Research Article Resampling to Speed Up Consolidation of Point...

Research ArticleResampling to Speed Up Consolidation of Point Clouds

Huanyu Yang12 Kuangrong Hao13 and Yongsheng Ding13

1College of Information Sciences and Technology Donghua University Shanghai 201620 China2College of Information and Engineering Shanghai Open University Shanghai 200233 China3Engineering Research Center of Digitized Textile amp Fashion Technology Ministry of Education Donghua UniversityShanghai 201620 China

Correspondence should be addressed to Kuangrong Hao krhaodhueducn

Received 23 January 2015 Accepted 10 March 2015

Academic Editor Carla Roque

Copyright copy 2015 Huanyu Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Processing of large-scale scattered point clouds has currently become a hot topic in the field of computer graphics research Asupposedly valid tool in producing a set of denoised outlier-free and evenly distributed particles over the original point cloudsWeighted Locally Optimal Projection (WLOP) algorithm has been used in the consolidation of unorganized 3D point clouds bymany researchers However the algorithm is considered relatively ineffective due to the large amount of the point clouds data andthe iteration calculation In this paper a resampling method applied to the point set of 3D model which significantly improvesthe computing speed of the WLOP algorithm In order to measure the impact of error which will increase with the improvementof calculation efficiency on the accuracy of the algorithm we define two quantitative indicators that is the projection error anduniformity of distribution The performance of our method will be evaluated by using both quantitative and qualitative analysesOur experimental validation demonstrates that this method greatly improves calculating efficiency notwithstanding the slightlyreduced projection accuracy in comparison to WLOP

1 Introduction

As a popular topic of growing interest in the fundamentalresearch of computer graphics reverse engineering in recon-structing 3D models from unorganized point cloud data hasbeen considered by many authors and various results on sur-face reconstruction from point clouds have been publishedin recent years Various methods have been proposed in thisregard including RBF-based approaches [1ndash4] integrationof Voronoi diagrams and variational method [5] Poissonsurface reconstruction technique [6] and smooth signeddistance method [7] The local moving least squares (MLS)surface by Levin [8] and its variants have been provento be a powerful surface representation of point set dataGuennebaud and Gross proposed algebraic point set surfaces(APSS) [9] and Oztireli et al proposed a robust implicit MLS(RIMLS) variant to project a point set (or a mesh) onto theMLS surface [10]

During the past two decades preprocessing of unorga-nized point cloud data has received considerable attention

mainly in virtue of their theoretical importance as well asthe extensive applications Many problems still have yet tobe addressed however In particular it is noteworthy thatthe 3D surface represented by unorganized point clouds istypically noisy as it contains holes with high variations inpoint density caused by acquisition errors ormisalignment ofmultiple scans The preprocessing for surface reconstructionincludes denoising outlier removal thinning orientationand redistribution of the input points (see eg [11ndash15])which has resorted to resampling for point cloud consolida-tionWLOP (Weighted Locally Optimal Projection) operator[15] and LOP (Locally Optimal Projection) operator [14] haveproven better immunity against noise and outlier of rawscanneddata in addition to their advantage in creating evenlyredistributed point cloudsHowever thesemethods are foundlimited in preserving geometrical features of the model in theprojection (see [16 17]) Such limitations however need to befurther addressed with concrete quantitative evidence whichis obtainable via computationally expensive reconstruction oflarge point set data The open problem we intend to solve in

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 646410 12 pageshttpdxdoiorg1011552015646410

2 Mathematical Problems in Engineering

this paper is the slow computing speed caused by multipleiterations in previously proposed models and limitation inresults obtained from reconstructing large point set data dueto the mathematical complexity

In this paper we present the resamplingmethod for pointclouds processing a new and more efficient organization ofdata We downsample the point set with 119896-nearest neighboralgorithm and the normal vectorial angle which reduces theamount of computation that is required and upsample thecomputing result using WLOP A numerical simulation onaccuracy uniformly distributed setting and computing timeis given to illustrate the applicability Our method can reducethe noise and create evenly redistributed points like WLOPFor more experiments we demonstrate that time complexityis significantly reduced using our method

Fourmain contributions of this paper can be summarizedas follows

(1) We review the application and progress of WLOPalgorithm for geometry reconstruction

(2) We design an optimization process to reduce thecomputational complexity of WLOP In the proposedalgorithm the surface points are classified by the 119896-nearest neighbor algorithm and the normal vector isused to choose the point from every group to eachsubsampled point set The WLOP operator is appliedto the point clouds of every subset This approachspeeds up the computation two to three times

(3) We contribute to the scarce literature on quantitativeanalysis for point cloud processing offering a defi-nition of the projection error indicators reference tothemethod of error estimation analysis of point cloudsimplification algorithm and the uniformdistributionindicators reference to the method of point density

(4) Our proposed ResamplingWLOP approach has beenevaluated on models with various shape complexitiesand noise levels The effectiveness and performanceof our method are validated and illustrated throughexperimental results and comparison with theWLOPmethod

The rest of this paper is organized as follows Section 2reviews the related studies in the consolidation of unorga-nized 3D point clouds from which the proposed methodis derived Section 3 provides a detailed description of theresampling method The evaluation criteria are describedin Section 4 The proposed method is implemented andevaluated against theWLOP algorithm in Section 5 followedby the concluding comments in Section 6

2 Related Work

In this section we briefly review the application and researchof Weighted Locally Optimal Projection (WLOP) algo-rithm among the various methods hitherto developed forthe processing of point clouds Lipman et al proposed aparameterization-free projection operator that is LocallyOptimal Projection (LOP) for geometry reconstruction rem-iniscent of the well-known multivariate 119871

1median problem

[14] This method is robust to noise outliers and nonunifor-mities of raw scanned data LOP operates well on raw datawithout relying on a local parameterization of the points oron their local orientation Given an unorganized set of points119875 = 119901

119895119895isin119869

sub 1198773 LOP defines a set of projected points 119883 =

119909119894119894isin119868

sub 1198773 by a fixed point iterationwhere given the current

iterate119883119896 119896 = 0 1 119898 the next iterate119883

119896+1 is tominimize

119909119896+1

119894= sum

119895isin119869

119901119895

120572119896

119894119895

sum119895isin119869

120572119896119894119895

+ 120583 sum

1198941015840isin119868119894

120575119896

1198941198941015840

120573119896

1198941198941015840

sum119895isin119869

1205731198961198941198941015840

(1)

where 120572119896

119894119895= 120579(120585

119896

119894119895)120585119896

119894119895 and 120573

119896

1198941198941015840 = 120579(120575

119896

1198941198941015840)|1205781015840(120575119896

1198941198941015840)|120575

119896

1198941198941015840

with 120585119896

119894119895= 119909119896

119894minus 119901119895and 120575

119896

1198941198941015840 = 119909119896

119894minus 119909119896

1198941015840

However LOP may not work well when the distributionof the input points is highly nonuniform Huang et al [15]improved this operator by incorporating locally adaptivedensity weights into LOP to improve its ability to deal withnonuniformity The improved operator named as WLOP(Weighted Locally Optimal Projection) is supported by arobust normal estimation method which they presented toassign consistent normal to each projected point

The projection for point 119909119896+1

119894is

119909119896+1

119894= sum

119895isin119869

119901119895

120572119896

119894119895V119895

sum119895isin119869

120572119896119894119895V119895

+ 120583 sum

1198941015840isin119868119894

120575119896

1198941198941015840

120596119896

1198941015840120573119896

1198941198941015840

sum119895isin119869

12059611989611989410158401205731198961198941198941015840

(2)

where 120578(119903) = minus119903 120579(119903) = 119890minus1199032(ℎ4)

