WSNs Compressed Sensing Signal Reconstruction Based on Improved Kernel...

Research ArticleWSNs Compressed Sensing Signal ReconstructionBased on Improved Kernel Fuzzy Clustering and DiscreteDifferential Evolution Algorithm

Zhou-zhou Liu 12 and Shi-ning Li1

1School of Computer Science Xirsquoan Aeronautical University Xirsquoan China2School of Computer Science Northwestern Polytechnical University Xirsquoan China

Correspondence should be addressed to Zhou-zhou Liu liuzhouzhou8192126com

Received 21 February 2019 Accepted 15 May 2019 Published 16 June 2019

Academic Editor Stelios M Potirakis

Copyright copy 2019 Zhou-zhou Liu and Shi-ning Li 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

To reconstruct compressed sensing (CS) signal fast and accurately this paper proposes an improved discrete differential evolution(IDDE) algorithm based on fuzzy clustering for CS reconstruction Aiming to overcome the shortcomings of traditional CSreconstruction algorithm such as heavy dependence on sparsity and low precision of reconstruction a discrete differentialevolution (DDE) algorithm based on improved kernel fuzzy clustering is designed In this algorithm fuzzy clustering algorithm isused to analyze the evolutionary population which improves the pertinence and scientificity of population learning evolution whilerealizing effective clusteringThe differential evolutionary particle coding method and evolutionarymechanism are redefined Andthe improved fuzzy clustering discrete differential evolution algorithm is applied to CS reconstruction algorithm in which signalwith unknown sparsity is considered as particle coding Then the wireless sensor networks (WSNs) sparse signal is accuratelyreconstructed through the iterative evolution of population Finally simulations are carried out in the WSNs data acquisitionenvironment Results show that compared with traditional reconstruction algorithms such as StOMP the reconstruction accuracyof the algorithm proposed in this paper is improved by 364-519 and the reconstruction time is reduced by 151-313

1 Introduction

The rapid development of wireless sensor networks (WSNs)has binging human society into the era of big data Howeverhuge amount of data collected from sensor network is alsoaccompanied by the sharp increase of the signal bandwidth[1] If we still adopt the traditional Nyquist sampling theoremfor data acquisition in this case it will bring unprecedentedchallenges to the hardware system [2] The advent of com-pressive sensing (CS) [3] solved this problem and created awhole new approach for the signal information processing[4 5] Through a series of nonlinear optimization algorithmCS is able to accurately reconstruct compressible signal froma small amount of data and greatly reduced the requirementfor the sampling rate [6] Thus the theory of compressedsensing is gradually and widely applied to image processingwireless sensor networks radio communication etc

Sparse representation observation matrix and recon-struction algorithm are the core contents in compressedsensing and the reconstruction algorithm is the most criticalpart [7] It is commonly divided into three kinds convexoptimization algorithm greedy algorithm and their com-bination The key problem in CS reconstruction is how touse a small amount of observed signal to reconstruct theoriginal signal rapidly and accurately Furthermore severalproblems such as high computational complexity largeamount of data and preset sparsity still need to be solvedReference [7] proposed a SL0 reconstruction algorithm forcompressive sensing based on BFGS quasi Newton methodwhich effectively improved the iterative efficiency of greedymatching pursuit algorithm by taking advantage of therapid convergence of quasi Newton method Wang et al [2]introduced Dice coefficient matching metric into SWOMPreconstruction algorithm to overcome the inherent defect

HindawiJournal of SensorsVolume 2019 Article ID 7039510 9 pageshttpsdoiorg10115520197039510

2 Journal of Sensors

that the inner product matching criterion of SWOMP issensitive to residual signals Simulation results show thattheir algorithm has higher signal reconstruction successrate In [8] researchers discussed the signal reconstructionalgorithm for compressed sensing without prior informationof signal sparsity But the signal reconstruction accuracy ofthis algorithm needs to be improved

Currently most compressed sensing algorithms recon-struct the original signal by minimizing its 1198970 or 1198971 norm[9] This is NP-hard problemTherefore we propose a WSNscompressed sensing signal reconstruction algorithm basedon fuzzy clustering and discrete differential evolution Inthis algorithm we consider signal with unknown sparsityas differential evolution particle coding and reconstruct thesignal through population iterative evolution To improve theefficiency and accuracy of signal reconstruction fuzzy clus-tering is utilized while the differential evolution mechanismis redefined Finally simulation experiments are carried outto verify its validity

2 Differential Evolution AlgorithmBased on Fuzzy Clustering

21 Differential Evolution Differential evolution (DE) [10]is a kind of intelligent optimization algorithm with randomsearch It preserves the individuals with better fitness throughmutation and competition to achieve the desired end Due toits excellent global convergence and good robustness fewernumber of control parameters and easy to be implementedDE has been widely used in solving complex optimizationproblems [11]

An iteration of the classical DE algorithm mainly consistsof three basic stepsmutation crossover and selection ForN-dimensional optimization there exists a differential evolutionpopulation composed of 119875 particles 119883119905119894(1199091199051198941 sdot sdot sdot 119909119905119894119873) (119894 isin[1 119875]) for the current generation 119905The process of performingmutation on119883119905119894 (1199091199051198941 sdot sdot sdot 119909119905119894119873) can be expressed as

119881119905119894 = 119883119905119886 + 119865 times (119883119905119887 minus119883119905119888) (1)

where 119865 isin [04 1] denotes the scalar number 119881119905119894 denotesthe mutant particle The indices 119886 119887 119888 isin [1 119875] are mutu-ally exclusive integers randomly chosen from the range[1 119875] which are also different from 119894 In essence mutationrandomly selects three different particles from the currentpopulation multiplies the difference of any two of thesethree particles with the scalar number 119865 and adds the scaleddifference to the third one to obtain the mutant particle tofuse more individual information

To enhance the potential diversity of the populationcrossover operation is then performed It is a process ofselecting components from119881119905119894(V1199051198941 sdot sdot sdot V119905119894119873) and119883119905119894 accordingto certain probability to obtain the trial particle described as

119880119905119894 (1199061199051198941 sdot sdot sdot 119906119905119894119873) lArr997904 119906119905119894119895=

V119905119894119895 (119903119886119899119889119887 (119895) le 119862119877 119900119903 119895 = 119903119899119887119903 (119894))119909119905119894119895 (119903119886119899119889119887 (119895) gt 119862119877 119900119903 119895 = 119903119899119887119903 (119894))(2)

where 119862119877 isin (0 1) is called the crossover rate 119903119886119899119889119887(119895) isin (0 1)is a random number lying between 0 and 1 and 119903119899119887119903(119894) is aninteger randomly selected from range [1119873] which ensuresthat 119880119905119894 gets at least one component from119881119905119894

Then the next generation particle 119883119905+1119894 is selected in theform of roulette to keep the population size constant oversubsequent generations Taking the minimum optimizationas an example the selection operation is described as

119883119905+1119894 = 119880119905119894 119894119891 119891 (119880119905119894) le 119891 (119883119905119894)119883119905119894 119890119897119904119890 (3)

where 119891(119883119905119894) is the fitness value of particle119883119905119894 We can see that in each iteration the population always

keeps the better individual with higher fitness so as to reachthe global optimal solution

22 Improved Kernel FCM Fuzzy C-means clustering algo-rithm (FCM) is one of the most widely used clustering algo-rithms It uses membership to describe the degree to whicha data belongs to a certain subclass for data classification[12] For a 119878-dimension dataset 119883 isin 119877119878 clustering is todivide 119899 samples 119909119896 isin 119883 119896 = 1 2 sdot sdot sdot 119899 into 119862 subclasseswhile making the value of the clustering objective functionminimum that is

min 119869 (119880119881) = 119862sum119894=1

119899sum119896=1

120583119898119894119896 1003817100381710038171003817119909119896 minus 11990711989410038171003817100381710038172 (4)

where 120583119894119896 is the membership of sample 119909119896 to the 119894th subclass119880 = [120583119894119896]119862times119899 is the membership matrix 119881 = 119907119894 (119894 =1 2 sdot sdot sdot 119862) is the set of clustering center and 119898 is the fuzzyweighted index Let 120597119869120597120583119894119896 = 0 120597119869120597V119894 = 0 we have

