 9 no 1pp 112ndash147 1998

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The DE algorithm was selected for multilevel imagethresholding It is simple to implement and produces goodresults However based on our experiments DE couldnot reach an optimal solution when it was applied to avery difficult problem Therefore a better DE algorithm isrequired We noticed that the mechanism of vector selectionand the size of the higher ranked population are an importantcriterion for success

The contribution of this paper is as followsDE with the onlooker and ranking-based mutation oper-

ation named 119874(120573)119877-DE is proposed to overcome the draw-back of the DE algorithm for multilevel image thresholdingespecially when the number of thresholds is large Theproposed algorithm homogenizes the onlooker phase of theABC algorithm and the ranking-based mutation operatorof the rank-DE [19] The main advantage of the proposedalgorithm is that a user can adjust the balancing of theexploitation and exploration capabilities of the algorithm

To verify the capabilities of the proposed 119874(120573)119877-DEalgorithm experiments to find the optimal solutions inthe multilevel image thresholding when the number ofthresholds ranged from two to 16 were set up It was foundthat the optimal solutions could be effectively reached usingthe proposed 119874(120573)119877-DE algorithm

The remainder of the paper is organized as followsSection 2 describes the multilevel thresholding problemSection 3 presents a brief review of the differential evolutionalgorithm (DE) In Section 4 the proposed new versionof the DE algorithm with the onlooker and ranking-basedmutation operator algorithm 119874(120573)119877-DE is described indetail Section 5 shows the experimental results of applyingthe proposed method to multilevel segmentation in differentimages Finally the conclusion of the paper is discussed inSection 6

2 Multilevel ThresholdingProblem Formulation

Otsursquos method [9] is based on the maximization of thebetween-class variance Consider a digital image having thesize 119867 times 119882 where 119882 is the width and 119867 is the height Thepixels of a given picture are represented in 119871 gray levels andthey are in 0 1 2 119871 minus 1 The number of pixels at level119894 is denoted by 119899

119894and the total number of pixels by 119873 =

1198991+ 1198992+ sdot sdot sdot + 119899

119871 The gray-level histogram is normalized

and regarded as a probability distribution and is written asfollows

119901119894=

119899119894

119873 119901119894ge 0

119871

sum

119894=1

119901119894= 1 (1)

The total mean of the image can be defined as

120583119879=

119871

sum

119894=1

119894 times 119901119894 (2)

The multilevel thresholding with respect to the given 119899 minus 1

threshold values 119905119895 119895 = 1 119899 minus 1 can be performed as

follows

119865 (119909 119910) =

0 119891 (119909 119910) le 1199051

1

2(1199051+ 1199052) 119905

1lt 119891 (119909 119910) le 119905

2

1

2(119905119899minus2

+ 119905119899minus1

) 119905119899minus2

lt 119891 (119909 119910) le 119905119899minus1

119871 119891 (119909 119910) gt 119905119899minus1

(3)

where (119909 119910) is the coordinate of a pixel and 119891(119909 119910) denotesthe intensity level of a pixel The pixels of a given image willbe divided into 119899 classes 119863

1 119863

119899in this regard

The optimal threshold can be determined by maximizingthe between-class variance function (BCV) 1205902

119861 which can be

defined by

1205902

119861=

119899

sum

119895=1

119908119895(120583119895minus 120583119879)2

(4)

where 119895 represents a specific class in such a way that 119908119895and

120583119895are the probability of occurrence and the mean of class

119895 respectively Equation (4) is also called Otsursquos functionThe probabilities of occurrence 119908

119895of classes 119863

1 119863

119899are

defined by

119908119895=

119905119895

sum

119894=1

119901119894 119895 = 1

119905119895

sum

119894=119905119895minus1+1

119901119894 1 lt 119895 lt 119899

119871

sum

119894=119905119895minus1+1

119901119894 119895 = 119899

(5)

The mean of each class 120583119895can be given by

120583119895=

119905119895

sum

119894=1

119894 times 119901119894

119908119895

119895 = 1

119905119895

sum

119894=119905119895minus1+1

119894 times 119901119894

119908119895

1 lt 119895 lt 119899

119871

sum

119894=119905119895minus1+1

119894 times 119901119894

119908119895

119895 = 119899

(6)

Thus the 119899-level thresholding problem is transformed to anoptimization problem The process is to search for 119899 minus 1

thresholds 119905119895that maximize the value 120593 which is generally

defined as

120593 = max1lt1199051ltsdotsdotsdotlt119905119899minus1lt119871

1205902

119861(119905119895) (7)

3 Differential Evolution Algorithm

The DE algorithm is an evolutionary optimization techniqueproposed by Storn and Price [11]Themain procedures of DEare briefly described as follows



119894119892=

(1199091198941 1199091198942 119909






follows

1199091198951198940

= 119909119895min + rndreal







according to

V119894119892+1

= 1199091199031 119892

+ 119865 sdot (1199091199032 119892

minus 1199091199033 119892

) (9)



1 1199032 and 119903




base vector1199091199032119892



1199032 119892minus 1199091199033 119892




119906119894119892+1

= (1199061119894119892+1

1199062119894119892+1

119906119863119894119892+1

) (10)


119906119895119894119892+1

=

V119895119894119892+1


119909119895119894119892




that ensures 119906119894119892+1



119909119894119892+1

=

119906119894119892

if 119891 (119906119894119892) le 119891 (119909

119894119892)

119909119894119892

if 119891 (119906119894119892) gt 119891 (119909

119894119892)

(12)


119894119892)


119894119892+1is set to 119906


119894119892








1199031




119894 is assigned






1= 1199032

= 1199033

= 119894

(6) 119895rand = rndint [1 119863]



(9) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(10) else(11) 119906

119894119895= 119909119894119895


119894


119894) then

(18) Replace 119909119894with 119906

119894




119894of the 119894th vector 119909

119894

is calculated as

119901119894=

119877119894

NP 119894 = 1 2 NP (14)




1199091199032





1199032vec-



1199031 which is



119905 of which





1199032and 119909

1199033




119894and 119909




1199091199032and 119909





minprop (120573)






(3) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr

(4) while minprop(120573) gt 1199011199031


(5) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr




2== 1199031do




(12) while 1199033== 1199032or 1199033== 1199031or 1199033== 119894 do


(14) end while



1= 0 lArr



1and 120573 lArr

(9) 119895rand = rndint[1 119863]



(12) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(13) else(14) 119906

119894119895= 119909119894119895


119894


119894) then

(21) Replace 119909119894with 119906

119894











119894 where





119903(119905) ge VTR



15 15 151205882



NG119903= ArgMin

119905

(BCV119903 (119905) ge VTR)



by NG as follows

NG =1



(17)