2

and where V119895

= 1 +

sum1198951015840isin119869119895

120579(119901119895minus1199011198951015840) and 120596

119896

119894= 1+sum

1198941015840isin119868119894

120579(120575119896

1198941198941015840) are adapt-

ive density weightsMany surface reconstruction algorithms can greatly ben-

efit from the output of their consolidation framework Huanget al presented a method to assign consistently orientednormal vectors to unorganized points with noises nonuni-formities and thin sharp features compared with the con-solidation approach [15] They presented a framework thatdistributes points by inserting samples into sparse regionsusing the WLOP operator which performs downsamplingand relaxation [18] Huang et al altered the LOP operator andmade it normal or edge-aware allowing for resampling awayfrom edges [19]

Although LOP and WLOP are both considered effectiveapproaches to reconstruct geometry several drawbacks arenot to be dismissed for instance limitation in guaranteeingthe smoothness of projection and in the reconstruction ofmodels which contain sharp features Li et al extended theWLOP operator by incorporating the normal informationinto the WLOP operator [16] In this modified approachneighbors across sharp features are regarded as outliers innormal space thus they are excluded from the projection tokeep the features intact Li et al later suggested incorporatinga normalmollification step into the extendedWLOPoperatorto get a more accurate result Liao et al presented an efficientand feature-preserving Locally Optimal Projection Operator(FLOP) for geometry reconstruction [17] which is based on abilateral-weighted LOP attending to information concerningboth spatial and geometric features FLOP has been greatly

Mathematical Problems in Engineering 3

Input

data setDownsampling Output

data setUpsamplingIterate

WLOP

Iterate

WLOPP1

P2

Xt1

Xt2

Figure 1 Resampling WLOP schema

accelerated by using the random sampling of the Kernel Den-sity Estimate (KDE) and extended for efficient and faithfulreconstruction of time-varying surface reconstruction FLOPmethod preserves the features better than LOP and WLOPoperators However it fails to accurately compute normalvectors for sparsely sampled models with sharp features andheavy outliers

Moving least squares (MLS) is a very attractive tool todesign effective meshless surface representations Oztireliet al reviewed robust implicit MLS surfaces in terms oflocal kernel regression RIMLS representation can handlesparse sampling generate a continuous surface with betterpreserved fine details and can naturally deal with any kindof sharp features with controllable sharpness [10] In a similarvein we validate ourmethod for surface reconstruction usingRIMLS representation

3 Resampling WLOP

31 ResamplingWLOP Schema TheWLOP algorithm suffersfrom high computational costs due to the large amount ofthe point clouds data and the iteration calculation Recentlymore attention is focused on how to speed up the computingtime of this algorithm Liao et al had greatly accelerated thecomputing time of FLOP [17]The subsampled point set byusing the random sampling of the Kernel Density Estimate(KDE) is a feature-aware approximation to the original inputpoint set 119875 The idea is similar to that in this paper Howeverthe number of is smaller than the number of 119875 The perfor-mance of their method which is actually data simplificationfor point cloud depends upon the simplified calculation ratioThe number of input point set 119875 is divided into two subsetsbut not to be reduced by using our method And so thecomparisons between our method and the approach in [17]cannot be made directly Existing solutions for speeding upthe process of point clouds using hierarchical-based methodare demonstrated in [20] Diez et al proposed a noveltechnique named as Hierarchical Normal Space Sampling(HNSS) to speed up coarce matching algorithms for pointclouds Two sets of points were grouped to points hierarchicalaccording to the distribution of their normal vectorsThedatahierarchy is traversed using RANSAC algorithm applied topoint matching for every corresponding hierarchy level of

two point sets Our resampling method involves only one setof points And so the comparisons between our method andthe approach in [20] cannot be made directly

In this section we present the ResamplingWeight LocallyOptimal Projection for a new and more efficient methodfor data organization The computational complexity of themultivariate 119871

1median problem which remains a bottleneck

of WLOP is superlinear in the number of input points 119875The purpose of organizing data is to reduce the numberof input points 119875 in each calculation Data after beingdownsampled is traversed using WLOP algorithm for everysubset The Resampling WLOP schema is shown in Figure 1and described in detail below

Step 1 We focus on the decomposition of set 119875 resulting intwo subsets119875

11198752The number of points on each subsampled

point set is the same Sections 32 and 33 will demonstratehow to choose points on each subset

Step 2 We use the WLOP algorithm for each of subsets 1198751

1198752sequentially resulting in 119883

1 1198832

Step 3 We create multiple output results by iterating overeach subset After 119905 iterations we can get 119883

119905

1 1198831199052

Step 4 Theultimate output point set is the sum of the119883119905

11198831199052

This may be regarded as an upsampling procedure

32 Simple Downsampling Let11987511198752be two subsets of119875The

main objective of this study is to determine a downsamplingmethod that brings a similar subset of 119875 The similar surfacecharacteristics could have been carried on the points of eachsubsampled point set to reduce the projection error caused bythe resampling algorithm

We classify the surface points according to their 119896-nearestneighbor We randomly choose 119901 from 119875 And the nearestneighbor points of 119901 are 119901

1015840 which are determined by KD-Tree algorithm Let 119901 be grouped into the first subset 119875

1 1199011015840 is

grouped into the second subset 1198752

33 Precise Downsampling To make the result more accu-rate we choose the points to each subset from every grouptaking into account the surface normal vectors The datadownsampling method is described as follows


Step 1 We randomly choose 119901 from 119875 And the 119896-nearestneighbor points of 119901 are 119875

1015840= 1199011015840

1 1199011015840

2 119901

1015840

119896 which are

determined by KD-Tree algorithm Let 119901 be grouped into thefirst subset 119875

1

Step 2 Let 119899 be the normal of point 119901 and 1198991015840

119894is the normal of

point 1199011015840

119894

cos (120601119894) =

119899 sdot 1198991015840

119894

|119899| times10038161003816100381610038161198991015840

119894

1003816100381610038161003816

(3)

where 120601119894is the normal vectorial angle between 119901 and 119901

1015840

119894 And

cos(120601119894) is the normal vectorial angle cosine between 119901 and 119901

1015840

119894

If cos(120601119895) is a minimum value of cos(120601

119894) (119894 = 1 2 119896)

that is 120601119895is a minimum value of cos(120601

119894) (119894 = 1 2 119896) we

regard that 1199011015840

119895has the similar characteristics as 119901 Then 119901

1015840

119895is


Step 3 119901 and 1199011015840

119895are removed from 119875 We will repeat the first

step until 119875 become the null setThe point set is downsampled by the nearest neighbor

algorithm without taking into account the surface normalvectors This is a Simple Resampling WLOP method inSection 32 We randomly choose 119901 from 119875 Our goal byusing our downsampling method is to find a more accurateand feature-aware approximation of 119901

1015840 to 119901 Neighborhoodis a factor to be considered the normal vectorial angle isalso a factor to be considered In this paper the points setis divided into two subsets So we choose 119896 = 2 and the119896-nearest neighbor points of 119901 are 119875

