A Novel 3D CAD Model Retrieval Method Based on Vertices ...

Research ArticleA Novel 3D CAD Model Retrieval Method Based onVertices Classification and Weights Combination Optimization

Ting Zhuang1 Xutang Zhang1 Zhenxiu Hou1 Wangmeng Zuo2 and Yan Liu3

1School of Mechatronics Engineering Harbin Institute of Technology Harbin China2School of Computer Science and Technology Harbin Institute of Technology Harbin China3School of Science Harbin Institute of Technology Harbin China

Correspondence should be addressed to Ting Zhuang zt hit126com

Received 18 June 2016 Revised 8 November 2016 Accepted 17 November 2016 Published 18 January 2017

Academic Editor Paolo Crippa

Copyright copy 2017 Ting Zhuang 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

3D shape retrieval is a problem of current interest in different fields especially in themechanical engineering domain According toour knowledgemultifeature based techniques achieve the best performance at presentHowever the practicability of thosemethodsis badly limited due to the high computational cost To improve the retrieval efficiency of 3D CAD model we propose a novel3D CAD model retrieval algorithm called VSC WCO which consists of a new 3D shape descriptor named VSC and WeightsCombinationOptimization schemeWCOVSC represents a 3Dmodel with three distance distribution histograms based on verticesclassification The weighted sum of 1198711 norm distances between corresponding distance histograms of two VSC descriptors isregarded as dissimilarity of two models For higher retrieval accuracy on a classified 3D model database WCO is proposed basedon Particle Swarm Optimization and existing class information Experimental results on ESB PSB and NTU databases show thatthe discriminative power of VSC is already comparable to or better than several typical shape descriptors AfterWCO is employedthe performance of VSC WCO is similar to the leading methods by all performance metrics and is much better by computationalefficiency

1 Introduction

The development of computer graphics technology gives abirth to the explosion in the number of 3D models which arenow widely used in many diverse applications Meanwhilemore and more 3Dmodel databases are available on the websuch as Princeton Shape Benchmark (PSB) [1] NationalTaiwan University Shape Benchmark (NTU) [2] and Engi-neering Shape Benchmark (ESB) [3] As Funkhouser et al [4]predicted the key question about 3D model has shifted fromldquoHow do we construct themrdquo to ldquoHow do we find themrdquoThere is no doubt that searching suitable 3D engineeringmodels from existing resource is extremely beneficial formechanical design because (1) more than 75 of designactivity comprises reuse of previous design knowledge toaddress a newdesign problem [5] and (2)modeling a complex3D model such as mechanical parts is still error-prone andtime-consuming Therefore it is very necessary to developefficient tools for 3D CAD model retrieval As a promising

technique the so-called content-based 3D model retrievalusing the characteristics of 3Dmodel itself has attracted a lotof research interests [6ndash13]The content-based approaches arebroadly classified into three categories shape-based view-based and multifeature based approaches

Shape-Based Approaches This kind of approaches uses thedistribution of 3D features to characterize the geometricproperties of a 3D model D2 [14] is a well known shape-based 3D model retrieval method D2 took samples on thedistances between two points on the model surface and thenaccumulated them into bins of different intervals to form adistance distribution histogram as the model shape descrip-tor D2 has many advantages robustness rotation invarianceand computational efficiency However the discriminativecapability of D2 is limited for complex 3D model such as3D engineering models Inspired by D2 several new shape-based methods were proposed Shih et al [15] investigateda descriptor named Grid D2 (GD2) In GD2 the 3D model

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 6049750 12 pageshttpsdoiorg10115520176049750

2 Mathematical Problems in Engineering

is decomposed by a voxel grid A voxel is regarded as validif there is a polygonal surface located within this voxel andinvalid otherwise The distribution of distance between twovalid voxels instead of two points on the surface is calculatedDifferent from GD2 the shape distributions in Volume D2(VD2) [16] are generated from Euclidean distance betweenrandomly selected voxels Not only surface voxels but alsointernal ones are involved In order to reduce the computa-tional complexity of D2Wu et al [17] proposed an algorithmcalled Quick D2 which modifies the sampling phase of D2Pan et al [18] proposed a new shape descriptor called PoissonHistogram which can be defined by the following two stepsFirstly a 3D shape signature was defined by Poisson equationSecondly a histogram was constructed by accumulating thevalues of the defined signature in bins

View-Based Approaches The main concept of visual repre-sentation in 3D model retrieval is to first convert the 3Dmodel into 2D projection image and then utilize matureimage processing techniques to extract various features [19]The original work was accomplished by Chen et al [2]who proposed a method named Light Field (LF) descriptorwhich is a typical view-based approach The main idea of LFis that if two 3D models are similar they also look similarfrom all viewing angles LF represents a 3D model as acollection of 2D images rendered from uniformly sampledpositions on a viewing sphere located around the modelThe distance between two descriptors is defined as the min-imum L1 distance taken over all rotations and all pairingof vertices on two dodecahedra LF had been shown toperform well on the ESB [3] database In [20] adaptiveviews clustering (AVC) selected the best characteristic viewsfrom more than 320 projected views The characteristicview selection algorithm is based on an adaptive clusteringalgorithm and uses statistical model distribution scoresto select the optimal number of views The view-basedmethods achieved better retrieval performance comparedto shape-based ones because they capture more of shapecontent However as pointed out in [2] view-based methodsinvolve more computation compared to the shape-basedones