NGAM =1

119873sum

All imagesNG (19)



0

400

800

1200

1600

0

400

800

1200

1600

0

400

800

1200

1600

0

1000

2000

3000

4000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

0 50 100 150 200 2500 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250 0 50 100 150 200 250

0 50 100 150 200 2500

400

800

1200

1600

0 50 100 150 200 250

0 50 100 150 200 250

0

400

800

1200

1600












0500

1000150020002500

0 50 100 150 200 2500

500

1000

1500

2000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 2500

400

800

1200

1600

2000

0 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0500

1000150020002500

0 50 100 150 200 250

0500

10001500200025003000

0

2000

4000

6000

8000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 250

(11) Bird (321 times 481)




(15) Fish (481 times 321)



(19) Snake (481 times 321)



(b)




SRHM =119873




05in Table 4


11989905


has SRHM ge 05)



(21)


TVD =


sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816


sum119899

119894=11119879lowast119903119894= 119879119898119903119894


(22)


119879lowast119903119894= 119879119898119903119894is


119903119894= 119879119898

119903119894and



2 4 6 8 10 12 14 160

200

400

600

800

1000

1200


Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =







Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

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HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

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0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

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0117

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0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

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HM

000

0000

0000

0000

0000

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0000

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0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

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D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

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nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119892=

(1199091198941 1199091198942 119909






follows

1199091198951198940

= 119909119895min + rndreal







according to

V119894119892+1

= 1199091199031 119892

+ 119865 sdot (1199091199032 119892

minus 1199091199033 119892

) (9)



1 1199032 and 119903




base vector1199091199032119892



1199032 119892minus 1199091199033 119892




119906119894119892+1

= (1199061119894119892+1

1199062119894119892+1

119906119863119894119892+1

) (10)


119906119895119894119892+1

=

V119895119894119892+1


119909119895119894119892




that ensures 119906119894119892+1



119909119894119892+1

=

119906119894119892

if 119891 (119906119894119892) le 119891 (119909

119894119892)

119909119894119892

if 119891 (119906119894119892) gt 119891 (119909

119894119892)

(12)


119894119892)


119894119892+1is set to 119906


119894119892








1199031




119894 is assigned






1= 1199032

= 1199033

= 119894

(6) 119895rand = rndint [1 119863]



(9) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(10) else(11) 119906

119894119895= 119909119894119895


119894


119894) then

(18) Replace 119909119894with 119906

119894




119894of the 119894th vector 119909

119894

is calculated as

119901119894=

119877119894

NP 119894 = 1 2 NP (14)




1199091199032





1199032vec-



1199031 which is



119905 of which





1199032and 119909

1199033




119894and 119909




1199091199032and 119909





minprop (120573)






(3) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr

(4) while minprop(120573) gt 1199011199031


(5) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr




2== 1199031do




(12) while 1199033== 1199032or 1199033== 1199031or 1199033== 119894 do


(14) end while



1= 0 lArr



1and 120573 lArr

(9) 119895rand = rndint[1 119863]



(12) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(13) else(14) 119906

119894119895= 119909119894119895


119894


119894) then

(21) Replace 119909119894with 119906

119894











119894 where





119903(119905) ge VTR



15 15 151205882



NG119903= ArgMin

119905

(BCV119903 (119905) ge VTR)



by NG as follows

NG =1



(17)





NGAM =1

119873sum

All imagesNG (19)



0

400

800

1200

1600

0

400

800

1200

1600

0

400

800

1200

1600

0

1000

2000

3000

4000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

0 50 100 150 200 2500 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250 0 50 100 150 200 250

0 50 100 150 200 2500

400

800

1200

1600

0 50 100 150 200 250

0 50 100 150 200 250

0

400

800

1200

1600












0500

1000150020002500

0 50 100 150 200 2500

500

1000

1500

2000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 2500

400

800

1200

1600

2000

0 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0500

1000150020002500

0 50 100 150 200 250

0500

10001500200025003000

0

2000

4000

6000

8000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 250

(11) Bird (321 times 481)




(15) Fish (481 times 321)



(19) Snake (481 times 321)



(b)




SRHM =119873




05in Table 4


11989905


has SRHM ge 05)



(21)


TVD =


sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816


sum119899

119894=11119879lowast119903119894= 119879119898119903119894


(22)


119879lowast119903119894= 119879119898119903119894is


119903119894= 119879119898

119903119894and



2 4 6 8 10 12 14 160

200

400

600

800

1000

1200


Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =







Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1= 1199032

= 1199033

= 119894

(6) 119895rand = rndint [1 119863]



(9) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(10) else(11) 119906

119894119895= 119909119894119895


119894


119894) then

(18) Replace 119909119894with 119906

119894




119894of the 119894th vector 119909

119894

is calculated as

119901119894=

119877119894

NP 119894 = 1 2 NP (14)




1199091199032





1199032vec-



1199031 which is



119905 of which





1199032and 119909

1199033




119894and 119909




1199091199032and 119909





minprop (120573)






(3) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr

(4) while minprop(120573) gt 1199011199031


(5) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr




2== 1199031do




(12) while 1199033== 1199032or 1199033== 1199031or 1199033== 119894 do


(14) end while



1= 0 lArr



1and 120573 lArr

(9) 119895rand = rndint[1 119863]



(12) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(13) else(14) 119906

119894119895= 119909119894119895


119894


119894) then

(21) Replace 119909119894with 119906

119894











119894 where





119903(119905) ge VTR



15 15 151205882



NG119903= ArgMin

119905

(BCV119903 (119905) ge VTR)



by NG as follows

NG =1



(17)





NGAM =1

119873sum

All imagesNG (19)



0

400

800

1200

1600

0

400

800

1200

1600

0

400

800

1200

1600

0

1000

2000

3000

4000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

0 50 100 150 200 2500 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250 0 50 100 150 200 250

0 50 100 150 200 2500

400

800

1200

1600

0 50 100 150 200 250

0 50 100 150 200 250

0

400

800

1200

1600












0500

1000150020002500

0 50 100 150 200 2500

500

1000

1500

2000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 2500

400

800

1200

1600

2000

0 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0500

1000150020002500

0 50 100 150 200 250

0500

10001500200025003000

0

2000

4000

6000

8000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 250

(11) Bird (321 times 481)




(15) Fish (481 times 321)



(19) Snake (481 times 321)



(b)




SRHM =119873




05in Table 4


11989905


has SRHM ge 05)



(21)


TVD =


sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816


sum119899

119894=11119879lowast119903119894= 119879119898119903119894


(22)


119879lowast119903119894= 119879119898119903119894is


119903119894= 119879119898

119903119894and



2 4 6 8 10 12 14 160

200

400

600

800

1000

1200


Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =







Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(3) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr

(4) while minprop(120573) gt 1199011199031


(5) 1199031= 1199031+ 1 if 119903

1gt NP then 119903

1= 1 end if lArr




2== 1199031do




(12) while 1199033== 1199032or 1199033== 1199031or 1199033== 119894 do


(14) end while



1= 0 lArr



1and 120573 lArr

(9) 119895rand = rndint[1 119863]



(12) 119906119894119895

= 1199091199031 119895

+ 119865 sdot (1199091199032 119895

minus 1199091199033 119895

)

(13) else(14) 119906

119894119895= 119909119894119895


119894


119894) then

(21) Replace 119909119894with 119906

119894











119894 where





119903(119905) ge VTR



15 15 151205882



NG119903= ArgMin

119905

(BCV119903 (119905) ge VTR)



by NG as follows

NG =1



(17)





NGAM =1

119873sum

All imagesNG (19)



0

400

800

1200

1600

0

400

800

1200

1600

0

400

800

1200

1600

0

1000

2000

3000

4000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

0 50 100 150 200 2500 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250 0 50 100 150 200 250

0 50 100 150 200 2500

400

800

1200

1600

0 50 100 150 200 250

0 50 100 150 200 250

0

400

800

1200

1600












0500

1000150020002500

0 50 100 150 200 2500

500

1000

1500

2000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 2500

400

800

1200

1600

2000

0 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0500

1000150020002500

0 50 100 150 200 250

0500

10001500200025003000

0

2000

4000

6000

8000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 250

(11) Bird (321 times 481)




(15) Fish (481 times 321)



(19) Snake (481 times 321)



(b)




SRHM =119873




05in Table 4


11989905


has SRHM ge 05)



(21)


TVD =


sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816


sum119899

119894=11119879lowast119903119894= 119879119898119903119894


(22)


119879lowast119903119894= 119879119898119903119894is


119903119894= 119879119898

119903119894and



2 4 6 8 10 12 14 160

200

400

600

800

1000

1200


Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =







Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

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000

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000

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1817

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74

1310

107

76

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000

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413

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000

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1817

1614

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64

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106

66

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16

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lt2

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0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

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9 15

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6 90263

7 10

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7 10

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9 12

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9 12

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9 12

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9 13

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12

16

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12

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16

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1817

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eaverage

ofthresholdvalued

istortio

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ndrank

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metho

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thresh

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DE

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DE

O(04)R-

DE

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DE

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DE

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DE

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DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

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171

11

11

116

11

11

11

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D163333

41753

2968

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0150

000

00270

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0075

0246

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1715

1614

81

1113

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109

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34770

36891

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2184

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1817

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46

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97216

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21565

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513

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1817

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285862

66259

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1816

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382076

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3120734

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1817

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511

98

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34

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1816

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119

65

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117664

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136935

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1817

1516

1210

67

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119

58

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21

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D1070300

592750

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31605280

157169

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1615

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2808

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466508

2990

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178394

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1815

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56

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1815

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Avg

for119899=2

to16

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7398

99387764

378828

402687

1091

93110080

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6652

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122229

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rate(120575)

8686

5525

6113

6798

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2176

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1348

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1815

1617

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1413

129

65

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21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894 where





119903(119905) ge VTR



15 15 151205882



NG119903= ArgMin

119905

(BCV119903 (119905) ge VTR)



by NG as follows

NG =1



(17)





NGAM =1

119873sum

All imagesNG (19)



0

400

800

1200

1600

0

400

800

1200

1600

0

400

800

1200

1600

0

1000

2000

3000

4000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

0 50 100 150 200 2500 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250 0 50 100 150 200 250

0 50 100 150 200 2500

400

800

1200

1600

0 50 100 150 200 250

0 50 100 150 200 250

0

400

800

1200

1600












0500

1000150020002500

0 50 100 150 200 2500

500

1000

1500

2000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 2500

400

800

1200

1600

2000

0 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0500

1000150020002500

0 50 100 150 200 250

0500

10001500200025003000

0

2000

4000

6000

8000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 250

(11) Bird (321 times 481)




(15) Fish (481 times 321)



(19) Snake (481 times 321)



(b)




SRHM =119873




05in Table 4


11989905


has SRHM ge 05)



(21)


TVD =


sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816


sum119899

119894=11119879lowast119903119894= 119879119898119903119894


(22)


119879lowast119903119894= 119879119898119903119894is


119903119894= 119879119898

119903119894and



2 4 6 8 10 12 14 160

200

400

600

800

1000

1200


Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =







Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

400

800

1200

1600

0

400

800

1200

1600

0

400

800

1200

1600

0

1000

2000

3000

4000

0

500

1000

1500

2000

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

0 50 100 150 200 2500 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250 0 50 100 150 200 250

0 50 100 150 200 2500

400

800

1200

1600

0 50 100 150 200 250

0 50 100 150 200 250

0

400

800

1200

1600












0500

1000150020002500

0 50 100 150 200 2500

500

1000

1500

2000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 2500

400

800

1200

1600

2000

0 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0500

1000150020002500

0 50 100 150 200 250

0500

10001500200025003000

0

2000

4000

6000

8000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 250

(11) Bird (321 times 481)




(15) Fish (481 times 321)



(19) Snake (481 times 321)



(b)




SRHM =119873




05in Table 4


11989905


has SRHM ge 05)



(21)


TVD =


sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816


sum119899

119894=11119879lowast119903119894= 119879119898119903119894


(22)


119879lowast119903119894= 119879119898119903119894is


119903119894= 119879119898

119903119894and



2 4 6 8 10 12 14 160

200

400

600

800

1000

1200


Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =







Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0500

1000150020002500

0 50 100 150 200 2500

500

1000

1500

2000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 2500

400

800

1200

1600

2000

0 50 100 150 200 250

0

400

800

1200

1600

2000

0 50 100 150 200 250

0 50 100 150 200 250

0

500

1000

1500

2000

2500

0 50 100 150 200 250

0500

1000150020002500

0 50 100 150 200 250

0500

10001500200025003000

0

2000

4000

6000

8000

0 50 100 150 200 250

0

400

800

1200

1600

0 50 100 150 200 250

(11) Bird (321 times 481)




(15) Fish (481 times 321)



(19) Snake (481 times 321)



(b)




SRHM =119873




05in Table 4


11989905


has SRHM ge 05)



(21)


TVD =


sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816


sum119899

119894=11119879lowast119903119894= 119879119898119903119894


(22)


119879lowast119903119894= 119879119898119903119894is


119903119894= 119879119898

119903119894and



2 4 6 8 10 12 14 160

200

400

600

800

1000

1200


Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =







Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











SRHM =119873




05in Table 4


11989905


has SRHM ge 05)