1015840= 119901

1015840

1 1199011015840

2 1199011015840 has

been chosen by comparing with 1206011and 120601

2according to Step

2 If the points set is divided into three or more subsets byusing the downsampling algorithm we will choose 119896 gt 2This is a Simple Resampling WLOP method In this paperwe only focus on two subsets cases in the division of subsetsprimarily because experimental result has shown that toomany subsampled point sets can lead to increasing projectionerror Comparison of two resampling methods is shown inSection 5

4 Projection Criteria

In this section we analyze the impact on the accuracy of theprojection algorithm We define two quantitative indicatorserror estimation and uniformity of distribution

41 Error Estimation Pauly et al designed a method forestimating numerical and visual error between a simplifiedpoint set and its original point set by resampling points onan MLS local surface [21] The distance function of a smoothsurface 119878 which assigns to a point 119901 the shortest distancefrom 119901 to 119878 has been investigated by Pottmann and Hofer[22] The following point-to-surface distance approximationis a variant of the previously adopted Taylor approximant ofthe distance function [22 23] which approximates the localshape of 119878 at the normal footpoint (the point on 119878 that hasthe shortest distance to 119901) by a tangent plane rather than aquadric surface

P

xipj

Figure 2 The minimum distance between 119909119894and 119875 is the distance

between 119909119894and the nearest point of 119875 lying on one of its sides

LOP and WLOP operators are actually the simplifiedmethods for point clouds data To make the distribution ofpoints more uniform the number of projected point cloudsis smaller than that of original point clouds as presented in[14] Therefore we refer to error estimation analysis of pointcloud simplification algorithmHowever how to compute thedistances efficiently and accurately in the context of pointcloud simplification still remains a challenge

The quality of the simplification can be measured by asuitable error metric such as the one-sided Hausdorff dis-tance [24 25] the two-sided Hausdorff distance [21 26] thequadratic error metric [27 28] and the minimum distanceThe evaluation metrics in the one-sided Hausdorff distancefamily such as theHausdorff fraction theHausdorff quantileand spatially coherent matching are also commonly appliedto measurement of the difference between two surfaces in alarge number of applications [29] Differences in definitionsof the point-based surfaces and of error metrics will result indifferent simplification error values

Given twofinite point sets119860 = 1198861 1198862 119886

119899(representing

a model image) and 119861 = 1198871 1198872 119887

119899(representing a test

image) Hausdorff distance is defined as [25]

119867 (119860 119861) = max (ℎ (119860 119861) ℎ (119861 119860)) (4)

where ℎ(119860 119861) = max119886isin119860

min119887isin119861

119886 minus 119887The function 119867(119860 119861) is called two-sided Hausdorff

distance and the function ℎ(119860 119861) is called one-sided Haus-dorff distance Hausdorff distance measures the degree ofsimilarity between one point set and another Dissimilar tomost of alternative methods of shape comparison Hausdorffdistance is not based on finding the corresponding modeThus it is more tolerant towards the perturbations of pointlocations since it deals with proximity more than withexact superposition However Hausdorff distance is highlysensitive to outliers

We suppose an original point set as 119875 and the point setafter projection as119883 Let 119901

119895isin 119875 (119895 = 1 2 119899) be a nearest

neighbor point of 119909119894

isin 119883 (119894 = 1 2 119898) in 119875 Then theminimum distance of 119909

119894isin 119883 (119894 = 1 2 119898) to the original

point sets 119875 is

120576119894=

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (5)

It is shown as a solid line in Figure 2


We define the one-sided Hausdorff distance as follows

119867 (119883 119875) = max119894isin119868

(120576119894) = max119894isin119868

(10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817) (6)

To reduce the sensitivity of outliers and better evaluatethe similarity between the two point sets we propose otherevaluation criteria using minimum distance error metric Wedefine Sum Error Value (SEV) as follows

SEV = sum

119894isin119868

120576119894= sum

119894isin119868

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (7)

Sum Error Value is the sum of minimum distancebetween 119909

119894isin 119883 (119894 = 1 2 119898) and 119901

119895isin 119875 (119895 = 1 2 119899)

which can reflect and give the projection error for all pointson the point set Regardless of the evaluation criteria whenthe value is less the projection error is smaller

42 Uniform Distribution Uniformity of point clouds can berepresented by the densityThere are twomajor approaches tothe expression of density (i) point clouds density extractionmethod based on block In this approach all points incloud are distributed on the surface Therefore the numberof points per unit area can be regarded as the density ofthree-dimensional point cloud And there is (ii) point cloudsdensity extraction method based on distance An alternativeof the expression of point clouds density is the distancesamong cloud points The distance between the point cloudsis smaller more concentrated distribution of the points andtherefore the density of the point cloud is relatively largeFrom a statistical point of view variance is a powerful toolfor uniform distribution In this paper data distributionuniformity is measured by using the distance variance ofthe point clouds The uniform distribution of point clouds isbetter with the distance variance value being smaller

Let 119870 = (1 2 119896) and 119868 = (1 2 119898) Let 1198831015840

119894=

1199091015840

1 1199091015840

2 119909

1015840

119896 be a 119896-nearest neighbor of 119909

119894 119894 isin 119868 in119883Then

the Euclidean distance between 119909119894and a point of its 119896-nearest

neighbor in 119883 is

119889119894(119895) =

10038171003817100381710038171003817119909119894minus 1199091015840

119895

10038171003817100381710038171003817 119895 isin 119870 (8)

The average distance between 119909119894and its 119896-nearest neigh-

bor is

119889119894=

1

119896sum

119895isin119870

119889119894(119895) 119895 isin 119870 (9)

The distance variance between 119909119894and its 119896-nearest neigh-

bor is

120590119894= sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

119895 isin 119870 (10)

The average distance between all points in 119883 and its 119896-nearest neighbor is

119889 =1

119898sum

119894isin119868

119889119894 (11)

So we define the Local Uniform Distribution (LUD) asfollows

LUD =1

119898sum

119894isin119868

120590119894=

1

119898sum

119894isin119868

sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

(12)

We define the Global Uniform Distribution (GUD) asfollows

GUD =1

119898sum

119894isin119868

(119889119894minus 119889)2

(13)

LUD reflects Local Uniform Distribution of the pointin its neighborhood range GUD reflects Global UniformDistribution of the surface

5 Experiments and Result

We compare the numerical results of our method usingdifferent established models with various shape complexitiesand noise levels the faceYo model the foot model the egeamodel the rocker-arm model and the Stanford Armadillomodel We reconstruct the model using the RIMLS methodimplemented on MeshLab developed by the Visual Comput-ingGroup [30]This section presents empirical evidence fromtests on a desktop computer with a 293GHz Intel Core i3-320CPU and 2GByte of RAM We implemented our prototypeprogram in Visual C++ 2010

After WLOP projection the position update consistsof two terms The first term sum

119895isin119869119901119895((120572119896

119894119895V119895)(sum119895isin119869

120572119896

119894119895V119895))

attracts the particle to the given point set The second termsum1198941015840isin119868119894

120575119896

1198941198941015840(120596119896

1198941015840120573119896

1198941198941015840 sum119895isin119869

120596119896

1198941015840120573119896

1198941198941015840) is the repulsion term which

repulses the particles away from other particles to make thepoint set more uniform 120583 isin [0 05) and ℎ are two user-selected parameters where 120583 controls the repulsion force todistribute points uniformly in119883 and ℎ is the support radius todetermine the size of the influence neighborhood Howeverthese two parameters aremanually specified by trial and errorin the existing literature [14ndash16] Only Liao et al employeda bilateral weight average algorithm to decide ℎ which wasadaptively determined with respect to local features [17]