Multifeature Based Approaches According to [3] the shaperepresentations that hold more shape content are betterat retrieving more relevant models Therefore combiningdifferent 3D shape features is an efficient way to enhance theretrieval accuracy [21] Papadakis et al [22] presented a novelhybrid 3D model shape descriptor named PANORAMAusing a set of panoramic views of a 3D model A panoramicview is generated by projecting the model to the lateral sur-face of a cylinder parallel to one of its three principal axes andcentered at the centroid of the model The 3D model is pro-jected to three perpendicular cylinders each one alignedwithone of its principal axes in order to capture the global shapeof the model They used Fourier and wavelet transforms toextract the features for each panoramic view Leng and Xiong[23] proposed a composite shape descriptor called TUGEwhich is obtained from a combination of two-view versionof depth buffer-based shape descriptor in [24] and GEDT

shape descriptor in [4] Li and Johan [25] proposed a hybrid3D shape descriptor ZFDR which comprises four featuresZernikemoments Fourier descriptor depth information andray-based features

Multifeatures based methods have more powerfulretrieval capability than shape-based and view-basedtechniques however the time consumption is much higherthan the latter because of involving more computationsIn ZFDR example [25] the average retrieval time on ESBdatabase (contains 866 3D CAD models) is 137 sec If thetarget database is 100 times bigger than ESB the averageretrieval time will be more than 10 sec High time cost limitsthe practicability of multifeature based techniques seriously

In addition to those methods mentioned above machinelearning technologies are also used to address 3D modelsretrieval In [26] 3D models are represented as probabilitydistributions of binary variables on a 3D voxel grid andthe proposed method uses a Convolutional Deep BeliefNetwork to learn the distribution of complex 3D shapesacross different categories and arbitrary poses from rawCAD data In [27] the distance between 3D models iscomputed based on distance histogram features and 3Dmoment features Using this distance measure the relation-ships between all 3D models in dataset are formulated as agraph structure A semisupervised learning process is thenconducted to estimate the relevance among the 3D modelsLeng et al [28] proposed a 3Dmodel recognitionmechanismbased on Deep Boltzmann Machines (DBM) The high-levelabstraction representation can be obtained from a DBMwhich is trained by depth images of 3D model and thefeature is used in semisupervised classification method Qinet al [29] proposed a deep learning approach to automati-cally classify 3D CAD models according to the mechanicalpart catalogue The designed deep neural network classifieris based on the latest machine learning technique deeplearning

In this paper to improve the retrieval efficiency of 3DCAD model we propose a novel retrieval algorithm namedVSC WCO which consists of a new shape-based 3D modeldescriptor called VSC and Weights Combination Optimiza-tion scheme WCO The VSC descriptor represents a 3Dmodel with three Euclidean distance distribution histogramswhich are calculated based on vertices classification Theweighted sum of L1 norm distances of corresponding his-tograms in two VSC descriptors is regarded as the dissim-ilarity of two models To avoid risks caused by using fixedweights in VSC distance computation and further improvethe retrieval performance on classified 3D model databaseWCO is proposed based on Particle Swarm Optimization(PSO) and existing class information of 3D model databaseThe objective ofWCO is searching optimal weights combina-tions for different query models

The rest of this paper is organized as follows The pro-posed 3D shape descriptor VSC is presented in Section 2 InSection 3 we describe the details of WCO and our 3D CADmodels retrieval method VSC WCO Experimental resultsare demonstrated in Section 4 Finally conclusions and futurework are presented in Section 5

Mathematical Problems in Engineering 3

2 3D Shape Descriptor VSC Based onVertices Classification

In this section we define a new shape-based 3D modeldescriptor named VSC based on classification of verticesA VSC descriptor consists of three distance distributionhistograms which are computed by the following steps (i)according to Tensor Voting Theory the vertices of a 3Dmodel are divided into three categories Face Sharp Edgeand Corner (ii) the Euclidean distances between pairs ofvertices in the same category are calculated (iii) the distancedistribution histograms of distinct categories are formed

21 Vertices Classification Based on Tensor Voting TheoryTensor voting theory has great advantages in computer visiontasks such as segmentation and object recognition Accordingto [30] the normal voting tensor is defined as

119879V = sum119905119894isin119873119905(V)

120583119905119894 119905119894 119879119905119894 (1)

where V is vertex of 3D mesh model 119879V is the normal votingtensor of V 119873119905(V) denotes a collection of 1-ring neighbortriangles of v 119905119894 is 119894th 1-ring neighbor triangle in 119873119894(V) 119905119894represents the unit normal vector of 119905119894 120583119905119894 is calculated basedon [30]

120583119905119894= ( area (119905119894)

max (area (119873119905 (V)))) exp(minus10038171003817100381710038171003817 119888119905119894 minus V

10038171003817100381710038171003817max (10038171003817100381710038171003817 119888119905119894 minus V

10038171003817100381710038171003817)) (2)

where area (119905119894) is area of triangle 119905119894 max(area(119873119905(V))) is max-imum area among119873119905(V) 119888119905119894 denotes barycenter of triangle 119905119894V represents the position of V

Because 119879V is a rank-3 positive semidefinite matrix it canbe diagonalized as follows