(21)


TVD =


sum119899

119894=1

1003816100381610038161003816119879lowast

119903119894minus 119879119898

119903119894

1003816100381610038161003816


sum119899

119894=11119879lowast119903119894= 119879119898119903119894


(22)


119879lowast119903119894= 119879119898119903119894is


119903119894= 119879119898

119903119894and



2 4 6 8 10 12 14 160

200

400

600

800

1000

1200


Tres

hold

val

ue d

istor

tion

PSO 120575 = 6798

ABC 120575 = 1925

DE 120575 = 1561

Rank-DE 120575 = 2176

DPSO 120575 = 6113

FODPSO 120575 = 5525

O R 120575 = 1348

O( )R 120575 = 4 3

O( )R 120575 = 3

O( )R 120575 = 2

O( )R 120575 = 769

O( )R 120575 = 494

O( )R 120575 =







Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table2Th

ebestvalueso

fthe

between-cla

ssvaria

ncea

ndthresholds

oftheldquomou

ntainrdquo

imagep

rodu

cedby

them

ultithreshfunctio

nof

theM

atlabtoolbo

xandby

thep

ropo

sedO(00)R-D

Ealgorithm

Then

umbero

fthresho

ldsv

ariesfrom

2to

16

Image

(1)

119899 (2)

Prod

uced

bymultithresh

Prod

uced

bytheO

(00)R-D

Ealgorithm

Objdiff

(7)=

(5)minus(3)

Threshdiff

(8)=

sum|(4)minus(6)|

Between-

class

varia

nce

(3)

Thresholds

(4)

Between-

class

varia

nce

(5)

Thresholds

(6)

Mou

ntain

22372886

60127

2372923

61128

0037

23

2495852

3276130

2496113

3377131

0261

34

2551778

3272109145

2551955

3373109147

0177

45

258015

4316898124158

258033

6326999125159

0182

56

2588264

357598118152175

2596956

244674101126160

8692

122

72598800

337098118141166207

2608807

24467398119

145175

10007

153

82605785

316891105121140

163192

2616294

24467397115135160191

10509

709

2619512

27527595111129148168197

2622314

2037547698116136160191

2802

11410

262039

223467397116

135157175204

232

262719

42036537493106121140

163194

6802

258

112625275

23457293105119

137156175204

228

2630496

2036537493105118134152172201

5221

199

122629641

2239587895109123139156177207247

263318

9183145597693105118

134152172201

3548

246

13263137

22038547494107121139157173193224248

2635290

183145597692103115129144161179207

3918

283

142633865

1936537491103115129144

160174190217245

2637088

17294256728696106117130145161179207

3223

307

15263519

0193350698596107121137152167178190212240

2638449

1728395061758897107118131146162179207

3259

351

162636357

163045597793105118133148162176192209230244

2639616

1727384960748695104113123135149164181209

3259

415


Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table3Th

eaverage

ofmeannu

mbero

fgeneration(N

GAM)a

ndrank

softhe

metho

ds

No

maxGen

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

119899=2

119866=50

NGAM

1833

715713

13218

13202

1053

814713

10117

6394

6903

7636

8208

9147

9261

9901

10709

11631

20850

Rank

1615

1312

914

81

23

45

67

1011

17119899=3

119866=100

NGAM

35200

36210

27479

33210

19488

29049

18958

10492

12296

13587

14807

16969

17107

1810

019546

2174

944

192

Rank

1516

1214

913

81

23

45

67

1011

17119899=4

119866=150

NGAM

4539

158683

49596

63251

31060

4813

029595

14733

1852

620885

22815

26632

26456

28058

3017

533499

7210

0Ra

nk12

1514

1610

138

12

34

65

79

1117

119899=5

119866=200

NGAM

55664

81877

87441

108131

4415

769088

41283

19508

2452

428097

30963

36935

3674

838647

41455

45631

100693

Rank

1214

1517

1013

81

23

46

57

911

16119899=6

119866=300

NGAM

NA(2)

10066

41419

37157328

61935

99700

56405

24623

32687

3816

742608

51003

50281

52089

55729

6174

8138716

Rank

1713

1516

1112

91

23

46

57

810

14119899=7

119866=400

NGAM

NA(1)

NA(1)

192267

236928

86938

142096

75340

2976

341691

49408

5619

367619

67063

69696

74389

8332

6184371

Rank

1716

1415

1112

91

23

46

57

810

13119899=8

119866=600

NGAM

NA(7)

NA(6)

NA(2)

NA(1)

117092

198986

100944

33967

51573

62263

7452

391836

89666

9519

8102552

115583

244880

Rank

1716

1514

1112

81

23

46

57

910

13119899=9

119866=800

NGAM

NA(11)

NA(13)

NA(12)

NA(3)

1510

04264253

125299

41228

6374

279432

90201

113530

110416

11439

7122555

137538

2994

92Ra

nk17

1615

1411

129

12

34

65

78

1013

119899=10

119866=1000

NGAM

NA(11)

NA(16)

NA(16)

NA(7)

NA(1)

3213

731477

70NA(3)

73564

9112

1105020

132968

128999

132691

1418

251579

97356546

Rank

1716

1514

1210

813

12

36

45

79

11119899=11

119866=1500

NGAM

NA(16)

NA(17)

NA(19

)NA(9)

NA(1)

405090

185020

NA(3)

NA(2)

NA(2)

127135

162822

157184

160123

175152

198522

434877

Rank

1716

1514

128

613

1010

14

23

57

9119899=12

119866=2000

NGAM

NA(18)

NA(20)

NA(20)

NA(9)

278743

535542

2299

65NA(4)

NA(5)

135266

16004

7205266

1973

86203814

222094

2515

35574104

Rank

1716

1514

128

613

1110

14

23

57

9119899=13

119866=3000

NGAM

NA(20)

NA(19

)NA(20)

NA(14

)NA(3)

NA(3)

275860

NA(12)

NA(5)

NA(1)

NA(1)

NA(1)

NA(1)

2615

22278082

31646

8726521

Rank

1716

1514

129

213

1110

66

61

34

5119899=14

119866=4000

NGAM

NA(20)

NA(20)

NA(20)

NA(14

)NA(4)

NA(4)

330194

NA(11)

NA(8)

NA(4)

NA(1)

NA(2)

296414

NA(1)

3316

26385951

889711

Rank

1716

1514

129

113

1110

78

65

23

4119899=15

119866=5000

NGAM

NA(20)

NA(20)

NA(20)

NA(19

)NA(13)

NA(5)

3992

28NA(11)

NA(10)

NA(8)

NA(6)