120597119869120597120583119894119896 = 0 997904rArr120583119894119896 = 1003817100381710038171003817119909119896 minus 1199071198941003817100381710038171003817minus2(119898minus1)sum119862119895=1 10038171003817100381710038171003817119909119896 minus 11990711989510038171003817100381710038171003817minus2(119898minus1) 120597119869120597V119894 = 0 997904rArrV119894 = sum119899119896=1 120583119898119894119896119909119896sum119899119896=1 120583119898119894119896st

119862sum119894=1

120583119894119896 = 1 120583119894119896 isin [0 1] 119899sum119896=1

120583119894119896 isin (0 1)

(5)

It can be seen that FCM is a kind of local searchalgorithm and is sensitive to initial values [13] Since FCMuses Euclidean distance to evaluate the similarity it is onlysuitable for processing data with appropriate compactness

Journal of Sensors 3

and good dispersion Therefore kernel fuzzy C-means clus-tering algorithm (KFCM) is used to replace the Euclideandistance with kernel-induced distance expressed as

min 119869 (119880119881) = 119862sum119894=1

119899sum119896=1

120583119898119894119896 1003817100381710038171003817Φ (119909119896) minusΦ (119907119894)10038171003817100381710038172 (6)

Φ(119909119896) minus Φ(119907119894)2 in (6) can be further derived into thefollowing form

1003817100381710038171003817Φ (119909119896) minusΦ (119907119894)10038171003817100381710038172= (Φ (119909119896) minusΦ (119907119894))119879 (Φ (119909119896) minusΦ (119907119894))= Φ119879 (119909119896)Φ (119909119896) minusΦ119879 (119909119896)Φ (119907119894)

minusΦ (119909119896)Φ119879 (119907119894) +Φ119879 (119907119894)Φ (119907119894)= 119870 (119909119896119909119896) minus 2119870 (119909119896 119907119894) + 119870 (119907119894 119907119894)

(7)

where 119870(119909 119907) = Φ119879(119909)Φ(119907) is the inner product ofkernel function In this paper we select Gaussian functionas 119870(119909 119907) which means 119870(119909 119907) = exp(minus119909119896 minus 11990711989421205902)119870(119909119896119909119896) = 1 119870(119907119896 119907119896) = 1 and Φ119879(119909119896)Φ(119907119894) =Φ(119909119896)Φ119879(119907119894) Thus the clustering objective function mem-bership 120583119894119896 and clustering center 119907119894 can be calculated accord-ing to the following equations

min 119869 (119880119881)= 2 119862sum119894=1

119899sum119896=1

120583119898119894119896 [1 minus 119870 (119909119896 119907119894) + 119870 (119907119894 119907119894)] (8)

120583119894119896 = [1 minus 119870 (119909119896 119907119894)]minus1(119898minus1)sum119862119895=1 [1 minus 119870(119909119896 119907119895)]minus1(119898minus1) V119894 = sum119899119896=1 120583119898119894119896119870 (119909119896 119907119894)119909119896sum119899119896=1 120583119898119894119896119870(119909119896 119907119894)

(9)

Although KFCM adopts kernel-induced distance tobroaden its application range it still has many problemswaiting to be solved Firstly the clustering number C shouldbe set in advance but the preset value will affect the finalclustering result directly Secondly KFCM usually generatesthe initial clustering center through random initializationwhich reduces the clustering effectiveness Finally the valueof fuzzy weighted index 119898 to a great extent affects theclustering result

23 Implementation of Improved Kernel FCM In this partwe introduce validity index based on compactness anddispersion to realize the automatic classification of clusteringnumber and initialization of the cluster center with ldquomaxi-mum benefitrdquo to increase the clustering performance

Validity index an appropriate validity index will wellreflect the clustering quality The number of clusters thatmakes the validity index reaches the optimum is called the

optimum number of clustersTherefore we use compactness119881119886119903 and dispersion 119878119890119901 to define validity index 119881119878(119880119881) as119881119878 (119880119881) = 119881119886119903 (119880119881)119878119890119901 (119880 119862) (10)

119881119886119903 (119880119881) = 119881119886119903 (119880119881)119881119886119903max119881119886119903max = max⏟⏟⏟⏟⏟⏟⏟

119862

119881119886119903 (119880119881)119878119890119901 (119880119881) = 119878119890119901 (119880 119862)119878119890119901max

119878119890119901max = max⏟⏟⏟⏟⏟⏟⏟

119862

119878119890119901 (119880 119862)(11)

where 119862 = 2 3 sdot sdot sdot 119862max and 119862max is the maximum numberof clusters The compactness 119881119886119903 and dispersion 119878119890119901 arecomputed in the following way

119881119886119903 (119880119881) = [[119862sum119894minus1

119899sum119895=1

1205831198941198961198892 (119909119895 119907119894)119899 (119894) ]]radic(119862 + 1119862 minus 1)119889 (119909119895 119907119894) = radic1 minus exp (minus120573 10038171003817100381710038171003817119909119895 minus 119907119894100381710038171003817100381710038172)

120573 = (1119899119899sum119895=1

10038171003817100381710038171003817100381710038171003817100381710038171003817119909119895 minus (1119899119899sum119895=1

119909119895)100381710038171003817100381710038171003817100381710038171003817100381710038172)minus1

(12)

119878119890119901 (119880 119862) = 1 minus max119894 =119895

(max119894 =119895

(min (120583119894119896 120583119895119896))) (13)

where 119899(119894) is the sample number of the 119894th subclass From thecalculation formulas above it can be seen that a smaller 119881119886119903means better homogeneity within a class and a larger 119878119890119901indicates greater separation between classes In summary thesmaller the value of validity index is the better the clusteringquality

Initialization of the cluster center with ldquomaximum benefitrdquoto reduce the influence of the random initialization clustercenter has on the clustering results the initialization of thecluster center with ldquomaximum benefitrdquo is designed In sampleset 119883 = 11990911199092 sdot sdot sdot 119909119899 randomly select one sample 119909119896 119896 =1 2 sdot sdot sdot 119899 as the first classification clustering center writtenas 1199071 = 119909119896 Then select another sample 119909119895 (119895 = 119896) withprobability 119875(119909119895 997888rarr 1199072) from the remaining samples as thesecond classification clustering center expressed as 1199072 = 119909119895Here the calculation formula for 119875(119909119895 997888rarr 1199072) is119875 (119909119895 997888rarr 1199072) = 10038171003817100381710038171003817119909119895 minus 1199071100381710038171003817100381710038172sum119899119894=1 1003817100381710038171003817119909119894 minus 119907110038171003817100381710038172 119894 = 1 2 sdot sdot sdot 119899 and 119894 = 119896

(14)


(1) Set the original number of clusters119862 = 2 andinitialize parameters maximum number of clusters119862max fuzzy weighted index 119898 maximum number ofiterations 119879max and the stopping criterion Θ

(2) Initialize the clustering center119881 according to theldquomaximum benefitrdquo clustering center initializationmethod Calculate the initial membership matrixaccording to (9)For 119862 = 2 119862max

(3) While (|119907119905119896 minus 119907119905minus1119896 | le Θ119905 le 119879max) do(4) Calculate membership matrix119880119905 and clustering

center119881119905 according to (9)(5) 119905 + 1 997888rarr 119905 (6) Calculate validity index 119881119878119862(119880119881) according to

(10)-(13)End for

(7) Output results the 119862 corresponding to the minimum119881119878119862(119880119881) is the optimal number of clusters and thecorresponding clustering center and membership is theoptimal 119881and119880 respectively