119879V = 1205821 1198901 1198901198791 + 1205822 1198902 1198901198792 + 1205823 1198903 1198901198793 (3)

where 1198901 1198902 and 1198903 are the corresponding unit eigenvectorsof 1205821 1205822 and 1205823 (1205821 ge 1205822 ge 1205823) respectively According tothe eigenvalues vertices can be divided into Face Sharp Edgeand Corner as follows [31]

(i) Face 1205821 is dominant 1205822 and 1205823 are close to 0(ii) Sharp Edge 1205821 and 1205822 are dominant 1205823 is close to 0(iii) Corner 1205821 1205822 and 1205823 are approximately equal

Figure 1 shows an example of vertices classification (a)is a 3D CAD model called advgr01 in ESB Marked points in(b) (c) and (d) denote the vertices of Face Sharp Edge andCorner respectively The numbers of vertices in the distinctcategories are 670 127 and 16 respectively

22 Distance Distribution Histogram After classification ofvertices the Euclidean distances between any two verticesin the distinct categories are measured To eliminate the

influence of quantity of points the distances histogramcontaining n bins is defined as

ℎ = 1198871 1198872 1198873 119887119899119873 (119873 minus 1) 2 (4)

where 119887119894 is the number of distances within the range of the119894th bin119873 is the quantity of each type of vertices ℎ denotes ahistogram constructed by counting how many distances fallinto each bin The width of bin is determined by

width = 1003816100381610038161003816119889max minus 119889min1003816100381610038161003816119899 (5)

where 119889max and 119889min are maximum and minimum distancebetween pairs of vertices in the same class n is the number ofbins

Three distance distribution histograms compose the 3Dshape descriptor VSC which is defined as follows

VSC = ℎ119891 ℎ119890 ℎ119888 (6)

where ℎ119891 ℎ119890 and ℎ119888 denote distance distribution histogramsof vertices in Face Sharp Edge and Corner respectivelyTable 1 is a comparison between feature extraction resultsof D2 and VSC for six models of ESB database the firsttwo models are taken from different classes of ESB thethird and fourth models are taken from ESBFlat-Thin WallComponentsBack Doors and the fifth and sixth are takenfrom ESBSolid Of RevolutionGear-like Parts As shown inTable 1 the first and second models are completely differentin shape but theirD2 descriptors are very similar Different toD2 those two models are quite different in the view of VSCThe third and fourth models are taken from the same classand their shapes are very similar There are almost no differ-ences between theVSCdescriptors of those twomodels so dothe D2 descriptorsThe fifth and sixth models are taken fromthe same class also but there is a little difference between theirshapesThrough comparing twoVSCdescriptors of those twomodels it is clear that the VSC descriptor reflects differencesbetween two models successfully Generally speaking due tothe subdivision of distances between sample points the shapediscrimination capacity of VSC is better than D2

23 Distance between Two VSC Descriptors After VSCdescriptors are constructed the similarity comparisonbetween two 3D models is mapped into the comparison ofVSC descriptors The distance between two VSC descriptorsis defined as the weighted sum of L1 norm distances ofcorresponding histograms

Dis (VSC119898119886 VSC119898119887) = 1199081 times 1198711 (ℎ119891119898119886 ℎ119891119898119887) + 1199082times 1198711 (ℎ119890119898119886 ℎ119890119898119887) + 1199083times 1198711 (ℎ119888119898119886 ℎ119888119898119887)

(7)

where119898119886 and119898119887 denote two 3D models VSC119898119886 and VSC119898119887are VSC descriptors of 119898119886 and 119898119887 respectively L1 is theL1 norm distance measure 1199081 1199082 and 1199083 are weights The

4 Mathematical Problems in Engineering

(a) (b) (c) (d)

Figure 1 An example of vertices classification (a) 3D CADmodel ESBRectangular-Cubic PrismBearing Blocksadvgr01 813 vertices (b)vertices of Face (c) vertices of Sharp Edge (d) vertices of Corner

Table 1 Comparison of feature extraction results of D2 and VSC on two ESB models

Model D2 VSC

Front0

2

4

0 300 600

times10minus3 D2

0

001

002

003

0 35 70

Face

0

001

002

003

0 35 70

Shape Edge

0

002

004

0 35 70

Corner

DEMO_REF0

2

4

0 300 600

times10minus3 D2

0

001

002

003

0 35 70

Face

0

001

002

003

0 35 70

Shape Edge

0 35 700

005

01Corner

Back Door 10

2

4

0 300 600

times10minus3 D2

0

02

04

0 35 70

Face

0 35 700

5

10Shape Edge

0 35 700

10

20Corner

Back Door 2

D2

0

2

4

0 300 600

times10minus3 Face

0

02

04

0 35 70

Shape Edge

0 35 700

5

10Corner

0 35 700

10

20

16t_2d_05w0

2

4

0 300 600

D2times10minus3

0 35 700

05

1Face

0 35 700

2

4Shape Edge

0 35 700

01

02Corner

22t_275d_05w0

2

4

0 300 600

D2times10minus3

0

02

04

0 35 70

Face

0 35 700

2

4

6Shape Edge

0 35 700

02

04Corner

Mathematical Problems in Engineering 5

three distance histograms reflect the different shape charac-teristics of a model and they have the same contribution forVSC distance computation Therefore we linearly combinethem In addition the distances between the correspondinghistograms fall in the same range as such we assign the sameweight for each histogram 1199081 = 1199082 = 1199083 = 13 Fortwo 3D models the distance between corresponding VSCdescriptor is regarded as the dissimilarity and the smaller theVSC distance is the more similar they are According to (7)the VSC distance between first two models shown in Table 1is 171