363506

362665

383788

4179

584818

301086001

Rank

1716

1514

138

112

1110

97

65

23

4119899=16

119866=6000

NGAM

NA(20)

NA(20)

NA(20)

NA(18)

NA(10)

NA(8)

NA(4)

NA(18)

NA(11)

NA(9)

NA(5)

NA(1)

NA(2)

4475

704917

72552856

1374438

Rank

1716

1514

129

413

1110

86

75

12

3

Avgfor

119899=2to

16

119899max

NA(119909)

NGAM

5NA(14

6)

38648

6NA(152)

58629

7NA(14

9)

8532

3

7NA(94)

102008

9NA(32)

88995

12

NA(20)

193456

15

NA(4)

144713

9NA(62)

22589

10

NA(41)

3616

7

10

NA(24)

52586

12

NA(13)

66593

12

NA(4)

106519

12

NA(3)

119204

13

NA(1)

143971

16

NA(0)

1677

08

16

NA(0)

190391

16

NA(0)

436499

Rank

1716

1514

129

413

1110

87

65

12

3


Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

1

3TV

D163333

41753

2968

7424

1510

0150

000

00270

1004

0519

0075

0246

0171

0075

000

0000

0000

0000

0Ra

nk18

1715

1614

81

1113

126

109

61

11

1

4TV

D2570

8398848

34770

36891

8716

7340

1428

2184

25499

13609

7318

7360

7398

4303

1672

0728

0130

0257

Rank

1817

1516

129

46

1413

810

117

53

12

5TV

D516582

148344

97216

62319

21565

5373

1864

3241

20441

10064

10422

4584

3904

3289

244

13385

1195

000

0Ra

nk18

1716

1514

103

513

1112

98

64

72

1

6TV

D534226

2194

371876

20166775

45728

1953

62528

7761

89938

43666

30923

13010

6524

6023

4136

6129

2421

0050

Rank

1817

1615

1310

38

1412

119

75

46

21

7TV

D5992

50265029

242194

285862

66259

58877

27976

35481

12504

075011

49851

44610

28039

29841

26075

17950

4776

0150

Rank

1816

1517

1211

58

1413

109

67

43

21

8TV

D475458

382076

380124

3096

357276

3120734

90287

89406

218266

143225

130898

99991

76353

82886

57251

57665

26822

2103

Rank

1817

1615

511

98

1413

1210

67

34

21

9TV

D862458

394744

375229

410710

113892

82099

44407

4653

9250470

154662

96417

6819

744

223

3533

326627

23406

12757

0780

Rank

1816

1517

1210

78

1413

119

65

43

21

10TV

D8477

03458536

445765

498328

117664

84541

43991

48474

249147

136935

93935

72288

43883

44110

36700

27978

20213

0164

Rank

1816

1517

1210

68

1413

119

57

43

21

11TV

D9779

76536258

528329

5312

111379

4187267

44944

44973

296513

152195

104092

75115

44461

47564

34944

26231

18782

0083

Rank

1817

1516

1210

67

1413

119

58

43

21

12TV

D1070300

592750

5899

31605280

157169

174580

1091

381012

82388491

258312

196098

150932

101112

82365

79966

51281

23429

000

0Ra

nk18

1615

1710

118

714

1312

96

54

32

1

13TV

D114

2808

5674

425794

957113

82183451

204665

1472

51123438

466508

2990

40216656

178394

114244

114491

9610

481076

52578

1975

Rank

1815

1617

1011

87

1413

129

56

43

21

14TV

D114

864

1632482

660908

770319

205714

21840

81414

31116716

524783

325936

2419

97177121

113426

104073

88593

65908

39647

0564

Rank

1815

1617

1011

87

1413

129

65

43

21

15TV

D124552

97078

35722126

816158

256477

3078

732195

5618246

4605623

428370

343439

2615

22174700

166309

138710

100296

62940

000

0Ra

nk18

1516

179

118

714

1312

106

54

32

1

16TV

D1182989

768714

835747

82800

42490

422797

532116

14195676

605790

416711

3113

142514

08174118

167316

133922

113776

76661

2539

Rank

1815

1716

911

87

1413

1210

65

43

21

Avg

for119899=2

to16

TVD

7398

99387764

378828

402687

1091

93110080

72428

6652

72578

38163884

122229

93652

6217

05919

948476

38387

22823

0578

Growth

rate(120575)