Pseudocode 1

The third classification clustering center 1199073 = 119909119908 (119908 = 119895 = 119896)is selected from the remaining samples with the probability119875(119909119908 997888rarr 1199073)119875 (119909119908 997888rarr 1199073) = 1003817100381710038171003817119909119908 minus 119907110038171003817100381710038172 + 1003817100381710038171003817119909119908 minus 119907210038171003817100381710038172sum119899119894=1 1003817100381710038171003817119909119894 minus 119907110038171003817100381710038172 + sum119899119894=1 1003817100381710038171003817119909119894 minus 119907210038171003817100381710038172 119894 = 1 2 sdot sdot sdot 119899 and 119894 = 119896 = 119895

(15)

and so on until the C clustering centers are all selected Suchprocedure ensures enough differences between classes andeffectively reduces the probability of the algorithm falling intolocal optimal

The improved kernel FCM (IKFCM) can realize theautomatic classification of clustering number Its pseudocodeis shown in Pseudocode 1

24 Implementation of Differential Evolution Based onIKFCM Like other intelligent optimization algorithms thepopulation diversity of DE decreases with the increase ofiteration times and the random selection of individuals formutation may accelerate the population trapped in localoptimal [14] Thus in order to further improve the globaloptimization ability and convergence accuracy of DE we usefuzzy clustering to analyze the population making it morescientific for selecting particles formutation in each iteration

As one of the most widely used clustering algorithmsFCM classifies data through analyzing how closely a data isrelative to different classes making data samples with moresimilarity be classified into the same class In this paperwe utilize IKFCM for cluster analysis before DE and itsclassification result is considered as the basis of selectingparticles for mutation After a certain number of iterations

End

Start

Random Generation of Initial Population

Clustering Analysis of Population using IKFCM

i = 1

Random Selection of ree Particles for

NY

Y

N

Algorithm Terminate

N

Y

Xti and Vt

i Perform Crossover Operationsto getUt

i

X ti and Ut

i Perform Crossover Operations

Xt+1i larrlarr Xt

i X t+1i larr Xt

i

i larr i+1 i gt P

t larr t+1

f (U ti ) le f (X t

i )

Mutation in Different Classed with Xti

Figure 1 Flow chart of IDDE

the population is divided into C subclasses by IKFCMand there are more similarities within classes and greaterdifferences between classes When it comes to mutationparticles 119883119905119886 119883

119905119887 and 119883

119905119888 will be selected from different

classes thus effectively expanding the diversity of mutantparticle and ensuring that particleswith larger differenceswillbe participated in the populationrsquos evolution and eventuallyimproves the convergence performance of DE The specificprocess of the DE based on fuzzy clustering named IDDE ispresented in Figure 1


3 Compressed Sensing SignalReconstruction Based on IDDE

31 Compressed Sensing Sparse Signal Reconstruction Incompressed sensing Ψ119873times119873 is called sparse matrix if it can beused to linearly describe signal119883119873times1 sparselyWe can expressit as

119883119873times1 = Ψ119873times119873119904119873times1 (16)

where 119904119873times1(1199041 sdot sdot sdot 119904119873) denotes the linear description of119883119873times1under Ψ119873times119873 and its sparsity satisfies 119870 ltlt 119873 There existsa measurement matrix Φ119872times119873 that allows 119883119873times1 be describedwith a small amount of data as

119910119872times1 = Φ119872times119873119883119873times1 (17)

where119910119872times1 is the measurement vector and satisfies 119872 ltlt 119873From (6) and (7) we have

119910119872times1119860119872times119873=Φ119872times119873Ψ119873times119873= 119860119872times119873119904119873times1 = 119860119904 (18)

Once 119860119872times119873 satisfies RIP condition [12] the original signalcan be reconstructed by solving the 1198970 norm that is

min 1199041198970st 119910 = 119860119904 = ΦΨ119904 (19)

Since the process of solving (9) is a NP-hard consideringthe signal with unknown sparsity as the differential evolutionparticle coding we propose a compressive sensing signalreconstruction algorithm based on IDDE which accuratelyreconstructed the unknown sparsity signal through popula-tion evolution

32 Implementation of Compressed Sensing Signal Recon-struction Based on IDDE DE is mainly used for contin-uous optimization but for discrete optimization it willgenerate large numbers of solutions that do not conformto the requirements severely reducing its convergence effi-ciency In this case taking the CS reconstruction algo-rithm into account the differential evolutionary particlecoding method and evolutionary mechanism in IDDE areredefined

Definition 1 (particle coding) In CS reconstruction algo-rithm optimization define particle coding as

119883119894 (1199091198941 sdot sdot sdot 119909119894119873) larr997888 119909119894119895 = 1 119894119891 119904119895 = 00 119890119897119904119890 (20)

It is obvious that 119883119894(1199091198941 sdot sdot sdot 119909119894119873) is correspondence with thenonzero elements in the sparse signal 119904119873times1(1199041 sdot sdot sdot 119904119873)Definition 2 Randomly select 1198651015840 (1 le 1198651015840 le 119873) code bitsin particle 119883119895 and replace the corresponding code bits in

particle 119883119894 (119894 = 119895) with it The exchange process is denotedas 1198651015840(119883119894 larr997888 119883119895)

1198651015840 (119883119894 larr997888 119883119895) = 1199091198941 larr997888 1199091198951 sdot sdot sdot 119909119895119896 larr997888 119909119895119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1198651015840

(21)

Then the mutation step in IDDE can be refreshed as

1198811199051198941 = 11986510158401 (119883119894 larr997888 119883119886) 1198811199051198942 = 11986510158402 (119883119894 larr997888 119883119887) 1198811199051198943 = 11986510158403 (119883119894 larr997888 119883119888)

(22)

where 119883119886 119883119887 and 119883119888 are three particles randomly selectedfrom three different clusters which are also different from thecluster that contain 119883119894

As can be seen from (22) by performing crossover onthe three mutant particles respectively we will obtain threetrial particles 1198801199051198941 119880

1199051198942 and 119880

1199051198943 Then a particle with better

fitness will be selected between the three trial particles and119883119894as

119883119905+1119894 = 119880119905119894119895 119880119905119894119895 = min

119891( )(119880119905119894111988011990511989421198801199051198943)

119883119905119894 119890119897119904119890 (23)

Definition 3 (objective function) In CS reconstruction algo-rithm optimization define the objective function as

min 119891 (119883119894) = 1003817100381710038171003817119910 minus ΦΨ11988311989410038171003817100381710038172 (24)

As the objective function of CS reconstruction algorithmoptimization is defined IDDE would locate the nonzeroelements of the sparse signal code after certain times ofevolution and obtain the amplitude of the nonzero elementsthrough least square method Finally the signal is preciselyreconstructed The performance of the CS reconstructionalgorithm is superior due to its utilization of fuzzy clusteringand improved discrete differential evolution algorithm Thepseudocode of the CS reconstruction algorithm based onIDDE is shown in Pseudocode 2

33 Application of IDDE in WSNs In this part we study thesparse event detection [13] in the context of wireless sensornetworks (WSNs) 119876 wireless sensor nodes are placed withinthemonitoring scope ofWSNs tomonitor119873 event sources119870network events randomly occurred from the119873 event sourcesat some timeWe use vector 119904119873times1 = [1199041 1199042 sdot sdot sdot 119904119873]119879 to denotethe energy signal of the event sources (119904119894 = 0 means that the119894th event source did not happen any event) Obviously if119870 ≪119873 119904119873times1 is a sparse signal with sparsity of119870 Assuming119876 = 119873signals received by 119873 nodes 119909119873times1 = [1199091 1199092 sdot sdot sdot 119909119873]119879 can bewritten as


(1) Initialize parameters maximum number of iterations119879max cluster number 119862 and the stopping criterion(2) Initialize the population of IDDE and set 119905 = 0(3) While (the requirement of IDDE is not satisfied) do(4) While (the stopping criterion of IKFCM is not

satisfied) do cluster analysis the population(5) Calculate119867 = [ℎ119894119896]119862times119875 and119874 = 119900119894 according to (3)(6) For 119894 = 1 119875 update the particle(7) Randomly select 3 particles from different clusters

which do not contain119883119905119894 Perform mutationaccording to (22) to obtain 1198811199051198941 119881