For a querymodel 119902 different weights combinations usedin (7) will produce different dissimilarities between 119902 andmodels in database and lead to different retrieval resultseventually The influence of weights combination againstretrieval result provides an opportunity to achieve moredesirable retrieval result on a classified 3D model database(details are discussed in the next section)

3 3D CAD Model Retrieval AlgorithmVSC_WCO Based on VSC andWeights Combination OptimizationScheme WCO

In this section to further improve retrieval performance in aclassified 3D model database we propose a 3D CAD modelretrieval algorithm called VSC WCO based on 3D shapedescriptorVSCpresented in Section 2 andWeights Combina-tion Optimization scheme WCO In order to avoid the riskscaused by using fixed weights combination in VSC distancecomputation WCO takes into account the class informationof 3D model database and utilizes Particle Swarm Optimiza-tion (PSO) to search optimal weights combination for eachclass

31 Weights Combination Optimization In order to avoidthe risks caused by fixed weights and improve retrieval per-formance we define Weights Combination Optimizationmethod named WCO which takes into account the classinformation of 3D model database and utilizes ParticleSwarm Optimization (PSO) [32] to search the optimalweights combination for each classThemotivations ofWCOare described as follows

(1) For a query model q different weights combina-tions used in (7) will produce different dissimilari-ties between q and models in database and lead todifferent retrieval results eventually The influence ofweights combination against retrieval result providesan opportunity to achieve more desirable retrievalresult by searching more suitable weights combina-tion for q

(2) In fact those three distance histograms ℎ119891 ℎ119890 andℎ119888 in the proposed shape descriptor VSC depict a 3Dmodel from different aspects respectively and theycan be regarded as three single 3D shape descriptorsIt is demonstrated in [3] that no single 3D modeldescriptor performs well on all models and differentdescriptors have different strengths and weaknesses

According to this conclusion fixed weights combina-tion used in (7) for VSC distance computation is notsuitable for all models

In this subsection we introduce the PSO first and thendescribe the details of WCO

311 Particle Swarm Optimization (PSO) PSO has severaladvantages over other artificial intelligence algorithms Forexample it is better at global optimization and is easier toapply to multiple-objective problems [28] PSO is initializedwith a population of random potential solutions and thealgorithm searches the optimal solution according to itsperformance The goodness of a particle is determined by afunction called fitness function in PSO The fitness functiontakes position of a particle as input and returns a singlenumber which denotes the goodness of the particle for theoptimization problem

Consider a group ofN particles that are searching a globaloptimal solution in a D dimensions space The position andvelocity of 119875119894 (119894th particle) can be expressed as

119883119894 = (1199091119894 1199092119894 119909119863119894 ) (8)

119881119894 = (V1119894 V2119894 V119863119894 ) (9)

119881119905+1119894 = 119908 times 119881119905119894 + 1198881 times 1199031 times (119901best119894 minus 119883119905119894) + 1198882 times 1199032times (119892best119905 minus 119883119905119894)

(10)

119883119905+1119894 = 119883119905119894 + 119881119905+1119894 (11)

where119881119905119894 and119881119905+1119894 denote the velocity of 119875119894 in iterations 119905 and119905 + 1 119883119905119894 and 119883119905+1119894 represent the positions of 119875119894 in iterations 119905and 119905+1119901best119894 is the best position of119875119894 until iteration 119905119892best119905is the best position among all the particles until iteration 1199051198881 and 1198882 are the personal learning coefficient and the sociallearning factor respectively 1199031 and 1199032 are random numbersin the range of (0 1) 119908 is an inertia factor from 08 to 12

In WCO PSO is used for searching the optimal weightscombination for each class in a classified 3D model databasethus the position and velocity of each particle consist of threenumbers respectively In order to avoid negative numbersduring the optimization process the relationship betweenpositions and weights is expressed as

119908119894 = exp (119909119894) (119894 = 1 2 3) (12)where 119908119894 is the 119894th weight and 119909119894 is the 119894th component ofparticlersquos position

It is obvious that if the weights combination used in (7) ismore reasonable to querymodel 119902 the VSC distance betweenq and all models in database will be more accurate and theretrieval performance will be better In other words the morereasonable weights combination the better retrieval perfor-manceThus the retrieval performance achieved by a particlecan be used as its fitnessWe employ the performancemetricsFirst Tier (FT) [1] to evaluate the retrieval performance ofeach particle FT is expressed as

FT (119877) = 119899119903|119862| minus 1 (13)

6 Mathematical Problems in Engineering

where R represents a retrieval result which is formed bysorting all models in ascending order based on their VSCdistances from query model 119902 119862 denotes the class of 119902 and|119862| is the cardinality of 119862 119899119903 is the number of models of 119862 intop |119862| minus 1 list of 119877312 Implementation Process of WCO By controlling weightalterations one ensures the evaluation of retrieval resultbetter which is the main idea of WCO The optimizationprocess ofWCO is summarized in Algorithm 1 and notationsused inAlgorithm 1 are explained in theNotations Accordingto Algorithm 1 the time and space complexity of WCO are119874(119873 times 119879 times 119875 times |119862| times |119872|) and 119874(1)32 Implementation of VSC WCO To improve the retrievalperformance on a classified 3D CAD model database wepropose a new 3D CAD model retrieval algorithm namedVSC WCO based on VSC descriptor described in Section 2and Weights Combination Optimization scheme WCO pre-sented in Section 31 Our VSC WCO is composed of twoparts online and offline parts which are described as followsand the whole process of VSC WCO is illustrated in Figure 2