8686

5525

6113

6798

1925

2176

1561

1348

4843

3225

2391

1893

1252

1186

999

769

494

009

Rank

1815

1617

1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table4Th

eaverage

successrate(SR

HM)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2SR

HM

00977

11

11

11

0999

11

11

11

11

1Ra

nk18

171

11

11

116

11

11

11

11

1

3SR

HM

000

00635

0972

0991

0979

0998

1000

0997

0986

0993

0999

0997

0998

0999

1000

1000

1000

1000

Rank

1817

1613

158

110

1412

610

86

11

11

4SR

HM

000

00379

0742

0934

0833

0987

0995

0986

0897

0945

0972

0971

0986

0993

0992

0991

0998

0997

Rank

1817

1613

157

38

1412

1011

84

56

12

5SR

HM

000

00185

0391

0767

0706

0936

0977

0960

0782

0878

0874

0944

0950

0959

0968

0960

0985

1000

Rank

1817

1614

1510

35

1311

129

87

45

21

6SR

HM

000

0000

00187

0433

0453

0863

0968

0900

0601

0706

0826

0878

0934

0922

0944

0935

0970

0999

Rank

1817

1615

1410

38

1312

119

67

45

21

7SR

HM

000

0000

0000

00153

0164

0696

0884

0835

0366

0561

0675

0741

0840

0851

0883

0914

0949

0998

Rank

1817

1615

1410

48

1312

119

76

53

21

8SR

HM

000

0000

0000

0000

0000

00300

0458

0530

0105

0256

0351

0502

0592

0628

0724

0757

0838

0972

Rank

1817

1614

1411

97

1312

108

65

43

21

9SR

HM

000

0000

0000

0000

0000

00495

0695

0681

0083

0274

0372

0548

0721

0725

0769

0815

0852

0999

Rank

1817

1614

1410

78

1312

119

65

43

21

10SR

HM

000

0000

0000

0000

0000

0000

00381

0375

000

00073

0218

0311

040

90370

046

00578

0686

0998

Rank

1817

1614

1412

67

1311

109

58

43

21

11SR

HM

000

0000

0000

0000

0000

0000

00271

0267

000

0000

0000

00144

040

70316

0424

0584

0704

0999

Rank

1817

1614

1412

78

1310

109

56

43

21

12SR

HM

000

0000

0000

0000

0000

00090

0270

0423

000

0000

00089

0177

040

0040

70501

064

10810

1000

Rank

1817

1614

1412

85

1311

109

76

43

21

13SR

HM

000

0000

0000

0000

0000

0000

0000

00151

000

0000

0000

0000

0000

0000

00254

0267

0381

0974

Rank

1817

1614

1412

65

1310

106

66

43

21

14SR

HM

000

0000

0000

0000

0000

0000

0000

00102

000

0000

0000

0000

0000

00110

000

00253

0494

0993

Rank

1817

1614

1412

74

1310

107

76

53

21

15SR

HM

000

0000

0000

0000

0000

0000

0000

00090

000

0000

0000

0000

00095

0117

0149

0288

046

610

00Ra

nk18

1716

1414

128

413

1010

87

65

32

1

16SR

HM

000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

00110

0192

0288

0977

Rank

1817

1614

1412

64

1310

106

66

53

21

Avgfor

119899=2to

16

11989905

119899max

SRHM

lt2

lt2

000

0

3 50376

4 6044

3

5 70454

5 7046

4

7 90665

9 12

0554

9 15

0304

6 90263

7 10

0313

7 10

0530

9 12

0420

9 12

0654

9 12

0625

9 13

0619

12

16

0498

12

16

0656

16

16

0994

Rank

1817

1615

1412

84

1311

109

67

53

21


Table5Th

eaverage

ofthresholdvalued

istortio

n(TVD)a

ndrank

softhe

metho

ds

No

Multi-

thresh

FODPS

ODPS

OPS

OABC

Rank

-DE

DE

O(rand)R-

DE

O(09)R-

DE

O(08)R-

DE

O(07)R-

DE

O(06)R-

DE

O(05)R-

DE

O(04)R-

DE

O(03)R-

DE

O(02)R-

DE

O(01)R-

DE

O(00)R-

DE

2TV

D7415

32217

000

0000

0000

0000

0000

0000

00050

000

0000

0000

0000

0000

0000

0000

0000

0000

0Ra

nk18

171

11

11

116

11

11

11

11

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Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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7398

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1011

87

1413

129

65

43

21




Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Starfish

2 2546885 85 1573 2779925 68 119 1774 2865707 60 101 138 1875 2912859 52 86 117 150 1946 2941728 47 77 105 132 162 2017 2960158 44 71 95 118 142 170 2068 2972356 43 68 90 110 131 153 180 2129 2981138 38 58 78 97 116 136 157 183 21410 2988206 37 56 75 93 110 128 146 167 192 21911 2993348 35 53 71 88 103 119 135 152 172 196 22112 2997352 34 51 68 84 98 112 127 142 158 177 200 22313 3000480 33 48 64 79 93 106 119 133 147 162 181 203 22514 3003076 32 47 62 76 89 101 114 127 140 154 169 187 207 22715 3005235 30 43 56 70 83 95 107 119 131 143 156 171 189 209 22816 3007060 30 42 55 68 80 92 103 114 126 138 150 163 178 195 213 230

Mountain

2 2372923 61 1283 2496113 33 77 1314 2551955 33 73 109 1475 2580336 32 69 99 125 1596 2596956 24 46 74 101 126 1607 2608807 24 46 73 98 119 145 1758 2616294 24 46 73 97 115 135 160 1919 2622314 20 37 54 76 98 116 136 160 19110 2627194 20 36 53 74 93 106 121 140 163 19411 2630496 20 36 53 74 93 105 118 134 152 172 20112 2633189 18 31 45 59 76 93 105 118 134 152 172 20113 2635290 18 31 45 59 76 92 103 115 129 144 161 179 20714 2637088 17 29 42 56 72 86 96 106 117 130 145 161 179 20715 2638449 17 28 39 50 61 75 88 97 107 118 131 146 162 179 20716 2639616 17 27 38 49 60 74 86 95 104 113 123 135 149 164 181 209

Cactus

2 1816448 73 1513 1970112 64 106 1734 2042275 55 87 125 1875 2080884 49 76 102 138 1966 2104147 46 70 93 119 157 2087 2119670 44 65 85 105 131 168 2158 2129358 42 60 77 94 113 138 174 2189 2136162 41 58 74 89 105 125 152 186 22410 2141579 40 56 71 85 99 115 135 161 193 22811 2145397 39 53 66 79 91 104 119 139 165 196 22912 2148282 38 51 64 76 88 100 114 131 152 176 204 23313 2150740 37 49 61 72 83 94 105 118 135 155 178 205 23314 2152626 36 47 58 68 78 88 98 109 122 138 158 181 207 23415 2154162 36 46 56 66 76 85 94 104 115 128 144 164 187 212 23716 2155471 36 46 56 66 75 84 93 102 112 124 138 155 174 195 217 239


Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 6 Continued


Butterfly

2 3873222 84 1553 3990855 81 144 1994 4051357 77 121 167 2075 4092929 60 89 129 172 2096 4119448 59 87 122 161 193 2217 4135937 53 74 98 128 165 195 2228 4148556 53 73 96 123 156 183 203 2289 4156528 52 72 94 118 144 170 190 207 23110 4163794 48 63 80 99 121 147 172 191 208 23111 4168489 47 62 79 98 118 141 164 183 198 213 23412 4172517 46 59 73 89 104 122 144 167 185 199 214 23513 4175487 46 59 72 87 102 118 137 157 175 189 202 216 23614 4177893 44 55 66 79 93 106 121 141 161 178 191 203 217 23715 4179908 44 55 66 78 92 105 119 137 156 173 186 197 208 221 23916 4181559 42 52 61 71 83 95 107 121 139 157 173 186 197 208 221 239

Circus

2 1651257 118 1723 1760512 105 150 1874 1817487 93 132 165 1955 1850243 87 122 152 177 2036 1870083 82 113 141 164 185 2087 1883966 77 104 129 151 171 190 2128 1893450 73 98 122 143 161 178 195 2169 1900592 70 92 114 134 152 168 183 199 21910 1905992 68 89 109 128 145 160 174 188 203 22211 1909887 66 86 105 123 139 153 166 179 192 206 22512 1913079 63 81 98 114 130 145 158 170 182 194 208 22613 1915676 62 79 95 111 126 140 152 164 175 186 197 210 22814 1917756 61 78 94 109 123 136 148 159 170 180 190 201 214 23115 1919524 59 74 88 102 115 128 140 151 161 171 181 191 202 214 23116 1920958 58 73 87 100 113 126 138 149 159 169 178 187 196 206 218 234