Using the above two quantitative indicators we analyzethemain parameters ofWLOP algorithmquantitativelyWiththe increase of 120583 the accuracy of the projection decreasesgradually and the point distribution on the surface is moreuniform In our experiments we use the repulsion parameter120583 as 120583 = 045 An exceedingly large value of ℎ might causeoversmoothed results contrarily if the value of ℎ is too smallwe cannot achieve the desired results either in [17] Howeverwe found that the change of two quantitative indicators issmall with different values of ℎ when 120583 = 045 We setℎ = 4radic119889bb119898 the same as that in [15] where 119889bb is thediagonal length of the bounding box of the input model and119898 is the number of points in 119883 Not only can we compare theprojection result after each iteration calculation with that ofthe previous calculation throughout quantitative analysis ofevaluation but also we can expect to choose the appropriateiteration times Details of the data analysis are omitted here


Table 1 Computational time

Data set Number oforiginal points

Number ofprojected particles

Time forWLOP(s)

Resampling WLOPResampling time(s) WLOP time(s) Total time(s) Acceleration ratio

FaceYo 13746 6872 5684 057 2885 2942 518Egea 8268 4134 2803 031 1417 1448 517Foot 10010 5005 3857 041 1267 1308 358Rocker-arm 10044 5022 3884 043 1829 1872 482Armadillo 172974 43243 35397 588 8581 86398 244Cane 501500 125374 259091 2025 9826 984625 38

Table 2 Comparison of error estimation among models SEV (Sum Error Value) HD (Hausdorff distance)

Data setSEV (Sum Error Value) HD (Hausdorff distance)

WLOP Simple ResamplingWLOP

Precise ResamplingWLOP WLOP Simple Resampling

WLOPPrecise Resampling

WLOPFaceYo 44649 51348 49861 18049 2136 19284Egea 142524 358709 262916 00927 10727 07968Foot 298816 397296 390547 12406 13621 12964Rocker-arm 14908 20686 204996 10793 15823 14308Armadillo 81101 92312 85243 067 076 071Cane 4256 483 456 052 061 058

WLOP Resampling WLOP0

10

20

30

40

50

60

Tim

e (s)

Running time for faceYo model

Figure 3 Comparison of the computational time for faceYo model

51 Computational Time Whatever downsampling methodis chosen computational time of Resampling WLOP is thesame By contrast the time consumed in resampling is farless than in WLOP The number of points in each testdata set and the resulting computational time consumed inusing WLOP and Resampling WLOP are listed in Table 1We set acceleration ratio as TimeRWLOPTimeWLOP whereTimeRWLOP is the computational time for ResamplingWLOPand TimeWLOP is the computational time for WLOP Theresults in Table 1 illustrate that adopting Resampling WLOPcan reduce the computational time down to 24 to 52of the original time using WLOP Figure 3 illustrates resultsof comparison on point cloud model of the faceYo with13746 points Figure 4 illustrates results of comparison onpoint cloud model of the Armadillo with 172974 points It


500

1000

1500

2000

2500

3000

3500

Tim

e (s)

Running time for Armadillo model

Figure 4 Comparison of the computational time for Armadillomodel

shows that by using our method the computational time issignificantly reduced And with the increase in the number ofpoints the efficiency of speeding up the point clouds processis improved by our method Hence the proposed method ofResamplingWLOP is a very efficient method in cases when acomparatively large amount of points needs to be projected

52 Projection Error The results of comparison betweenWLOP and Resampling WLOP in terms of the two quanti-tative indicators are shown in Tables 2 and 3 The projectionerror estimation is shown in Table 2 Table 3 illustrates theresults of point uniform distribution

Our experimental validation in Section 51 demonstratesthat our method greatly improves calculating efficiency But


Table 3 Comparison of point distribution among models LUD (Local Uniform Distribution) GUD (Global Uniform Distribution)

Data setLUD (Local Uniform Distribution) GUD (Global Uniform Distribution)




WLOPFaceYo 02326 02942 02829 26604 28183 26744Egea 0006 00554 00088 00199 00246 00208Foot 00074 00077 00076 756 lowast 10

minus4921 lowast 10

minus4824 lowast 10

minus4

Rocker-arm 00047 00056 00054 00135 0016 00143Armadillo 533 559 542 003 004 004Cane 21 23 215 024 027 025

1 2 3 4 5 63

35

4

45

5

55

6

Iteration times

Sum Error Value

WLOPSimple Resampling WLOPPrecise Resampling WLOP

SEV

(lowast1000

)

(a) Sum Error Value (SEV)

1 2 3 4 5 61

12

14

16

18

2

22

24

26

28

3

Iteration times

Hau

sdor

ff di

stan

ce

Hausdorff distance value


(b) Hausdorff distance

Figure 5 Error estimation for faceYo model

through comparison result in Tables 2 and 3 the accuracy ofthe projection is reduced using resamplingmethod accordingto two quantitative indicators The noise may be generatedusing our method These errors include Sum Error ValueHausdorff distance Local Uniform Distribution and GlobalUniform Distribution It is shown that adopting the preciseresampling method the projection error estimation is lessand the point uniform distribution is better than the SimpleResampling method

Taking the faceYo model as an example the Hausdorffdistance by using Precise Resampling WLOP is increasedto 68 (from 18049 to 19284) compared to that by usingWLOP and the Sum Error Value by using Precise Resam-pling WLOP is increased to 116 (from 44649 to 49861)compared to that by using WLOP in Table 2 Table 3 illus-trates that the Local Uniform Distribution by using PreciseResampling WLOP is increased to 216 (from 02326 to02829) compared to that by using WLOP and the GlobalUniform Distribution by using Precise Resampling WLOP isonly increased to 037 (from 26604 to 26744) compared tothat by using WLOP However Table 1 and Figure 3 illustrate

that the computational time by using Resampling WLOP isdecreased to 518 (from 5684ms to 2942) compared to thatby using WLOP

The reconstruction of the faceYo model by one iterationof WLOP is presented in Figure 9(a) and that of PreciseResampling WLOP is presented in Figure 9(b) The recon-struction of the Armadillo model by one iteration of WLOPis presented in Figure 10(a) and that of Precise Resam-pling WLOP is presented in Figure 10(b) The projectionerror of Resampling WLOP stays in a perfect or unalteredcondition compared with WLOP method And it is shownthat the reconstruction quality remains satisfactory withwell-preserved features and fairly distributed points usingResampling WLOP from the contrastive analysis

53 Effect of Iterations In order to get an improved recon-struction we conduct six iterations for the faceYo modeland the Armadillo model by WLOP Simple ResamplingWLOP and Precise Resampling WLOP As shown in [15]with increasing iteration times the consolidation of pointclouds of WLOP algorithm is better Our goal is to compare


1 2 3 4 5 62

21

22

23

24

25

26

27

28

29

Iteration times

LUD

Local Uniform Distribution value


(a) LUD

1 2 3 4 5 60

005

01

015

02

025

Iteration times

GU

D

Global Uniform Distribution value


(b) GUD

Figure 6 Uniform distribution for faceYo model

1 2 3 4 5 67

75

8

85

9

95

10

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 605

06

07

08

09

1

11

Iteration times

Hau

sdor

ff di

stanc

e




Figure 7 Error estimation for Armadillo model

two quantitative indicators between Resampling WLOP andWLOPThe test result illustrated that the two indicators withiteration times increased are similar in either algorithm Sowe use 6 iteration times