Offline

(1) VSC Feature Extraction We extract the VSC shapedescriptors of all the models in Train and Test setsbased on the method in Section 2

(2) OptimalWeights Calculation in Train SetWe calculatethe optimal weights combinations (OWC) for allclasses of Train set based on WCO described inSection 31 All optimal weights combinations arestored in a database with the corresponding classinformation

Online

(1) VSC Feature Extraction We extract the VSC shapedescriptor of query model q based on the method inSection 2

(2) Determine the Most Similar Class (MSC) of q in TrainSet First we compute the VSC distance between qandmodels in Train set according to (7) usingweightscombination 13 13 and 13 and then we selectthe nearest modelrsquos class as the most similar class of qwhich is denoted as MSCq

(3) Determine the Optimal Weights Combination (OWC)of q according to MSCq The optimal weights com-bination corresponding to qrsquos MSC is selected fromdatabase which is denoted as OWCq

(4) VSC Distances Computation in Test Set We computethe VSC distances between q and models in Test setaccording to (7) using weights combination OWCq

(5) Ranking and Output Sort all the models in Test set inascending order based on their VSC distances com-puted in step (4) and output retrieval lists accordingly

4 Experiments

In this paper all retrieval experiments are implemented inMATLAB R2011b and performed in a PC with configurationCPU Intel Pentium Dual-Core E540027GHz memory20GB OS Windows XP To investigate the performanceof VSC WCO for 3D engineering and generic models weselected the following three representative standard bench-mark databases

(i) Engineer Shape Benchmark (ESB) [3] It is developedby Purdue University for evaluating the search meth-ods relevant to the mechanical engineering domainThere are 866 3D engineering models in ESB andthose models are classified into three superclassesnamely Flat-Thin Wall Component Rectangular-Cubic and Solids of Revolution Within each super-class models are further classified into clusters ofsimilar shapes Figure 3 shows some example views of3Dmodels in ESB In order to ensure the usefulness ofweights combinations calculated byWCO we equallydivide ESB database into two parts Train and Testsets the former is used forWCO and the latter is usedas target database for retrieval experiments

(ii) Princeton Shape Benchmark (PSB) [1] It contains 18143D models totally which are classified into Test andTrain parts In our experiments Train database isused for WCO and Test database is used for retrievalexperiments

(iii) National TaiwanUniversity (NTU)Database [2]NTUdatabase contains 1833 3D models and only 549 3Dmodels are grouped into 47 classes and the remaining1284 models are assigned as the ldquomiscellaneousrdquoThus we only use the 549 classified ones for ourexperiments

41 Retrieval Performance Evaluation Metrics To compre-hensively evaluate the 3D model retrieval results we employfive metrics including precision-recall curve Nearest Neigh-bor (NN) First Tier (FT) Second Tier (ST) and DiscountedCumulative Gain (DCG) [1] Precision indicates how muchpercentage of the top K models belongs to the same class asthe query model while recall means how much percentageof a class has been retrieved among the top K retrieval listThe precision-recall curve comprehensively demonstratesretrieval performance which is assessed in terms of averagerecall and average precision NN is defined as the percentageof the closestmatches that are relevantmodels FT is the recallof the top 119862 minus 1 list where 119862 is the cardinality of the relevantclass of the query model ST is the recall of the top 2(119862 minus 1)list DCG is defined as the summed weighted value relatedto it which combines precision and recall as well as rankingpositions

42 Comparison Algorithms To compare the performance ofour retrieval algorithmVSC WCOwe consider the followingfour algorithms

Mathematical Problems in Engineering 7

Input A classified 3D model databaseMOutput Optimal weights combinations for all classes inM(1) Initialize a population of particles with random positions and velocities(2) FOR (119894 = 1 119894 le 119873 119894 + +)(3) WHILE (119905 le 119879 fitness119892119887119890119904119905 = 1)(4) FOR (119895 = 1 119895 le 119875 119895 + +)(5) FOR (119896 = 1 119896 le |119862119894| 119896 + +)(6) 119902 = 119898119862119894

119896 the kth model in 119862119894 is used as query model

(7) FOR (119897 = 1 119897 le |119872| 119897 + +)(8) 119889119897 = exp(1199091199011198951 ) times 1198711(ℎ119891119902 ℎ119891119898119897 ) + exp(1199091199011198952 ) times 1198711(ℎ119890119902 ℎ119890119898119897 ) + exp(1199091199011198953 ) times 1198711(ℎ119888119902 ℎ119888119898119897 )(9) End FOR(11) 119877119896 = Ascending (1198891 1198892 119889|119872|minus1)(12) End FOR(13) fitness119905119901119895 = sum|119862119894 |

119896=1FT(119877119896)|119862119894|

(14) IF (fitness119905119901119895 gt fitness119901best119901119895 )