1199051198942 and119881

1199051198943

(8) Generate11988011990511989411198801199051198942 and119880

1199051198943 through crossover in the

following way

119880119905119894 (1199061199051198941 sdot sdot sdot 119906119905119894119873) lArr997904 119906119905119894119895 = V119905119894119895 (119903119886119899119889119887 (119895) le 119862119877 119900119903 119895 = 119903119899119887119903 (119894))119909119905119894119895 (119903119886119899119889119887 (119895) gt 119862119877 119900119903 119895 = 119903119899119887119903 (119894))

(9) Evaluate the trial particle119880119905119894If 119880119905119894119895 = min119891()(1198801199051198941 1198801199051198942 1198801199051198943) then 119883119905+1119894 = 119880119905119894119895Else 119883119905+1119894 = 119883119905119894End if

(10) End for(11) 119905 + 1 997888rarr 119905(12) Output results

Pseudocode 2

[[[[[[[

11990911199092119909119873

]]]]]]]=

[[[[[[[[

1003816100381610038161003816ℎ111003816100381610038161003816 (11988911)minus120572 1003816100381610038161003816ℎ121003816100381610038161003816 (11988912)minus120572 sdot sdot sdot 1003816100381610038161003816ℎ11198731003816100381610038161003816 (1198891119873)minus1205721003816100381610038161003816ℎ211003816100381610038161003816 (11988921)minus120572 1003816100381610038161003816ℎ221003816100381610038161003816 (11988922)minus120572 sdot sdot sdot 1003816100381610038161003816ℎ21198731003816100381610038161003816 (1198892119873)minus120572 d1003816100381610038161003816ℎ11987311003816100381610038161003816 (1198891198721)minus120572 1003816100381610038161003816ℎ11987321003816100381610038161003816 (1198891198722)minus120572 sdot sdot sdot 1003816100381610038161003816ℎ1198731198731003816100381610038161003816 (119889119872119873)minus120572

]]]]]]]]

[[[[[[[

11990411199042119904119873

]]]]]]]= Ψ119873times119873119904 (25)

where 119889119894119895 is the distance the signal transmitted ℎ119894119895 is theenergy attenuation model According to CS theory sparseevent can be detected by the data received by 119872 nodes

119910119872times1 =[[[[[[[

11991011199102119910119872

]]]]]]]=

[[[[[[[[

1 0 sdot sdot sdot 0 sdot sdot sdot 00 1 sdot sdot sdot 0 sdot sdot sdot 0 d0 0 sdot sdot sdot 1 0 0

]]]]]]]]

[[[[[[[

11990911199092119909119873

]]]]]]]= Φ119872times119873119909119873times1

(26)

By (25) and (26) we have 119910119872times1 = Φ119872times119873119909119873times1 =Φ119872times119873Ψ119873times119873119904119873times1 = 119860119872times119873119904119873times1 It can be shown that when119872 ge 119874(119888(119870 + 1) ln(119873119870)) 119860 can meet the requirementof RIP But the CS reconstruction algorithm based on IDDE

requires that an accurate measurement matrix should beguaranteedThuswe transform matrix119860119872times119873 in the followingway

119860 = 119880Σ119881T (27)

where Σ119872times119873 is semipositive definite diagonal matrix 119880 and119881 is orthogonal matrix Therefore (27) can be rewritten as

119910 = Φ119872times119873119909119873times1 = 119860119904 = 119880Σ119881119879119904

= 119880[[[[[[[[[

119872times119872⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1205901 0 sdot sdot sdot 00 1205902 sdot sdot sdot 0 d0 0 sdot sdot sdot 120590119872

(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 sdot sdot sdot 00 sdot sdot sdot 0 00 sdot sdot sdot 0

]]]]]]]]]119881119879119904

(28)


Equation (28) can be further transformed into

Σminus11 119880119879119910

= Σminus11 119880119879119880[119872times119872⏞⏞⏞⏞⏞⏞⏞Σ1

(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞0 ] [119872times119872⏞⏞⏞⏞⏞⏞⏞1198811

(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞1198812

] 119904= 1198811119879119904 997904rArr1199101015840 = Σminus11 119880119879119910 = 1198811119879119904

(29)

Equation (29) represents a newmeasurement system As seenfrom the derivation process the measurement matrix 1198811

119879 isrow orthogonal matrix thus RIP condition is satisfied

In this case sparse event detection in WSNs is trans-formed into the problem of using 119872 wireless sensor nodesto detect 119870 event sources The detailed steps include trainingparameters with the experimental data in the deployment teststage ofWSNs to obtain the optimal parameter configurationprocessing received signal by the use of reconstructionalgorithm based on IDDE to obtain the sparse signal vector119904119873times1 in the detection stage tomonitor sparse events effectivelyand timely

4 Simulation

Simulations are performed in MATLAB to demonstrate theeffectiveness of our algorithm We deploy 119876 = 500 sensornodes to monitor the temperature of 119873 = 120 places Theevent source sequence corresponded to event energy signal119904119873times1 = [1199041 1199042 sdot sdot sdot 119904119873]119879 Reconstruction success rate 119877119878reconstruction time 119879 and reconstruction relative error 119877119864are selected to evaluate the reconstruction performance Thereconstruction relative error 119877119864 is calculated as follows

119877119864 = radic [sum119873119894=1 (119904119894 minus 119904119894)][sum119873119894=1 1199041198942] times 100 (30)

where 119904119873times1(1199041 sdot sdot sdot 119904119873) denotes the original signal and119904119873times1(1199041 sdot sdot sdot 119904119873) stands for the reconstructed signal Thereconstruction success rate 119877119878 refers to the probability that119877119864 is less than the threshold 12057541 Reconstruction Performance Selecting typical discretesignal as the original signal and the reconstruction per-formance of the reconstruction algorithm based on IDDEunder the assumption that 119870 is unknown is analyzed Theperformance of our algorithm is also compared to the blindsparsity reconstruction algorithm in [8] SWOMP in [2]and the classical StOMP Run each algorithm independentlyfor 50 times calculate the average reconstruction successrate 119877119878 the average reconstruction time 119879 and the averagereconstruction relative error 119877119864 for performance analysisand set different 120575 for different signal The reconstructionresult of the CS reconstruction algorithm based on IDDEis shown in Figure 2 The evaluation index results of the 4reconstruction algorithms are compared in Table 1

It can be seen from Figure 2 that the CS reconstructionalgorithm based on IDDE can obtain enough recovered data

Sign

al am

plitu

de

0

02

04

06

08

1

0 10 20 30 40 50 60

Original signalReconstructed signal

Signal length

Figure 2 Reconstruction result of the IDDE reconstruction algo-rithm

points from the compressed sensing data and reconstructto the original signal indicating that the algorithm weproposed in this paper is capable of reconstructing signalwith unknown sparsity In Table 1 the reconstruction successrates of the algorithm we propose reached 100 for differentoriginal signals the algorithm in [8] reached above 97and the algorithm in [2] was only about 50 and StOMPcould hardly reconstruct the original signal In terms ofreconstruction time our algorithm is significantly faster thanthe other 3 algorithms with 119879 reduced by 151-313 As forreconstruction error compared with the algorithm in [2]which has relatively better reconstruction success rate the119877119864of our algorithm decreased by about 364-519 In summarycompared with previous reconstruction algorithms such asStOMP the CS reconstruction algorithm based on IDDEproposed in this paper performs better much more suitablefor reconstructing signal with unknown sparsity All theseadvantages on one hand could be accredited to the utilizationof IDDE Through IDDE the signal sparsity is transformedinto particle coding which makes the differential evolutionalgorithm be able to find the global optimum solution whileobtaining the location of nonzero elements On the otherhand fuzzy clustering is used to analyze the populationmaking the learning target of the population evolutionmore rational and scientific which ultimately improves thereconstruction performance of our algorithm