The result of the comparison among three algorithms interms of projection error and the uniform distribution valueis shown in Figures 5ndash8 With increased iteration times thesum value error becomes smaller regardless of the projectionmethod being used However the Hausdorff distance valueincreases slightly because of the noise generated by algorithmerror Variation trends of point distribution by Resampling

WLOP are similar to those by WLOP It is also shown thatadopting the precise resamplingmethod the projection errorestimation is less and the point uniform distribution is betterthan the Simple Resampling method

The reconstruction of faceYo models created by sixiterations of WLOP and Precise ResamplingWLOP is shownin Figures 9(c) and 9(d) The reconstruction of Armadillomodels created by six iterations ofWLOP and Precise Resam-pling WLOP is shown in Figures 10(c) and 10(d) Becauseof the noise generated by algorithm error with increasediteration times the reconstruction result of the faceYo model


1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD

Figure 8 Uniform distribution for Armadillo model

(a) (b)

(c) (d)

Figure 9 FaceYo model (a) (c) Model created by one iteration and six iterations of WLOP (b) (d) model created by one iteration and sixiterations of Precise Resampling WLOP


(a) (b)

(c) (d)

Figure 10 Armadillo model (a) (c) Model created by one iteration and six iterations of WLOP (b) (d) model created by one iteration andsix iterations of Precise Resampling WLOP

by Precise Resamping WLOP has not been satisfactoryHowever it is shown that the reconstruction of Armadillomodel remains satisfactory with well-preserved features andfairly distributed points using Resampling WLOP It is seenthat the effect of projection error and uniform distribution byresampling algorithm is gradually decreased as the number ofthe points of the model increases

The WLOP operator achieves better point distributionand noise-resistance than Resampling WLOP However thecomputational cost of Resampling WLOP is significantlyreduced as shown in Section 51 If the iteration method isused the effect of our acceleration algorithm becomes moreremarkable

6 Conclusion and Future Work

In this paper we introduce an improved algorithm for consol-idation of point clouds It is based on classical WLOP con-sidering resampling algorithms We also present empirical

evidence for its effectiveness and advantages in comparisonto the classical WLOP algorithm

Besides our implementation is evaluated on models withvarious shape complexities andnoise levelsThe experimentalresults show that the computation time consumed by usingResampling WLOP is about one-half of that by using WLOPA combination of quantitative and qualitative analyses isperformed to further explore the relationship between thepoint projection and accuracy of the surface reconstructionThe resampling processing method is proven to enjoy adistinct advantage when the number of point clouds is verylarge

However one major potential drawback of the modelis the observed increase in error and noise as a result ofthe reduction in computation cost In this regard severalavenues for further research can be proposed for instance(i) further improvement in the quality of the projection (ii)adaptation of the Resampling WLOP model to various kindsof point-based models with different surface properties and(iii) improvement in the robustness of the algorithm


Conflict of Interests

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

Acknowledgments

This work was supported in part by the Key Project ofthe National Natural Science Foundation of China (no61134009) the National Natural Science Foundation of China(nos 61473077 61473078) Cooperative Research Funds ofthe National Natural Science Funds Overseas and HongKong and Macao Scholars (no 61428302) Program forChangjiang Scholars and Innovation Research Team in Uni-versity from the Ministry of Education (no IRT1220) Spe-cialized Research Fund for Shanghai Leading Talents Projectof the Shanghai Committee of Science and Technology (nos13JC1407500 11JC1400200) Innovation Program of ShanghaiMunicipal Education Commission (no 14ZZ067) and theFundamental Research Funds for the Central Universities(no 2232012A3-04)

References

[1] J C Carr R K Beatson J B Cherrie et al ldquoReconstructionand representation of 3D objects with radial basis functionsrdquoin Proceedings of the Computer Graphics Annual Conference(SIGGRAPH rsquo01) pp 67ndash76 August 2001

[2] Y Ohtake A Belyaev M Alexa G Turk and H P SeidelldquoMulti-level partition of unity implicitsrdquo ACM Transactions onGraphics vol 22 no 3 pp 463ndash470 2003

[3] I Tobor P Reuter and C Schlick ldquoReconstructing multi-scalevariational partition of unity implicit surfaces with attributesrdquoGraphical Models vol 68 no 1 pp 25ndash41 2006

[4] S Liu and C C Wang ldquoQuasi-interpolation for surface recon-struction from scattered data with radial basis functionrdquo Com-puter Aided Geometric Design vol 29 no 7 pp 435ndash447 2012

[5] P Alliez D Cohen-Steiner Y Tong andMDesbrun ldquoVoronoi-based variational reconstruction of unoriented point setsrdquo inProceedings of the 5th Eurographics Symposium on GeometryProcessing (SGP rsquo07) pp 39ndash48 2007

[6] MKazhdanM Bolitho andHHoppe ldquoPoisson surface recon-structionrdquo in Proceedings of the 4th Eurographics Symposium onGeometry Processing (SGP rsquo06) pp 61ndash70 2006

[7] F Calakli and G Taubin ldquoSSD smooth signed distance surfacereconstructionrdquo Computer Graphics Forum vol 30 no 7 pp1993ndash2002 2011

[8] D Levin ldquoMesh-independent surface interpolationrdquo in Geo-metric Modeling for Scientific Visualization vol 3 pp 37ndash49Springer Berlin Germany 2003

[9] G Guennebaud and M Gross ldquoAlgebraic point set surfacesrdquoACM Transactions on Graphics vol 26 no 3 Article ID1276406 2007

[10] A C Oztireli G Guennebaud andM Gross ldquoFeature preserv-ing point set surfaces based on non-linear kernel regressionrdquoComputer Graphics Forum vol 28 no 2 pp 493ndash501 2009

[11] A C Oztireli M Alexa and M Gross ldquoSpectral sampling ofmanifoldsrdquoACMTransactions on Graphics vol 29 no 6 article168 2010

[12] M Alexa J Behr D Cohen-Or S Fleishman D Levin and CT Silva ldquoComputing and rendering point set surfacesrdquo IEEE

Transactions onVisualization andComputer Graphics vol 9 no1 pp 3ndash15 2003

[13] Y Miao P Diaz-Gutierrez R Pajarola M Gopi and J FengldquoShape isophotic error metric controllable re-sampling forpoint-sampled surfacesrdquo in Proceedings of the IEEE Interna-tional Conference on ShapeModeling andApplications (SMI rsquo09)pp 28ndash35 IEEE Beijing China June 2009

[14] Y Lipman D Cohen-Or D Levin and H Tal-Ezer ldquoParame-terization-free projection for geometry reconstructionrdquo ACMTransactions onGraphics vol 26 no 3 Article ID 1276405 2007

[15] H Huang D Li H Zhang U Ascher and D Cohen-OrldquoConsolidation of unorganized point clouds for surface recon-structionrdquo ACM Transactions on Graphics vol 28 no 5 pp176ndash176 2009

[16] B Li W Jiang Z Cheng G Dang and S Jin ldquoFeaturepreserving consolidation for unorganized point cloudsrdquo inProceedings of the 1st IEEE International Conference on Progressin Informatics and Computing (PIC rsquo10) pp 892ndash895 IEEEShanghai China December 2010

[17] B Liao C Xiao L Jin and H Fu ldquoEfficient feature-preserv-ing local projection operator for geometry reconstructionrdquoComputer-Aided Design vol 45 no 5 pp 861ndash874 2013