Snow

2 5261705 80 1693 5624289 71 139 2074 5729116 50 92 144 2085 5785138 49 91 140 192 2316 5819333 45 81 111 148 194 2327 5835770 43 76 101 127 159 196 2328 5850190 30 55 83 107 133 163 197 2339 5862437 30 55 82 106 129 157 185 211 23710 5870284 29 52 75 94 111 132 159 186 212 23711 5875817 29 52 75 94 111 132 158 183 206 227 24412 5880335 29 52 74 93 109 127 149 169 188 209 228 24413 5884513 23 39 57 76 94 110 128 150 170 188 209 228 24414 5887355 22 37 54 70 84 98 112 129 151 170 188 209 228 24415 5889953 21 36 53 69 83 97 110 124 141 159 175 192 211 229 24516 5891998 21 36 53 69 83 97 110 124 141 159 174 189 207 222 236 248


Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 6 Continued


Palace

2 2623440 99 1653 2791488 84 132 1864 286098 70 103 143 1915 2908165 69 101 138 177 2186 2934330 64 89 117 147 181 2207 2953745 54 75 99 126 153 183 2208 2966250 50 69 89 111 134 158 185 2219 2974719 47 65 81 100 120 141 163 188 22210 2981875 47 64 80 98 117 137 157 178 199 22611 2986898 45 61 75 90 106 123 141 159 179 200 22712 2990579 43 58 69 82 96 111 126 143 160 180 201 22713 2993809 43 58 69 81 95 110 125 141 157 173 190 208 23114 2996112 43 57 68 80 94 108 123 138 153 167 182 199 218 23915 2998213 42 56 66 77 88 100 113 126 140 154 167 182 199 218 23916 2999918 42 55 65 75 86 98 110 123 136 149 162 176 191 205 222 241

Flower

2 1489281 61 1303 1627897 39 77 1414 1685956 36 67 105 1605 1715220 28 49 75 111 1646 1736423 27 47 71 102 143 1927 1752512 26 43 61 83 111 151 1998 1761968 25 42 58 77 99 126 161 2049 1768354 24 40 54 70 88 109 135 167 20710 1772903 23 37 48 60 75 92 112 138 169 20811 1776470 23 36 47 58 72 87 104 124 149 177 21112 1779106 22 34 44 54 65 78 92 108 128 152 179 21213 1781251 20 30 39 48 58 70 83 96 112 131 154 180 21314 1783049 19 29 38 47 56 66 78 90 104 120 139 161 185 21515 1784478 19 29 38 46 54 63 74 85 97 111 128 147 168 190 21816 1785641 19 28 37 45 53 62 72 83 94 106 120 136 155 175 196 222

Wherry

2 3313161 108 1893 3543272 102 161 2184 3599924 83 121 163 2185 3634708 81 118 152 184 2246 3656048 72 103 130 156 186 2257 3668313 68 94 120 139 161 189 2268 3678216 60 83 110 132 152 175 198 2309 3686001 56 77 100 122 138 156 179 201 23110 3691730 56 76 98 120 136 153 174 194 219 24311 3696416 55 74 94 114 130 142 158 178 197 222 24512 3699866 54 72 91 110 126 138 151 168 185 202 225 24613 3702619 52 68 83 100 117 130 140 153 169 185 202 225 24614 3705033 52 68 83 100 117 130 140 152 167 182 197 215 235 24915 3706937 51 66 79 94 110 123 133 142 154 169 184 199 216 235 24916 3708484 49 63 75 89 104 118 129 138 147 159 173 186 200 217 235 249


Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 6 Continued


Waterfall

2 4512801 88 1703 4646137 72 115 1824 4711019 67 103 150 2045 4752267 58 87 119 164 2126 4777063 53 78 104 134 176 2177 4793510 48 70 92 116 147 186 2218 4805756 46 66 86 107 131 163 199 2269 4814097 43 61 79 97 117 141 173 205 22810 4820724 40 57 73 90 108 128 152 182 210 23011 4825739 38 54 69 84 100 117 137 162 191 215 23212 4829689 37 53 67 81 96 111 128 148 173 198 218 23313 4832826 35 50 63 76 89 103 117 133 153 177 201 220 23414 4835371 32 46 58 70 82 95 108 122 138 158 182 204 221 23415 4837537 32 46 57 68 80 92 104 117 131 148 169 191 210 224 23616 4839226 30 43 54 64 74 85 96 108 120 134 151 172 193 211 225 236

Bird

2 901450 71 1223 975230 64 111 1404 1027509 61 104 131 1645 1051482 54 93 119 138 1696 1067992 47 82 108 127 142 1727 1077304 40 70 94 113 129 143 1738 1086005 39 69 93 112 128 141 159 1929 1091341 37 65 88 105 119 131 142 160 19310 1095355 37 64 86 103 117 129 139 149 167 20011 1098475 33 56 77 93 107 119 130 140 150 169 20212 1100749 33 56 77 93 107 119 129 138 146 158 178 20913 1102639 31 52 71 87 100 111 121 130 139 147 159 179 21014 1104132 29 48 66 82 95 107 118 127 134 141 149 161 181 21115 1105446 28 45 62 77 90 101 111 120 128 135 142 150 162 182 21216 1106500 28 45 62 77 90 101 111 120 128 135 141 148 157 171 191 219

Police

2 3647353 74 1503 3844314 70 135 1924 3966225 63 112 158 2095 4013875 61 104 140 174 2146 4047198 32 67 106 141 175 2147 4067996 32 67 104 133 161 186 2198 4084933 29 52 78 106 134 162 187 2199 4094705 29 52 78 103 125 147 169 190 22010 4101702 29 52 78 102 123 143 165 184 203 22811 4108018 28 49 71 90 107 126 146 166 185 204 22912 4112304 28 49 71 89 105 122 139 157 174 189 207 23113 4115165 28 46 62 78 92 106 123 140 158 174 189 207 23114 4117926 28 46 62 78 91 105 120 135 151 166 180 193 210 23215 4120045 28 46 62 78 91 103 116 129 143 158 172 185 197 214 23516 4121861 28 46 62 78 91 103 116 128 142 156 170 182 193 206 223 240


Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 6 Continued


Ostrich

2 1073452 75 1353 1139260 69 101 1494 1178650 65 92 125 1765 1203749 56 78 100 131 1796 1218643 47 65 85 103 133 1817 1228925 47 64 83 100 122 152 1928 1236023 45 59 75 90 104 125 155 1949 1240756 40 52 65 80 94 107 128 157 19510 1244909 40 52 64 78 91 103 119 141 168 20111 1247879 40 51 62 75 87 97 108 125 148 174 20512 1250313 37 48 57 68 80 91 101 112 128 150 175 20613 1252298 31 44 53 63 75 86 95 105 117 134 154 178 20714 1253956 29 42 50 58 68 79 89 98 108 120 137 157 181 20915 1255366 29 42 50 58 67 77 86 94 102 111 124 141 160 183 21016 1256490 28 41 49 57 66 76 85 93 101 110 122 137 155 174 196 219