(15) 119909119901best1198751198951 119909119901best1198751198952 119909119901best1198751198953 = 1199091198751198951 1199091198751198952 1199091198751198953 (16) fitness119901best119901119895 = fitness119905119875119895 (17) End IF(18) IF (fitness119905119901119895 gt fitness119892best)(19) 119909119892best1 119909119892best2 119909119892best3 = 1199091198751198951 1199091198751198952 1199091198751198953 (20) fitness119892best = fitness119905119875119895 (21) End IF(22) End FOR(23) t++(24) Update all particles according to Equation (10) and Equation (11)(25) EndWHILE(26) Output 119909119892best1 119909119892best2 119909119892best3 as optimal weights combination for 119862119894(27) End FOR

Algorithm 1 Search optimal weighs combinations

Query model qgiven by user

Train set VSCextraction

Ranking and outputresult to user

Train set of VSCdescriptors

Search MSC of q onTrain set based on

VSC distances usingequal weights

User

Weights combinationsoptimization (WCO)

Test set of VSCdescriptors

Online

Offline

Optimal weightscombinations for

each class

VSC extraction

Test set

(MSCq denotesthe most similar

class of q)

Compute the VSCdistances between qand models in Test

CornerSharp EdgeFace

CornerSharp EdgeFace(1) Vertices classification

(2) Distance histograms

0 10 20 30 40 0 10 20 30 40 0 10 20 30 40

01

008006

004

002

0

02

015

01

005

0

02

015

01

005

0

Wq = W(MSCq)

set using Wqmiddot middot middot

Figure 2 The process of the proposed algorithm VSC WCO

8 Mathematical Problems in Engineering

Figure 3 Example views of 3D models in ESB PSB and NTU

(i) D2 a classic shape-based retrieval method and themost related method of VSC We implemented D2based on the original paper [14]

(ii) LF a typical view-based retrieval method which per-formed well on ESB [3] We performed experimentsbased on their provided execution files [2]

(iii) ZFDR a state-of-the-art hybrid retrieval algorithmwhich integrates a 3D modelrsquos both visual and geo-metric information from different aspects For theretrieval performance of ZFDR on ESB PSB andNTU we refer to [25]

(iv) PANORAMA a state-of-the-art hybrid retrieval algo-rithm which utilizes a set of panoramic viewsAccording to Li and Johan [25] PANORAMA isarguably the best retrieval method reported to dateFor the retrieval performance of PANORAMA onESB PSB and NTU we refer to [22 25]

43 Experiments Results and Analysis In our experimentsevery model in Test set is used as query model and to avoidbias we exclude this model from Test set when computingthe VSC distance Figures 4 and 5 and Table 2 compare theperformance of our VSC WCO algorithm and the above-mentioned four methods To evaluate the effectiveness ofWCO for comparison we also list the performances whenusing only theVSC shape descriptor As can be seen in Figures4 and 5 and Table 2 firstly our shape descriptor VSC is betterthan D2 and comparable to LF with lower computationalcomplexity Secondly after applying the WCO VSC WCOachieves significantly better performance than VSC OnESB VSC WCO exceeds VSC in NN FT ST and DCG by24 134 122 and 61 On PSB those metrics were

Table 2 Performance comparison retrieval performance metricsof VSC VSC WCO D2 LF ZFDR and PANORAMA on ESB PSBand NTU databases

Database Methods NN FT ST DCG

ESB

PANORAMA 086 049 064 079ZFDR 084 047 061 077

VSC WCO 084 051 058 081VSC 082 040 048 076LF 082 041 054 075D2 073 027 037 070

PSB

PANORAMA 075 048 060 mdashZFDR 070 044 055 070

VSC WCO 069 042 054 072VSC 067 036 047 062LF 066 038 049 063

NTU

PANORAMA 080 049 061 076ZFDR 075 045 058 073

VSC WCO 067 043 054 069VSC 065 037 046 061LF 061 034 042 055

increased by 29 167 148 and 161 The correspond-ing increments are 31 162 174 and 131 on NTUThis increase indicates that fixed weights combination forVSC distance computation is not suitable for all querymodelsand the optimal weights combinations produced byWCO areuseful to improve the retrieval performance of VSC Finallythe performance of VSC WCO is substantially the sameas that achieved by the state-of-the-art algorithms such asPANORAMA and ZFDR Furthermore the average retrievaltime of VSC WCO on ESB PSB and NTU database are lessthan the latter Table 3 lists the timings of VSC WCO on

Mathematical Problems in Engineering 9

Table 3 Run time information of VSC WCO on ESB PSB andNTU

Database t1 t2 t3ESB 0293 0016 0017PSB 0364 0034 0038NTU 0416 0001 0001

ESB PSB and NTU databases In Table 3 1199051 denotes the timeconsumption of VSC extraction of query model 1199052 denotesthe time consumption of finding the most similar class ofquerymodel in Train set 1199053 is time consumption of searchingsimilar models in Test set On a common configurationcomputer (CPU Intel Pentium Dual-Core E540027GHzmemory 20GB) the average retrieval time of VSC WCOis less than 05 s which enables real-time retrieval from largerepositories This is a significant advantage of the VSC WCOas it can be rendered more efficient by reducing its storagerequirements and time complexity Altogether VSC WCOhas considerable retrieval performance and is very efficientin computation and comparison time