[18] S Liu K-C Chan and C C L Wang ldquoIterative consolidationof unorganized point cloudsrdquo IEEE Computer Graphics andApplications vol 32 no 3 pp 70ndash83 2012

[19] H Huang S Wu M Gong D Cohen-Or U Ascher and H RZhang ldquoEdge-aware point set resamplingrdquo ACM Transactionson Graphics vol 32 no 1 article 9 2013

[20] Y Diez J Martı and J Salvi ldquoHierarchical Normal SpaceSampling to speed up point cloud coarse matchingrdquo PatternRecognition Letters vol 33 no 16 pp 2127ndash2133 2012

[21] M Pauly M Gross and L P Kobbelt ldquoEfficient simplificationof point-sampled surfacesrdquo in Proceedings of the IEEE Visualisa-tion (VIS rsquo02) pp 163ndash170 IEEE BostonMass USANovember2002

[22] H Pottmann and M Hofer ldquoGeometry of the squared dis-tance function to curves and surfacesrdquo in Visualization andMathematics III H-C Hege and K Polthier Eds pp 221ndash242Springer New York NY USA 2003

[23] N J Mitra N Gelfand H Pottmann and L Guibas ldquoRegis-tration of point cloud data from a geometric optimization per-spectiverdquo in Proceedings of the EurographicsACM SIGGRAPHSymposium on Geometry Processing (SGP rsquo04) pp 22ndash31 July2004

[24] RKleinG Liebich andW Strasser ldquoMesh reductionwith errorcontrolrdquo in Proceedings of the 7th conference on Visualization(VIS rsquo96) pp 311ndash318 San Francisco Calif USA October 1996

[25] D P Huttenlocher G A Klanderman and W J RucklidgeldquoComparing images using the Hausdorff distancerdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence vol 15 no9 pp 850ndash863 1993

[26] P Cignoni C Rocchini and R Scopigno ldquoMetro measuringerror on simplified surfacesrdquo Computer Graphics Forum vol 17no 2 pp 167ndash174 1998

[27] M Garland and P S Heckbert ldquoSurface simplification usingquadric error metricsrdquo in Proceedings of the Conference onComputer Graphics (SIGGRAPH rsquo97) pp 209ndash216 August 1997

[28] M Garland and Y Zhou ldquoQuadric-based simplification in anydimensionrdquo ACM Transactions on Graphics vol 24 no 2 pp209ndash239 2005


[29] Z Yu H-S Wong H Peng and Q Ma ldquoASM an adaptivesimplification method for 3D point-based modelsrdquo Computer-Aided Design vol 42 no 7 pp 598ndash612 2010

[30] Group TVC MeshlabONLINE 2012 httpmeshlabsource-forgenet

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


this paper is the slow computing speed caused by multipleiterations in previously proposed models and limitation inresults obtained from reconstructing large point set data dueto the mathematical complexity

In this paper we present the resamplingmethod for pointclouds processing a new and more efficient organization ofdata We downsample the point set with 119896-nearest neighboralgorithm and the normal vectorial angle which reduces theamount of computation that is required and upsample thecomputing result using WLOP A numerical simulation onaccuracy uniformly distributed setting and computing timeis given to illustrate the applicability Our method can reducethe noise and create evenly redistributed points like WLOPFor more experiments we demonstrate that time complexityis significantly reduced using our method

Fourmain contributions of this paper can be summarizedas follows

(1) We review the application and progress of WLOPalgorithm for geometry reconstruction

(2) We design an optimization process to reduce thecomputational complexity of WLOP In the proposedalgorithm the surface points are classified by the 119896-nearest neighbor algorithm and the normal vector isused to choose the point from every group to eachsubsampled point set The WLOP operator is appliedto the point clouds of every subset This approachspeeds up the computation two to three times

(3) We contribute to the scarce literature on quantitativeanalysis for point cloud processing offering a defi-nition of the projection error indicators reference tothemethod of error estimation analysis of point cloudsimplification algorithm and the uniformdistributionindicators reference to the method of point density

(4) Our proposed ResamplingWLOP approach has beenevaluated on models with various shape complexitiesand noise levels The effectiveness and performanceof our method are validated and illustrated throughexperimental results and comparison with theWLOPmethod

The rest of this paper is organized as follows Section 2reviews the related studies in the consolidation of unorga-nized 3D point clouds from which the proposed methodis derived Section 3 provides a detailed description of theresampling method The evaluation criteria are describedin Section 4 The proposed method is implemented andevaluated against theWLOP algorithm in Section 5 followedby the concluding comments in Section 6

2 Related Work

In this section we briefly review the application and researchof Weighted Locally Optimal Projection (WLOP) algo-rithm among the various methods hitherto developed forthe processing of point clouds Lipman et al proposed aparameterization-free projection operator that is LocallyOptimal Projection (LOP) for geometry reconstruction rem-iniscent of the well-known multivariate 119871

1median problem

[14] This method is robust to noise outliers and nonunifor-mities of raw scanned data LOP operates well on raw datawithout relying on a local parameterization of the points oron their local orientation Given an unorganized set of points119875 = 119901

119895119895isin119869

sub 1198773 LOP defines a set of projected points 119883 =

119909119894119894isin119868

sub 1198773 by a fixed point iterationwhere given the current

iterate119883119896 119896 = 0 1 119898 the next iterate119883

119896+1 is tominimize

119909119896+1

119894= sum

119895isin119869

119901119895

120572119896

119894119895

sum119895isin119869

120572119896119894119895

+ 120583 sum

1198941015840isin119868119894

120575119896

1198941198941015840

120573119896

1198941198941015840

sum119895isin119869

1205731198961198941198941015840

(1)

where 120572119896

119894119895= 120579(120585

119896

119894119895)120585119896

119894119895 and 120573

119896

1198941198941015840 = 120579(120575

119896

1198941198941015840)|1205781015840(120575119896

1198941198941015840)|120575

119896

1198941198941015840

with 120585119896

119894119895= 119909119896

119894minus 119901119895and 120575

119896

1198941198941015840 = 119909119896

119894minus 119909119896

1198941015840

However LOP may not work well when the distributionof the input points is highly nonuniform Huang et al [15]improved this operator by incorporating locally adaptivedensity weights into LOP to improve its ability to deal withnonuniformity The improved operator named as WLOP(Weighted Locally Optimal Projection) is supported by arobust normal estimation method which they presented toassign consistent normal to each projected point

The projection for point 119909119896+1

119894is

119909119896+1

119894= sum

119895isin119869

119901119895

120572119896

119894119895V119895

sum119895isin119869

120572119896119894119895V119895

+ 120583 sum

1198941015840isin119868119894

120575119896

1198941198941015840

120596119896

1198941015840120573119896

1198941198941015840

sum119895isin119869

12059611989611989410158401205731198961198941198941015840

(2)

where 120578(119903) = minus119903 120579(119903) = 119890minus1199032(ℎ4)

2

and where V119895

= 1 +

sum1198951015840isin119869119895

120579(119901119895minus1199011198951015840) and 120596

119896

119894= 1+sum

1198941015840isin119868119894

120579(120575119896

1198941198941015840) are adapt-

ive density weightsMany surface reconstruction algorithms can greatly ben-

efit from the output of their consolidation framework Huanget al presented a method to assign consistently orientednormal vectors to unorganized points with noises nonuni-formities and thin sharp features compared with the con-solidation approach [15] They presented a framework thatdistributes points by inserting samples into sparse regionsusing the WLOP operator which performs downsamplingand relaxation [18] Huang et al altered the LOP operator andmade it normal or edge-aware allowing for resampling awayfrom edges [19]