42 Effects of Parameters on the Reconstruction AlgorithmPerformance Sparsity 119870 and observation time119872 are criticalto the reconstruction accuracy Simulations were carried outto examine the reconstruction success rate of the CS recon-struction algorithm based on IDDE under different 119870 and119872 respectively the blind sparsity reconstruction algorithmin [8] SWOMP in [2] and StOMP are also simulated asa comparison Each algorithm runs 50 times Comparisonresults are shown in Figures 3 and 4

Figure 3 shows that when the signal length and 119872 arefixed all the four reconstruction success rate curves decreasewith the increase of119870 Comparedwith the other 3 algorithmsour algorithm maintains a reconstruction success rate of over


Table 1 Evaluation index comparisons of the four reconstruction algorithms

Signal Index AlgorithmThis paper Literature[8] Literature[2] StOMP

Signal1119877119878 100 973 502 173119879s 017 038 028 037119877119864 0007 0012 1845 3330

Signal2119877119878 100 982 492 153119879s 025 044 036 082119877119864 0014 0018 1038 2143

Signal3119877119878 100 976 557 108119879s 056 092 066 127119877119864 0013 0017 1836 3045

Sparsity K

Reco

nstr

ucio

nsu

cces

s rat

e (

)

0 10 20 30 40 50 600

20

40

60

80

100

is paperLiterature[8]

Literature[2]StOMP

Figure 3 Sparsity 119870 versus reconstruction success rate (119872 = 100)

94 when 119870 is within the range of (25 35) while that of theother 3 algorithms decrease significantly

As shown in Figure 4 when the signal length and 119870 areunchanged all the 4 curves rise with the increase of 119872 andthe 119877119878 of the algorithm we proposed is much better than theother 3 algorithmswith the same119872 Especiallywhen119872 = 80the 119877119878 of our algorithm can reach 100

However large number of sensor nodes means large net-work energy consumption which is unfavorable to prolongthe survival time of WSNs Thus in the following part weanalyze the robustness and antinoise performance of ouralgorithm

43 Antinoise Performance Addwhite noise into the discretesignal to be detected and select the reconstruction algorithmin [8] and StOMP as a comparison We investigate theirantinoise performances under 119878119873119877 ranging from 20dB to45dB

From Figure 5 we can see that if the noise is notvery serious the reconstruction success rates of the threealgorithms change a little as the noise level changesWhen thenoise is at a high level the reconstruction success rates changesignificantly with the noise level The 119877119878 of the algorithm

40 60 80 100 12040

50

60

70

80

90

100

Reco

nstr

uctio

nsu

cces

s rat

e (

)


Literature[2]StOMP

Observation times M

Figure 4 Observation time 119872 versus reconstruction success rate(119870 = 20)

SNR (dB)

0102030405060708090

100

10 20 30 40 50

Reco

nstr

uctio

nsu

cces

s rat

e (

)

is paperLiterature[8]StOMP

Figure 5 SNRversus reconstruction success rate (119870 = 30119872 = 100119873 = 120)

we propose will still remain around 93 if the SNR decreaseto 15dB while the other two algorithms decrease obviouslyindicating that our algorithm is good at resisting noise


5 Conclusions

In this paper we propose a compressed sensing signal recon-struction algorithm based on IDDE The algorithm employsintelligent optimization and fuzzy clustering together trans-forming signal reconstruction into the global optimization ofdifferential algorithm to reconstruct signal with unknownsparsity through fuzzy cluster analysis particle coding anddifferential evolution Its application in WSNs sparse eventdetection is also discussed Simulation results testify that thealgorithm we propose performs excellent in signal recon-struction and is robust to noise and interference

Data Availability

The research data used to support the findings of this studywere supplied by government agencies such as laborato-ries under license and so cannot be made freely availableRequests for access to these data should be made to Zhou-zhou Liu via liuzhouzhou8192126com

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors disclosed receipt of the following financialsupport for the research authorship andor publicationof this article This work was supported in part by theNational Natural Science Foundation of China under Grant61871313 China Postdoctoral Science Foundation fundedproject China under Grant 2018M633573 and Shaanxi Post-doctoral Science Foundation funded project under Grant2018BSHQYXMZZ11

References

[1] TWang Y Liang W JiaM Arif A Liu andM Xie ldquoCouplingresource management based on fog computing in smart citysystemsrdquo Journal of Network and Computer Applications vol135 pp 11ndash19 2019

[2] D Wang Y Wu W Cao and J Xu ldquoImproved reconstructionalgorithm for compressed sensingrdquo Journal of NorthwesternPolytechnical University vol 35 no 5 pp 774ndash779 2017 (Chi-nese)

[3] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[4] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo Institute of Electrical and ElectronicsEngineers Transactions on Information Theory vol 52 no 2 pp489ndash509 2006

[5] E J Candes and T Tao ldquoNear-optimal signal recovery fromrandom projections universal encoding strategiesrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 52 no 12 pp 5406ndash5425 2006

[6] J Zeng T Wang W Jia S Peng and G Wang ldquoA survey onsensor-cloudrdquoComputer Research andDevelopment vol 54 no5 pp 925ndash939 2017

[7] N Sun J Liu and D Xiao ldquoSL0 reconstruction algorithm forcompressive sensing based on BFGS quasi Newton methodrdquoJournal of Communications vol 40 no 10 pp 2408ndash2414 2018(Chinese)

[8] Z-N Zhang R-T Huang and J-W Yan ldquoA blind sparsityreconstruction algorithm for compressed sensing signalrdquo ActaElectronica Sinica vol 39 no 1 pp 18ndash22 2011 (Chinese)

[9] HMohimaniM Babaie-Zadeh andC Jutten ldquoA fast approachfor overcomplete sparse decomposition based on smoothed 1198970normrdquo IEEE Transactions on Signal Processing vol 57 no 1 pp289ndash301 2009

[10] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011

[11] H Feng L Luo Y Wang and M Ye ldquoMulti-objective datacollecting strategies for wireless sensor network based on thetime variable multi-salesman problem and genetic algorithmrdquoJournal on Communication vol 38 no 3 pp 112ndash123 2017

[12] F Zhou X Y Zeng and L H Yang ldquoA regularized sparserepresentation methodrdquo Acta Mathematica Sinica vol 58 no4 pp 649ndash660 2015 (Chinese)

[13] B Sun X Shan K Wu and Y Xiao ldquoAnomaly detection basedsecure in-network aggregation for wireless sensor networksrdquoIEEE Systems Journal vol 7 no 1 pp 13ndash25 2013

[14] QWu C Zhang and L Gao ldquoDifferential evolution algorithmbased on hybrid crossoverrdquo Journal of Huazhong University ofScience and Technology (Natural Science Edition) vol 46 no 5pp 78ndash105 2018 (Chinese)

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that the inner product matching criterion of SWOMP issensitive to residual signals Simulation results show thattheir algorithm has higher signal reconstruction successrate In [8] researchers discussed the signal reconstructionalgorithm for compressed sensing without prior informationof signal sparsity But the signal reconstruction accuracy ofthis algorithm needs to be improved

Currently most compressed sensing algorithms recon-struct the original signal by minimizing its 1198970 or 1198971 norm[9] This is NP-hard problemTherefore we propose a WSNscompressed sensing signal reconstruction algorithm basedon fuzzy clustering and discrete differential evolution Inthis algorithm we consider signal with unknown sparsityas differential evolution particle coding and reconstruct thesignal through population iterative evolution To improve theefficiency and accuracy of signal reconstruction fuzzy clus-tering is utilized while the differential evolution mechanismis redefined Finally simulation experiments are carried outto verify its validity