In Table 4 we provide a few examples of queries andthe retrieved results to first 8 retrieval models from the ESBdatabase using D2 VSC and VSC WCO As illustrated inTable 4 the performance of VSC WCO is better than VSCand D2 in terms of both quantity and rankings of relevantmodels

44 Limitations Our 3D CAD model retrieval algorithmVSC WCOhas achieved good performance on ESB databaseHowever there are still some limitations Firstly since theclass information of 3Dmodel database is a key condition forthe proposed Weights Combination Optimization methodWCO VSC WCO only can be used to the already classified3D model database Secondly the optimal weights combina-tion to a query model q is determined by the most similarclass of q therefore if the most similar class is not the targetclass then our method VSC WCO will fail to get the rightmodels

5 Conclusions and Future Work

In this paper to improve the retrieval efficiency for 3D CADmodels in a classified 3D model database we have proposeda 3D model retrieval method called VSC WCO which isbased on the proposed 3D shape descriptor VSC andWeightsCombination Optimization scheme WCO VSC descriptorrepresents a 3D model with three distance histograms whichis computed by vertices classification based onTensorTheoryWCO utilizes the existing class information and ParticleSwarm Optimization (PSO) to search optimal weights com-binations for all classes The retrieval experiments are con-ducted on Engineering Shape Benchmark (ESB) We equallydivide the ESB database into two parts one is used as Trainset for WCO to compute the optimal weights combinationsfor every class in ESB and the other one is used as Test set forretrieval experiment Several metrics are selected to evaluatethe retrieval capacity of our algorithm Retrieval results onESB demonstrated that (1) the retrieval performance of the

proposed 3D shape descriptor VSC is much better thanshape-based descriptor D2 and comparable to view-baseddescriptor Light Field with lower computational complex-ity and (2) after WCO is employed the performance ofVSC WCO is close to the state-of-the-art algorithms such asPANORAMA and ZFDR Furthermore the average retrievaltime of VSC WCO on ESB database is much lower than thelatter

Further research will focus on the following aspects (1)in order to improve the accuracy of the most similar class ofquerymodel we need to enhance the discrimination capacityof proposed shape descriptor VSC (2) develop a 3D modelclustering algorithm for applying our method VSC WCO onunclassified 3D model databases

Notations

119872 Classified 3D models databasewhich has several classes|119872| Number of models in119872119873 Number of classes in119872119862119894 The 119894th class of119872|119862119894| Number of models in 119862119894119898119862119894

119896 The 119896th model of 119862119894119875 Number of particles in PSO119875119895 The 119895th particle of PSO119876 Query model119889119897 VSC distance between 119902 and119898119897

Ascending(1198891 1198892 119889|119872|minus1) Sort all the models in119872 inascending order based on VSCdistances119877119896 Retrieval result when119898119862119894

119896is used

as query model

119879 Maximum number ofiterations119905 Current number of iterations

fitness119905119901119895 Fitness of 119875119895 in the 119905thiteration

FT(119877119896) Retrieval performance of 119875119895when119898119862119894

119896is used as query

model measured by metricsFirst Tier which is expressedas (13)119901best119875119895 The best position achieved by119875119895 so far

119909119901best1198751198951 119909119901best1198751198952 119909119901best1198751198953 Three components of119901best119875119895 rsquos positionfitness119901best119901119895 Fitness of 119901best119875119895119892best The best position achieved by

any particle so far119909119892best1 119909119892best2 119909119892best3 Three components of 119892bestrsquosposition

fitness119892best Fitness of 119892bestCompeting Interests

The authors declare that they have no competing interests

10 Mathematical Problems in Engineering

Flat-Thin Wall Components

VSC_WCOLF

VSCD2

VSC_WCOLF

VSCD2

D2

VSC_WCOLF

VSCD2

Rectangular-Cubic

Solid of Revolution ESB

01 02 03 04 05 06 07 08 09 10Recall

0010203040506070809

1Pr

ecisi

on

0010203040506070809

1

Prec

ision

01 02 03 04 05 06 07 08 09 10Recall

01 02 03 04 05 06 07 08 09 10Recall

0010203040506070809

1

Prec

ision

0010203040506070809

1

Prec

ision

01 02 03 04 05 06 07 08 09 10Recall

PANORAMAZFDRVSC_WCO

VSCLF

Figure 4 Performance comparison precision-recall curve of VSC VSC WCO and other methods on each superclass of ESB and whole ESBdatabase

PSB

PANORAMAZFDRVSC_WCO

VSCLF

PANORAMAZFDRVSC_WCO

VSCLF

NTU

01 02 03 04 05 06 07 08 09 10Recall

0

01

02

03

04

05

06

07

08

09

1

Prec

ision

0

01

02

03

04

05

06

07

08

09

1

Prec

ision

01 02 03 04 05 06 07 08 09 10Recall

Figure 5 Performance comparison precision-recall curve of VSC VSC WCO and other methods on PSB and NTU databases

Mathematical Problems in Engineering 11

Table 4 Some retrieval examples on ESB database using D2 VSC and VSC WCO

Querymodel Method First 8 similar models

1 2 3 4 5 6 7 8

D2

VSC

VSC WCO

D2

VSC

VSC WCO

D2

VSC

VSC WCO

Acknowledgments

This study was supported by Natural Science Foundationof China (51001121) Fundamental Research Funds for theCenter Universities and Program for Innovation Researchof Science in Harbin Institute of Technology (PIRS OFHIT Q201503) China Postdoctoral Science Foundationfunded project (2015M581450) and Program for InnovationResearch of Harbin City (2015RAQXJ090)