Although LOP and WLOP are both considered effectiveapproaches to reconstruct geometry several drawbacks arenot to be dismissed for instance limitation in guaranteeingthe smoothness of projection and in the reconstruction ofmodels which contain sharp features Li et al extended theWLOP operator by incorporating the normal informationinto the WLOP operator [16] In this modified approachneighbors across sharp features are regarded as outliers innormal space thus they are excluded from the projection tokeep the features intact Li et al later suggested incorporatinga normalmollification step into the extendedWLOPoperatorto get a more accurate result Liao et al presented an efficientand feature-preserving Locally Optimal Projection Operator(FLOP) for geometry reconstruction [17] which is based on abilateral-weighted LOP attending to information concerningboth spatial and geometric features FLOP has been greatly


Input



WLOP

Iterate

WLOPP1

P2

Xt1

Xt2




3 Resampling WLOP











1 1198832


119905

1 1198831199052


11198831199052






1 1199011015840 is





1015840= 1199011015840

1 1199011015840

2 119901

1015840

119896 which are


1



point 1199011015840

119894

cos (120601119894) =

119899 sdot 1198991015840

119894

|119899| times10038161003816100381610038161198991015840

119894

1003816100381610038161003816

(3)


1015840

119894 And


1015840

119894


119894) (119894 = 1 2 119896)


119894) (119894 = 1 2 119896) we



1015840

119895is


Step 3 119901 and 1199011015840





1015840= 119901

1015840

1 1199011015840

2 1199011015840 has


2according to Step





P

xipj






119899(representing

a model image) and 119861 = 1198871 1198872 119887



119867 (119860 119861) = max (ℎ (119860 119861) ℎ (119861 119860)) (4)

where ℎ(119860 119861) = max119886isin119860

min119887isin119861




119895isin 119875 (119895 = 1 2 119899) be a nearest





120576119894=

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (5)




119867 (119883 119875) = max119894isin119868

(120576119894) = max119894isin119868

(10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817) (6)


SEV = sum

119894isin119868

120576119894= sum

119894isin119868

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (7)


119894isin 119883 (119894 = 1 2 119898) and 119901

119895isin 119875 (119895 = 1 2 119899)



Let 119870 = (1 2 119896) and 119868 = (1 2 119898) Let 1198831015840

119894=

1199091015840

1 1199091015840

2 119909

1015840


119894 119894 isin 119868 in119883Then



119889119894(119895) =

10038171003817100381710038171003817119909119894minus 1199091015840

119895

10038171003817100381710038171003817 119895 isin 119870 (8)


bor is

119889119894=

1

119896sum

119895isin119870

119889119894(119895) 119895 isin 119870 (9)


bor is

120590119894= sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

119895 isin 119870 (10)


119889 =1

119898sum

119894isin119868

119889119894 (11)


LUD =1

119898sum

119894isin119868

120590119894=

1

119898sum

119894isin119868

sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

(12)


GUD =1

119898sum

119894isin119868

(119889119894minus 119889)2

(13)





119895isin119869119901119895((120572119896

119894119895V119895)(sum119895isin119869

120572119896

119894119895V119895))


120575119896

1198941198941015840(120596119896

1198941015840120573119896

1198941198941015840 sum119895isin119869

120596119896

1198941015840120573119896








Time forWLOP(s)










10

20

30

40

50

60

Tim

e (s)





500

1000

1500

2000

2500

3000

3500

Tim

e (s)













minus4921 lowast 10

minus4824 lowast 10

minus4


1 2 3 4 5 63

35

4

45

5

55

6

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 61

12

14

16

18

2

22

24

26

28

3

Iteration times

Hau

sdor

ff di

stan

ce











1 2 3 4 5 62

21

22

23

24

25

26

27

28

29

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

005

01

015

02

025

Iteration times

GU

D



(b) GUD


1 2 3 4 5 67

75

8

85

9

95

10

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 605

06

07

08

09

1

11

Iteration times

Hau

sdor

ff di

stanc

e










1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD


(a) (b)

(c) (d)



(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Input



WLOP

Iterate

WLOPP1

P2

Xt1

Xt2




3 Resampling WLOP











1 1198832


119905

1 1198831199052


11198831199052






1 1199011015840 is





1015840= 1199011015840

1 1199011015840

2 119901

1015840

119896 which are


1



point 1199011015840

119894

cos (120601119894) =

119899 sdot 1198991015840

119894

|119899| times10038161003816100381610038161198991015840

119894

1003816100381610038161003816

(3)


1015840

119894 And


1015840

119894


119894) (119894 = 1 2 119896)


119894) (119894 = 1 2 119896) we



1015840

119895is


Step 3 119901 and 1199011015840





1015840= 119901

1015840

1 1199011015840

2 1199011015840 has


2according to Step





P

xipj






119899(representing

a model image) and 119861 = 1198871 1198872 119887



119867 (119860 119861) = max (ℎ (119860 119861) ℎ (119861 119860)) (4)

where ℎ(119860 119861) = max119886isin119860

min119887isin119861




119895isin 119875 (119895 = 1 2 119899) be a nearest





120576119894=

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (5)




119867 (119883 119875) = max119894isin119868

(120576119894) = max119894isin119868

(10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817) (6)


SEV = sum

119894isin119868

120576119894= sum

119894isin119868

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (7)


119894isin 119883 (119894 = 1 2 119898) and 119901

119895isin 119875 (119895 = 1 2 119899)



Let 119870 = (1 2 119896) and 119868 = (1 2 119898) Let 1198831015840

119894=

1199091015840

1 1199091015840

2 119909

1015840


119894 119894 isin 119868 in119883Then



119889119894(119895) =

10038171003817100381710038171003817119909119894minus 1199091015840

119895

10038171003817100381710038171003817 119895 isin 119870 (8)


bor is

119889119894=

1

119896sum

119895isin119870

119889119894(119895) 119895 isin 119870 (9)


bor is

120590119894= sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

119895 isin 119870 (10)


119889 =1

119898sum

119894isin119868

119889119894 (11)


LUD =1

119898sum

119894isin119868

120590119894=

1

119898sum

119894isin119868

sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

(12)


GUD =1

119898sum

119894isin119868

(119889119894minus 119889)2

(13)





119895isin119869119901119895((120572119896

119894119895V119895)(sum119895isin119869

120572119896

119894119895V119895))


120575119896

1198941198941015840(120596119896

1198941015840120573119896

1198941198941015840 sum119895isin119869

120596119896

1198941015840120573119896








Time forWLOP(s)










10

20

30

40

50

60

Tim

e (s)





500

1000

1500

2000

2500

3000

3500

Tim

e (s)













minus4921 lowast 10

minus4824 lowast 10

minus4


1 2 3 4 5 63

35

4

45

5

55

6

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 61

12

14

16

18

2

22

24

26

28

3

Iteration times

Hau

sdor

ff di

stan

ce











1 2 3 4 5 62

21

22

23

24

25

26

27

28

29

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

005

01

015

02

025

Iteration times

GU

D



(b) GUD


1 2 3 4 5 67

75

8

85

9

95

10

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 605

06

07

08

09

1

11

Iteration times

Hau

sdor

ff di

stanc

e










1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD


(a) (b)

(c) (d)