2 Differential Evolution AlgorithmBased on Fuzzy Clustering

21 Differential Evolution Differential evolution (DE) [10]is a kind of intelligent optimization algorithm with randomsearch It preserves the individuals with better fitness throughmutation and competition to achieve the desired end Due toits excellent global convergence and good robustness fewernumber of control parameters and easy to be implementedDE has been widely used in solving complex optimizationproblems [11]

An iteration of the classical DE algorithm mainly consistsof three basic stepsmutation crossover and selection ForN-dimensional optimization there exists a differential evolutionpopulation composed of 119875 particles 119883119905119894(1199091199051198941 sdot sdot sdot 119909119905119894119873) (119894 isin[1 119875]) for the current generation 119905The process of performingmutation on119883119905119894 (1199091199051198941 sdot sdot sdot 119909119905119894119873) can be expressed as

119881119905119894 = 119883119905119886 + 119865 times (119883119905119887 minus119883119905119888) (1)

where 119865 isin [04 1] denotes the scalar number 119881119905119894 denotesthe mutant particle The indices 119886 119887 119888 isin [1 119875] are mutu-ally exclusive integers randomly chosen from the range[1 119875] which are also different from 119894 In essence mutationrandomly selects three different particles from the currentpopulation multiplies the difference of any two of thesethree particles with the scalar number 119865 and adds the scaleddifference to the third one to obtain the mutant particle tofuse more individual information

To enhance the potential diversity of the populationcrossover operation is then performed It is a process ofselecting components from119881119905119894(V1199051198941 sdot sdot sdot V119905119894119873) and119883119905119894 accordingto certain probability to obtain the trial particle described as

119880119905119894 (1199061199051198941 sdot sdot sdot 119906119905119894119873) lArr997904 119906119905119894119895=

V119905119894119895 (119903119886119899119889119887 (119895) le 119862119877 119900119903 119895 = 119903119899119887119903 (119894))119909119905119894119895 (119903119886119899119889119887 (119895) gt 119862119877 119900119903 119895 = 119903119899119887119903 (119894))(2)

where 119862119877 isin (0 1) is called the crossover rate 119903119886119899119889119887(119895) isin (0 1)is a random number lying between 0 and 1 and 119903119899119887119903(119894) is aninteger randomly selected from range [1119873] which ensuresthat 119880119905119894 gets at least one component from119881119905119894

Then the next generation particle 119883119905+1119894 is selected in theform of roulette to keep the population size constant oversubsequent generations Taking the minimum optimizationas an example the selection operation is described as

119883119905+1119894 = 119880119905119894 119894119891 119891 (119880119905119894) le 119891 (119883119905119894)119883119905119894 119890119897119904119890 (3)

where 119891(119883119905119894) is the fitness value of particle119883119905119894 We can see that in each iteration the population always

keeps the better individual with higher fitness so as to reachthe global optimal solution

22 Improved Kernel FCM Fuzzy C-means clustering algo-rithm (FCM) is one of the most widely used clustering algo-rithms It uses membership to describe the degree to whicha data belongs to a certain subclass for data classification[12] For a 119878-dimension dataset 119883 isin 119877119878 clustering is todivide 119899 samples 119909119896 isin 119883 119896 = 1 2 sdot sdot sdot 119899 into 119862 subclasseswhile making the value of the clustering objective functionminimum that is

min 119869 (119880119881) = 119862sum119894=1

119899sum119896=1

120583119898119894119896 1003817100381710038171003817119909119896 minus 11990711989410038171003817100381710038172 (4)

where 120583119894119896 is the membership of sample 119909119896 to the 119894th subclass119880 = [120583119894119896]119862times119899 is the membership matrix 119881 = 119907119894 (119894 =1 2 sdot sdot sdot 119862) is the set of clustering center and 119898 is the fuzzyweighted index Let 120597119869120597120583119894119896 = 0 120597119869120597V119894 = 0 we have

120597119869120597120583119894119896 = 0 997904rArr120583119894119896 = 1003817100381710038171003817119909119896 minus 1199071198941003817100381710038171003817minus2(119898minus1)sum119862119895=1 10038171003817100381710038171003817119909119896 minus 11990711989510038171003817100381710038171003817minus2(119898minus1) 120597119869120597V119894 = 0 997904rArrV119894 = sum119899119896=1 120583119898119894119896119909119896sum119899119896=1 120583119898119894119896st

119862sum119894=1

120583119894119896 = 1 120583119894119896 isin [0 1] 119899sum119896=1

120583119894119896 isin (0 1)

(5)

It can be seen that FCM is a kind of local searchalgorithm and is sensitive to initial values [13] Since FCMuses Euclidean distance to evaluate the similarity it is onlysuitable for processing data with appropriate compactness



min 119869 (119880119881) = 119862sum119894=1

119899sum119896=1

120583119898119894119896 1003817100381710038171003817Φ (119909119896) minusΦ (119907119894)10038171003817100381710038172 (6)



minusΦ (119909119896)Φ119879 (119907119894) +Φ119879 (119907119894)Φ (119907119894)= 119870 (119909119896119909119896) minus 2119870 (119909119896 119907119894) + 119870 (119907119894 119907119894)

(7)


min 119869 (119880119881)= 2 119862sum119894=1

119899sum119896=1

120583119898119894119896 [1 minus 119870 (119909119896 119907119894) + 119870 (119907119894 119907119894)] (8)


(9)





119881119886119903 (119880119881) = 119881119886119903 (119880119881)119881119886119903max119881119886119903max = max⏟⏟⏟⏟⏟⏟⏟

119862

119881119886119903 (119880119881)119878119890119901 (119880119881) = 119878119890119901 (119880 119862)119878119890119901max

119878119890119901max = max⏟⏟⏟⏟⏟⏟⏟

119862

119878119890119901 (119880 119862)(11)


119881119886119903 (119880119881) = [[119862sum119894minus1

119899sum119895=1


120573 = (1119899119899sum119895=1

10038171003817100381710038171003817100381710038171003817100381710038171003817119909119895 minus (1119899119899sum119895=1

119909119895)100381710038171003817100381710038171003817100381710038171003817100381710038172)minus1

(12)

119878119890119901 (119880 119862) = 1 minus max119894 =119895

(max119894 =119895

(min (120583119894119896 120583119895119896))) (13)



(14)






(10)-(13)End for


Pseudocode 1


(15)





End

Start



i = 1


NY

Y

N

Algorithm Terminate

N

Y

Xti and Vt


i

X ti and Ut


Xt+1i larrlarr Xt

i X t+1i larr Xt

i

i larr i+1 i gt P

t larr t+1

f (U ti ) le f (X t

i )




119905119887 and 119883












min 1199041198970st 119910 = 119860119904 = ΦΨ119904 (19)








(21)


1198811199051198941 = 11986510158401 (119883119894 larr997888 119883119886) 1198811199051198942 = 11986510158402 (119883119894 larr997888 119883119887) 1198811199051198943 = 11986510158403 (119883119894 larr997888 119883119888)

(22)



1199051198942 and 119880



119883119905+1119894 = 119880119905119894119895 119880119905119894119895 = min

119891( )(119880119905119894111988011990511989421198801199051198943)

119883119905119894 119890119897119904119890 (23)


min 119891 (119883119894) = 1003817100381710038171003817119910 minus ΦΨ11988311989410038171003817100381710038172 (24)







1199051198942 and119881

1199051198943

(8) Generate11988011990511989411198801199051198942 and119880


following way




Pseudocode 2

[[[[[[[

11990911199092119909119873

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990411199042119904119873

]]]]]]]= Ψ119873times119873119904 (25)


119910119872times1 =[[[[[[[

11991011199102119910119872

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990911199092119909119873

]]]]]]]= Φ119872times119873119909119873times1

(26)



119860 = 119880Σ119881T (27)


119910 = Φ119872times119873119909119873times1 = 119860119904 = 119880Σ119881119879119904

= 119880[[[[[[[[[



]]]]]]]]]119881119879119904

(28)