Viaduct

2 7920458 77 1803 8117991 54 109 1934 8203807 42 84 131 2035 8246806 35 68 103 146 2106 8272775 31 59 88 120 160 2167 8287714 28 53 77 103 132 169 2208 8298322 27 51 75 100 128 164 212 2469 8308240 24 45 66 88 112 139 172 216 24710 8315133 22 41 60 79 99 121 146 178 219 24711 8320032 20 37 54 71 89 108 128 152 183 221 24812 8323799 20 36 52 68 84 101 119 139 162 190 224 24813 8326793 18 33 48 62 77 92 108 125 144 167 195 226 24814 8329119 17 31 44 57 70 84 98 113 129 148 170 197 227 24815 8330983 17 31 44 57 70 84 98 113 129 147 169 195 224 243 25116 8332744 16 29 42 54 66 78 91 104 118 133 151 172 196 224 243 251

Fish

2 3593389 64 1483 3870456 44 104 1774 3972731 34 81 127 1885 4024885 27 63 101 139 1946 4054836 24 56 90 123 156 2057 4075236 22 49 78 107 135 168 2138 4088300 20 44 68 93 118 142 173 2169 4097101 19 40 62 85 107 128 149 178 21810 4103194 18 38 58 78 98 118 137 158 185 22211 4107954 16 34 51 69 88 107 125 143 163 190 22412 4111678 15 31 47 64 82 100 117 133 149 169 195 22713 4114783 14 28 43 58 74 90 106 121 136 152 172 198 22814 4116954 14 28 43 58 73 88 103 117 131 145 160 179 203 23115 4118931 13 25 38 51 64 78 92 106 120 133 147 162 181 205 23216 4120522 13 25 37 49 62 75 89 102 115 127 139 152 168 188 211 235


Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 6 Continued


Houses

2 2543788 56 1163 2627230 53 105 1504 2663698 42 69 110 1525 2694902 39 65 96 130 1586 2708655 39 65 95 127 150 1697 2720263 35 55 74 98 128 151 1708 2726676 35 55 74 98 127 148 164 1849 2732853 32 48 65 80 101 128 148 164 18410 2736510 30 45 60 74 87 106 129 149 164 18411 2739566 30 45 60 73 86 105 128 145 156 168 18712 2742148 29 42 55 68 79 94 112 130 145 156 168 18713 2744071 28 39 51 63 74 84 98 115 131 145 156 168 18714 2745542 28 39 51 63 74 84 98 115 131 145 155 165 176 19315 2746817 27 37 47 58 68 77 86 100 116 131 145 155 165 176 19316 2747991 27 37 47 57 67 76 85 98 113 127 139 147 156 165 176 193

Mushroom

2 1988328 76 1453 2153037 65 110 1744 2237441 59 93 135 1935 2277427 52 78 106 145 1996 2301629 48 71 94 122 157 2057 2317538 46 67 87 110 138 171 2138 2328526 44 62 79 97 118 145 177 2169 2335812 42 58 74 90 107 127 152 181 21810 2340981 41 56 70 84 99 116 136 159 186 22011 2345160 39 52 65 78 92 107 124 144 166 191 22312 2348413 38 51 63 75 87 100 115 132 152 174 199 22713 2350988 38 50 62 74 85 97 110 125 142 160 180 203 22914 2353119 37 48 59 69 79 89 100 113 127 144 162 181 204 23015 2354748 36 46 56 66 75 84 94 105 117 130 146 163 182 205 23016 2356113 36 46 56 65 74 83 92 102 113 125 139 154 169 187 208 232

Snow mountain

2 1912613 85 1493 2135274 79 137 1974 2234669 70 119 154 2045 2288799 55 96 129 160 2066 2317224 52 90 118 142 167 2097 2333078 50 85 111 132 153 174 2128 2344629 45 76 100 120 139 158 177 2149 2352951 44 74 97 116 133 150 167 187 22010 2359175 41 68 90 108 124 140 156 172 193 22511 2364304 38 62 84 102 117 132 146 160 175 196 22712 2367971 34 56 78 96 111 125 139 152 165 179 199 22813 2371208 33 53 73 90 104 117 130 143 155 167 180 200 22914 2373567 32 52 72 89 103 115 127 139 150 161 172 185 205 23215 2375586 31 49 68 84 98 109 120 131 142 153 164 175 188 208 23316 2377255 26 42 59 75 89 101 111 121 132 143 154 165 176 189 209 234


Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 6 Continued


Snake

2 1118615 87 1343 1231320 76 114 1544 1286555 69 101 129 1665 1317027 63 91 115 140 1756 1336172 59 84 105 126 149 1827 1348933 55 78 97 115 133 155 1878 1357665 52 73 91 107 123 140 161 1929 1364159 50 70 87 102 116 131 148 169 19810 1368955 47 65 81 95 108 121 135 151 172 20011 1372626 46 63 78 91 103 115 127 140 156 176 20312 1375513 45 61 75 88 99 110 121 133 146 162 182 20813 1377847 43 58 72 84 95 106 116 127 138 151 166 185 21114 1379736 42 56 69 81 92 102 112 122 132 143 155 170 189 21415 1381282 41 54 66 77 87 97 106 115 124 134 145 157 172 191 21516 1382603 40 52 63 74 84 93 102 111 120 129 139 150 162 177 195 219

Riosanpablo

2 2667020 95 1603 2818660 75 121 1774 2892439 68 102 143 1895 2931654 62 89 121 158 1976 2957018 58 82 107 138 171 2047 2973269 53 74 95 119 147 177 2078 2984972 50 69 87 108 133 159 185 2129 2993359 48 66 83 101 122 145 168 191 21510 2999615 46 63 78 93 110 130 151 173 195 21711 3004374 45 61 75 89 104 121 140 160 180 200 22012 3008082 44 59 72 84 97 112 129 147 166 185 203 22213 3011101 43 57 69 81 93 107 122 138 155 172 189 206 22414 3013465 41 54 66 77 88 100 113 128 144 160 176 192 208 22515 3015423 40 53 64 74 84 95 107 121 135 150 165 180 195 210 22616 3017081 39 51 62 72 82 92 103 115 128 142 156 170 184 198 212 227







1 b1 c1) and (a

2 b2 c2) as


1and a

2 Since both a

1and a

2are


1and b

2 If a1is 16 then b

1must


1must be NA





1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1and c










05




(11989905




1 b1 c1) and (a

2

b2 c2) as follows

















6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6 Conclusions





Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Global Multilevel Thresholding Using ...

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