(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1015840= 1199011015840

1 1199011015840

2 119901

1015840

119896 which are


1



point 1199011015840

119894

cos (120601119894) =

119899 sdot 1198991015840

119894

|119899| times10038161003816100381610038161198991015840

119894

1003816100381610038161003816

(3)


1015840

119894 And


1015840

119894


119894) (119894 = 1 2 119896)


119894) (119894 = 1 2 119896) we



1015840

119895is


Step 3 119901 and 1199011015840





1015840= 119901

1015840

1 1199011015840

2 1199011015840 has


2according to Step





P

xipj






119899(representing

a model image) and 119861 = 1198871 1198872 119887



119867 (119860 119861) = max (ℎ (119860 119861) ℎ (119861 119860)) (4)

where ℎ(119860 119861) = max119886isin119860

min119887isin119861




119895isin 119875 (119895 = 1 2 119899) be a nearest





120576119894=

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (5)




119867 (119883 119875) = max119894isin119868

(120576119894) = max119894isin119868

(10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817) (6)


SEV = sum

119894isin119868

120576119894= sum

119894isin119868

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (7)


119894isin 119883 (119894 = 1 2 119898) and 119901

119895isin 119875 (119895 = 1 2 119899)



Let 119870 = (1 2 119896) and 119868 = (1 2 119898) Let 1198831015840

119894=

1199091015840

1 1199091015840

2 119909

1015840


119894 119894 isin 119868 in119883Then



119889119894(119895) =

10038171003817100381710038171003817119909119894minus 1199091015840

119895

10038171003817100381710038171003817 119895 isin 119870 (8)


bor is

119889119894=

1

119896sum

119895isin119870

119889119894(119895) 119895 isin 119870 (9)


bor is

120590119894= sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

119895 isin 119870 (10)


119889 =1

119898sum

119894isin119868

119889119894 (11)


LUD =1

119898sum

119894isin119868

120590119894=

1

119898sum

119894isin119868

sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

(12)


GUD =1

119898sum

119894isin119868

(119889119894minus 119889)2

(13)





119895isin119869119901119895((120572119896

119894119895V119895)(sum119895isin119869

120572119896

119894119895V119895))


120575119896

1198941198941015840(120596119896

1198941015840120573119896

1198941198941015840 sum119895isin119869

120596119896

1198941015840120573119896








Time forWLOP(s)










10

20

30

40

50

60

Tim

e (s)





500

1000

1500

2000

2500

3000

3500

Tim

e (s)













minus4921 lowast 10

minus4824 lowast 10

minus4


1 2 3 4 5 63

35

4

45

5

55

6

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 61

12

14

16

18

2

22

24

26

28

3

Iteration times

Hau

sdor

ff di

stan

ce











1 2 3 4 5 62

21

22

23

24

25

26

27

28

29

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

005

01

015

02

025

Iteration times

GU

D



(b) GUD


1 2 3 4 5 67

75

8

85

9

95

10

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 605

06

07

08

09

1

11

Iteration times

Hau

sdor

ff di

stanc

e










1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD


(a) (b)

(c) (d)



(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867 (119883 119875) = max119894isin119868

(120576119894) = max119894isin119868

(10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817) (6)


SEV = sum

119894isin119868

120576119894= sum

119894isin119868

10038171003817100381710038171003817119909119894minus 119901119895

10038171003817100381710038171003817 (7)


119894isin 119883 (119894 = 1 2 119898) and 119901

119895isin 119875 (119895 = 1 2 119899)



Let 119870 = (1 2 119896) and 119868 = (1 2 119898) Let 1198831015840

119894=

1199091015840

1 1199091015840

2 119909

1015840


119894 119894 isin 119868 in119883Then



119889119894(119895) =

10038171003817100381710038171003817119909119894minus 1199091015840

119895

10038171003817100381710038171003817 119895 isin 119870 (8)


bor is

119889119894=

1

119896sum

119895isin119870

119889119894(119895) 119895 isin 119870 (9)


bor is

120590119894= sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

119895 isin 119870 (10)


119889 =1

119898sum

119894isin119868

119889119894 (11)


LUD =1

119898sum

119894isin119868

120590119894=

1

119898sum

119894isin119868

sum

119895isin119870

(119889119894(119895) minus 119889

119894)2

(12)


GUD =1

119898sum

119894isin119868

(119889119894minus 119889)2

(13)





119895isin119869119901119895((120572119896

119894119895V119895)(sum119895isin119869

120572119896

119894119895V119895))


120575119896

1198941198941015840(120596119896

1198941015840120573119896

1198941198941015840 sum119895isin119869

120596119896

1198941015840120573119896








Time forWLOP(s)










10

20

30

40

50

60

Tim

e (s)





500

1000

1500

2000

2500

3000

3500

Tim

e (s)













minus4921 lowast 10

minus4824 lowast 10

minus4


1 2 3 4 5 63

35

4

45

5

55

6

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 61

12

14

16

18

2

22

24

26

28

3

Iteration times

Hau

sdor

ff di

stan

ce











1 2 3 4 5 62

21

22

23

24

25

26

27

28

29

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

005

01

015

02

025

Iteration times

GU

D



(b) GUD


1 2 3 4 5 67

75

8

85

9

95

10

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 605

06

07

08

09

1

11

Iteration times

Hau

sdor

ff di

stanc

e










1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD


(a) (b)

(c) (d)



(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Time forWLOP(s)










10

20

30

40

50

60

Tim

e (s)





500

1000

1500

2000

2500

3000

3500

Tim

e (s)













minus4921 lowast 10

minus4824 lowast 10

minus4


1 2 3 4 5 63

35

4

45

5

55

6

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 61

12

14

16

18

2

22

24

26

28

3

Iteration times

Hau

sdor

ff di

stan

ce











1 2 3 4 5 62

21

22

23

24

25

26

27

28

29

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

005

01

015

02

025

Iteration times

GU

D



(b) GUD


1 2 3 4 5 67

75

8

85

9

95

10

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 605

06

07

08

09

1

11

Iteration times

Hau

sdor

ff di

stanc

e










1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD


(a) (b)

(c) (d)



(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















minus4921 lowast 10

minus4824 lowast 10

minus4


1 2 3 4 5 63

35

4

45

5

55

6

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 61

12

14

16

18

2

22

24

26

28

3

Iteration times

Hau

sdor

ff di

stan

ce











1 2 3 4 5 62

21

22

23

24

25

26

27

28

29

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

005

01

015

02

025

Iteration times

GU

D



(b) GUD


1 2 3 4 5 67

75

8

85

9

95

10

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 605

06

07

08

09

1

11

Iteration times

Hau

sdor

ff di

stanc

e










1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD


(a) (b)

(c) (d)



(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 62

21

22

23

24

25

26

27

28

29

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

005

01

015

02

025

Iteration times

GU

D



(b) GUD


1 2 3 4 5 67

75

8

85

9

95

10

Iteration times

Sum Error Value


SEV

(lowast1000

)


1 2 3 4 5 605

06

07

08

09

1

11

Iteration times

Hau

sdor

ff di

stanc

e










1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD


(a) (b)

(c) (d)



(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 65

51

52

53

54

55

56

57

58

59

6

Iteration times

LUD



(a) LUD

1 2 3 4 5 60

001

002

003

004

005

006

007

008

009

01

Iteration times

GU

D



(b) GUD


(a) (b)

(c) (d)



(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Resampling to Speed Up Consolidation of Point...

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Transcript of Research Article Resampling to Speed Up Consolidation of Point...