Σminus11 119880119879119910

= Σminus11 119880119879119880[119872times119872⏞⏞⏞⏞⏞⏞⏞Σ1


(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞1198812

] 119904= 1198811119879119904 997904rArr1199101015840 = Σminus11 119880119879119910 = 1198811119879119904

(29)




4 Simulation





Sign

al am

plitu

de

0

02

04

06

08

1

0 10 20 30 40 50 60


Signal length








Signal1119877119878 100 973 502 173119879s 017 038 028 037119877119864 0007 0012 1845 3330

Signal2119877119878 100 982 492 153119879s 025 044 036 082119877119864 0014 0018 1038 2143

Signal3119877119878 100 976 557 108119879s 056 092 066 127119877119864 0013 0017 1836 3045

Sparsity K

Reco

nstr

ucio

nsu

cces

s rat

e (

)

0 10 20 30 40 50 600

20

40

60

80

100


Literature[2]StOMP







40 60 80 100 12040

50

60

70

80

90

100

Reco

nstr

uctio

nsu

cces

s rat

e (

)


Literature[2]StOMP

Observation times M


SNR (dB)

0102030405060708090

100

10 20 30 40 50

Reco

nstr

uctio

nsu

cces

s rat

e (

)





5 Conclusions


Data Availability




Acknowledgments


References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




min 119869 (119880119881) = 119862sum119894=1

119899sum119896=1

120583119898119894119896 1003817100381710038171003817Φ (119909119896) minusΦ (119907119894)10038171003817100381710038172 (6)



minusΦ (119909119896)Φ119879 (119907119894) +Φ119879 (119907119894)Φ (119907119894)= 119870 (119909119896119909119896) minus 2119870 (119909119896 119907119894) + 119870 (119907119894 119907119894)

(7)


min 119869 (119880119881)= 2 119862sum119894=1

119899sum119896=1

120583119898119894119896 [1 minus 119870 (119909119896 119907119894) + 119870 (119907119894 119907119894)] (8)


(9)





119881119886119903 (119880119881) = 119881119886119903 (119880119881)119881119886119903max119881119886119903max = max⏟⏟⏟⏟⏟⏟⏟

119862

119881119886119903 (119880119881)119878119890119901 (119880119881) = 119878119890119901 (119880 119862)119878119890119901max

119878119890119901max = max⏟⏟⏟⏟⏟⏟⏟

119862

119878119890119901 (119880 119862)(11)


119881119886119903 (119880119881) = [[119862sum119894minus1

119899sum119895=1


120573 = (1119899119899sum119895=1

10038171003817100381710038171003817100381710038171003817100381710038171003817119909119895 minus (1119899119899sum119895=1

119909119895)100381710038171003817100381710038171003817100381710038171003817100381710038172)minus1

(12)

119878119890119901 (119880 119862) = 1 minus max119894 =119895

(max119894 =119895

(min (120583119894119896 120583119895119896))) (13)



(14)






(10)-(13)End for


Pseudocode 1


(15)





End

Start



i = 1


NY

Y

N

Algorithm Terminate

N

Y

Xti and Vt


i

X ti and Ut


Xt+1i larrlarr Xt

i X t+1i larr Xt

i

i larr i+1 i gt P

t larr t+1

f (U ti ) le f (X t

i )




119905119887 and 119883












min 1199041198970st 119910 = 119860119904 = ΦΨ119904 (19)








(21)


1198811199051198941 = 11986510158401 (119883119894 larr997888 119883119886) 1198811199051198942 = 11986510158402 (119883119894 larr997888 119883119887) 1198811199051198943 = 11986510158403 (119883119894 larr997888 119883119888)

(22)



1199051198942 and 119880



119883119905+1119894 = 119880119905119894119895 119880119905119894119895 = min

119891( )(119880119905119894111988011990511989421198801199051198943)

119883119905119894 119890119897119904119890 (23)


min 119891 (119883119894) = 1003817100381710038171003817119910 minus ΦΨ11988311989410038171003817100381710038172 (24)







1199051198942 and119881

1199051198943

(8) Generate11988011990511989411198801199051198942 and119880


following way




Pseudocode 2

[[[[[[[

11990911199092119909119873

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990411199042119904119873

]]]]]]]= Ψ119873times119873119904 (25)


119910119872times1 =[[[[[[[

11991011199102119910119872

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990911199092119909119873

]]]]]]]= Φ119872times119873119909119873times1

(26)



119860 = 119880Σ119881T (27)


119910 = Φ119872times119873119909119873times1 = 119860119904 = 119880Σ119881119879119904

= 119880[[[[[[[[[



]]]]]]]]]119881119879119904

(28)



Σminus11 119880119879119910

= Σminus11 119880119879119880[119872times119872⏞⏞⏞⏞⏞⏞⏞Σ1


(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞1198812

] 119904= 1198811119879119904 997904rArr1199101015840 = Σminus11 119880119879119910 = 1198811119879119904

(29)




4 Simulation





Sign

al am

plitu

de

0

02

04

06

08

1

0 10 20 30 40 50 60


Signal length








Signal1119877119878 100 973 502 173119879s 017 038 028 037119877119864 0007 0012 1845 3330

Signal2119877119878 100 982 492 153119879s 025 044 036 082119877119864 0014 0018 1038 2143

Signal3119877119878 100 976 557 108119879s 056 092 066 127119877119864 0013 0017 1836 3045

Sparsity K

Reco

nstr

ucio

nsu

cces

s rat

e (

)

0 10 20 30 40 50 600

20

40

60

80

100


Literature[2]StOMP







40 60 80 100 12040

50

60

70

80

90

100

Reco

nstr

uctio

nsu

cces

s rat

e (

)


Literature[2]StOMP

Observation times M


SNR (dB)

0102030405060708090

100

10 20 30 40 50

Reco

nstr

uctio

nsu

cces

s rat

e (

)





5 Conclusions


Data Availability




Acknowledgments


References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







(10)-(13)End for


Pseudocode 1


(15)





End

Start



i = 1


NY

Y

N

Algorithm Terminate

N

Y

Xti and Vt


i

X ti and Ut


Xt+1i larrlarr Xt

i X t+1i larr Xt

i

i larr i+1 i gt P

t larr t+1

f (U ti ) le f (X t

i )




119905119887 and 119883












min 1199041198970st 119910 = 119860119904 = ΦΨ119904 (19)








(21)


1198811199051198941 = 11986510158401 (119883119894 larr997888 119883119886) 1198811199051198942 = 11986510158402 (119883119894 larr997888 119883119887) 1198811199051198943 = 11986510158403 (119883119894 larr997888 119883119888)

(22)



1199051198942 and 119880



119883119905+1119894 = 119880119905119894119895 119880119905119894119895 = min

119891( )(119880119905119894111988011990511989421198801199051198943)

119883119905119894 119890119897119904119890 (23)


min 119891 (119883119894) = 1003817100381710038171003817119910 minus ΦΨ11988311989410038171003817100381710038172 (24)







1199051198942 and119881

1199051198943

(8) Generate11988011990511989411198801199051198942 and119880


following way




Pseudocode 2

[[[[[[[

11990911199092119909119873

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990411199042119904119873

]]]]]]]= Ψ119873times119873119904 (25)


119910119872times1 =[[[[[[[

11991011199102119910119872

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990911199092119909119873

]]]]]]]= Φ119872times119873119909119873times1

(26)



119860 = 119880Σ119881T (27)


119910 = Φ119872times119873119909119873times1 = 119860119904 = 119880Σ119881119879119904

= 119880[[[[[[[[[



]]]]]]]]]119881119879119904

(28)



Σminus11 119880119879119910

= Σminus11 119880119879119880[119872times119872⏞⏞⏞⏞⏞⏞⏞Σ1


(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞1198812

] 119904= 1198811119879119904 997904rArr1199101015840 = Σminus11 119880119879119910 = 1198811119879119904

(29)




4 Simulation





Sign

al am

plitu

de

0

02

04

06

08

1

0 10 20 30 40 50 60


Signal length








Signal1119877119878 100 973 502 173119879s 017 038 028 037119877119864 0007 0012 1845 3330

Signal2119877119878 100 982 492 153119879s 025 044 036 082119877119864 0014 0018 1038 2143

Signal3119877119878 100 976 557 108119879s 056 092 066 127119877119864 0013 0017 1836 3045

Sparsity K

Reco

nstr

ucio

nsu

cces

s rat

e (

)

0 10 20 30 40 50 600

20

40

60

80

100


Literature[2]StOMP







40 60 80 100 12040

50

60

70

80

90

100

Reco

nstr

uctio

nsu

cces

s rat

e (

)


Literature[2]StOMP

Observation times M


SNR (dB)

0102030405060708090

100

10 20 30 40 50

Reco

nstr

uctio

nsu

cces

s rat

e (

)





5 Conclusions


Data Availability




Acknowledgments


References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











min 1199041198970st 119910 = 119860119904 = ΦΨ119904 (19)








(21)


1198811199051198941 = 11986510158401 (119883119894 larr997888 119883119886) 1198811199051198942 = 11986510158402 (119883119894 larr997888 119883119887) 1198811199051198943 = 11986510158403 (119883119894 larr997888 119883119888)

(22)



1199051198942 and 119880



119883119905+1119894 = 119880119905119894119895 119880119905119894119895 = min

119891( )(119880119905119894111988011990511989421198801199051198943)

119883119905119894 119890119897119904119890 (23)


min 119891 (119883119894) = 1003817100381710038171003817119910 minus ΦΨ11988311989410038171003817100381710038172 (24)







1199051198942 and119881

1199051198943

(8) Generate11988011990511989411198801199051198942 and119880


following way




Pseudocode 2

[[[[[[[

11990911199092119909119873

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990411199042119904119873

]]]]]]]= Ψ119873times119873119904 (25)


119910119872times1 =[[[[[[[

11991011199102119910119872

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990911199092119909119873

]]]]]]]= Φ119872times119873119909119873times1

(26)



119860 = 119880Σ119881T (27)


119910 = Φ119872times119873119909119873times1 = 119860119904 = 119880Σ119881119879119904

= 119880[[[[[[[[[



]]]]]]]]]119881119879119904

(28)



Σminus11 119880119879119910

= Σminus11 119880119879119880[119872times119872⏞⏞⏞⏞⏞⏞⏞Σ1


(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞1198812

] 119904= 1198811119879119904 997904rArr1199101015840 = Σminus11 119880119879119910 = 1198811119879119904

(29)




4 Simulation





Sign

al am

plitu

de

0

02

04

06

08

1

0 10 20 30 40 50 60


Signal length








Signal1119877119878 100 973 502 173119879s 017 038 028 037119877119864 0007 0012 1845 3330

Signal2119877119878 100 982 492 153119879s 025 044 036 082119877119864 0014 0018 1038 2143

Signal3119877119878 100 976 557 108119879s 056 092 066 127119877119864 0013 0017 1836 3045

Sparsity K

Reco

nstr

ucio

nsu

cces

s rat

e (

)

0 10 20 30 40 50 600

20

40

60

80

100


Literature[2]StOMP







40 60 80 100 12040

50

60

70

80

90

100

Reco

nstr

uctio

nsu

cces

s rat

e (

)


Literature[2]StOMP

Observation times M


SNR (dB)

0102030405060708090

100

10 20 30 40 50

Reco

nstr

uctio

nsu

cces

s rat

e (

)





5 Conclusions


Data Availability




Acknowledgments


References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






1199051198942 and119881

1199051198943

(8) Generate11988011990511989411198801199051198942 and119880


following way




Pseudocode 2

[[[[[[[

11990911199092119909119873

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990411199042119904119873

]]]]]]]= Ψ119873times119873119904 (25)


119910119872times1 =[[[[[[[

11991011199102119910119872

]]]]]]]=

[[[[[[[[


]]]]]]]]

[[[[[[[

11990911199092119909119873

]]]]]]]= Φ119872times119873119909119873times1

(26)



119860 = 119880Σ119881T (27)


119910 = Φ119872times119873119909119873times1 = 119860119904 = 119880Σ119881119879119904

= 119880[[[[[[[[[



]]]]]]]]]119881119879119904

(28)



Σminus11 119880119879119910

= Σminus11 119880119879119880[119872times119872⏞⏞⏞⏞⏞⏞⏞Σ1


(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞1198812

] 119904= 1198811119879119904 997904rArr1199101015840 = Σminus11 119880119879119910 = 1198811119879119904

(29)




4 Simulation





Sign

al am

plitu

de

0

02

04

06

08

1

0 10 20 30 40 50 60


Signal length








Signal1119877119878 100 973 502 173119879s 017 038 028 037119877119864 0007 0012 1845 3330

Signal2119877119878 100 982 492 153119879s 025 044 036 082119877119864 0014 0018 1038 2143

Signal3119877119878 100 976 557 108119879s 056 092 066 127119877119864 0013 0017 1836 3045

Sparsity K

Reco

nstr

ucio

nsu

cces

s rat

e (

)

0 10 20 30 40 50 600

20

40

60

80

100


Literature[2]StOMP







40 60 80 100 12040

50

60

70

80

90

100

Reco

nstr

uctio

nsu

cces

s rat

e (

)


Literature[2]StOMP

Observation times M


SNR (dB)

0102030405060708090

100

10 20 30 40 50

Reco

nstr

uctio

nsu

cces

s rat

e (

)





5 Conclusions


Data Availability




Acknowledgments


References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Σminus11 119880119879119910

= Σminus11 119880119879119880[119872times119872⏞⏞⏞⏞⏞⏞⏞Σ1


(119873minus119872)times119872⏞⏞⏞⏞⏞⏞⏞1198812

] 119904= 1198811119879119904 997904rArr1199101015840 = Σminus11 119880119879119910 = 1198811119879119904

(29)




4 Simulation





Sign

al am

plitu

de

0

02

04

06

08

1

0 10 20 30 40 50 60


Signal length








Signal1119877119878 100 973 502 173119879s 017 038 028 037119877119864 0007 0012 1845 3330

Signal2119877119878 100 982 492 153119879s 025 044 036 082119877119864 0014 0018 1038 2143

Signal3119877119878 100 976 557 108119879s 056 092 066 127119877119864 0013 0017 1836 3045

Sparsity K

Reco

nstr

ucio

nsu

cces

s rat

e (

)

0 10 20 30 40 50 600

20

40

60

80

100


Literature[2]StOMP







40 60 80 100 12040

50

60

70

80

90

100

Reco

nstr

uctio

nsu

cces

s rat

e (

)


Literature[2]StOMP

Observation times M


SNR (dB)

0102030405060708090

100

10 20 30 40 50

Reco

nstr

uctio

nsu

cces

s rat

e (

)





5 Conclusions


Data Availability




Acknowledgments


References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Signal1119877119878 100 973 502 173119879s 017 038 028 037119877119864 0007 0012 1845 3330

Signal2119877119878 100 982 492 153119879s 025 044 036 082119877119864 0014 0018 1038 2143

Signal3119877119878 100 976 557 108119879s 056 092 066 127119877119864 0013 0017 1836 3045

Sparsity K

Reco

nstr

ucio

nsu

cces

s rat

e (

)

0 10 20 30 40 50 600

20

40

60

80

100


Literature[2]StOMP







40 60 80 100 12040

50

60

70

80

90

100

Reco

nstr

uctio

nsu

cces

s rat

e (

)


Literature[2]StOMP

Observation times M


SNR (dB)

0102030405060708090

100

10 20 30 40 50

Reco

nstr

uctio

nsu

cces

s rat

e (

)





5 Conclusions


Data Availability




Acknowledgments


References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



5 Conclusions


Data Availability




Acknowledgments


References

















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


WSNs Compressed Sensing Signal Reconstruction Based on Improved Kernel...

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Transcript of WSNs Compressed Sensing Signal Reconstruction Based on Improved Kernel...