References

[1] P Shilane PMinMKazhdan andT Funkhouser ldquoThePrince-ton shape benchmarkrdquo in Proceedings of the Shape ModelingInternational (SMI rsquo04) pp 167ndash178 Genova Italy June 2004

[2] D-Y Chen X-P Tian Y-T Shen and M Ouhyoung ldquoOnvisual similarity based 3D model retrievalrdquo Computer GraphicsForum vol 22 no 3 pp 223ndash232 2003

[3] S Jayanti Y Kalyanaraman N Iyer and K Ramani ldquoDevelop-ing an engineering shape benchmark for CAD modelsrdquo CADComputer Aided Design vol 38 no 9 pp 939ndash953 2006

[4] T Funkhouser P Min M Kazhdan et al ldquoA search engine for3D modelsrdquo ACM Transactions on Graphics vol 22 no 1 pp83ndash105 2003

[5] W C Regli and V A Cicirello ldquoManaging digital libraries forcomputer-aided designrdquo CAD Computer Aided Design vol 32no 2 pp 119ndash132 2000

[6] B Bustos D A Keim D Saupe T Schreck and D V VranicldquoFeature-based similarity search in 3D object databasesrdquo ACMComputing Surveys vol 37 no 4 pp 345ndash387 2005

[7] T Funkhouser M Kazhdan P Min and P Shilane ldquoShape-based retrival and analysis of 3DmodelrdquoCommunications of theACM vol 48 no 6 pp 58ndash64 2005

[8] N Iyer S Jayanti K Lou Y Kalyanaraman and K RamanildquoThree-dimensional shape searching state-of-the-art reviewand future trendsrdquo CAD Computer Aided Design vol 37 no 5pp 509ndash530 2005

[9] A Del Bimbo and P Pala ldquoContent-based retrieval of 3D mod-elsrdquoACMTransactions onMultimedia Computing Communica-tions and Applications vol 2 no 1 pp 20ndash43 2006

[10] Y Yang H Lin and Y Zhang ldquoContent-based 3-D modelretrieval a surveyrdquo IEEE Transactions on Systems Man andCybernetics Part C Applications and Reviews vol 37 no 6 pp1081ndash1098 2007

[11] B Bustos D Keim D Saupe and T Schreck ldquoContent-based3D object retrievalrdquo IEEE Computer Graphics and Applicationsvol 27 no 4 pp 22ndash27 2007

[12] J W H Tangelder and R C Veltkamp ldquoA survey of contentbased 3D shape retrieval methodsrdquo Multimedia Tools andApplications vol 39 no 3 pp 441ndash471 2008

[13] A Foncubierta-Rodrıguez H Muller and A DepeursingeldquoRetrieval of high-dimensional visual data current state trendsand challenges aheadrdquo Multimedia Tools and Applications vol69 no 2 pp 539ndash567 2014

[14] R Osada T Funkhouser B Chazelle and D Bobkin ldquoShapedistributionrdquo ACM Trans Graphics vol 4 pp 807ndash832 2002

[15] J-L Shih C-H Lee and J TWang ldquo3D object retrieval systembased on grid D2rdquo Electronics Letters vol 41 no 4 pp 179ndash1812005

12 Mathematical Problems in Engineering

[16] Q W Bian J L Wang and Y J He ldquoRetrieval of 3D shapesusing volume D2rdquo Electronics Letters vol 45 no 23 pp 1163ndash1165 2009

[17] Y Wu L Tian and C Li ldquoHigh efficient methods of content-based 3D model retrievalrdquo Chinese Journal of MechanicalEngineering vol 26 no 2 pp 248ndash256 2013

[18] X Pan Q You Z Liu and Q H Chen ldquo3D shape retrieval byPoisson histogramrdquo Pattern Recognition Letters vol 32 no 6pp 787ndash794 2011

[19] Y Gao and Q Dai ldquoView-based 3D object retrieval challengesand approachesrdquo IEEEMultimedia vol 21 no 3 pp 52ndash57 2014

[20] T F Ansary M Daoudi and J-P Vandeborre ldquoA bayesian 3-Dsearch engine using adaptive views clusteringrdquo IEEE Transac-tions on Multimedia vol 9 no 1 pp 78ndash88 2007

[21] PDaras AAxenopoulos andG Litos ldquoInvestigating the effectsof multiple factors towards more accurate 3-D object retrievalrdquoIEEE Transactions on Multimedia vol 14 no 2 pp 374ndash3882012

[22] P Papadakis I Pratikakis T Theoharis and S PerantonisldquoPanorama a 3D shape descriptor based on panoramic viewsfor unsupervised 3d object retrievalrdquo International Journal ofComputer Vision vol 89 no 2-3 pp 177ndash192 2010

[23] B Leng and Z Xiong ldquoModelSeek an effective 3D modelretrieval systemrdquoMultimedia Tools and Applications vol 51 no3 pp 935ndash962 2011

[24] D V Vranic 3d model retrieval [PhD thesis] University ofLeipzig Leipzig Germany 2004

[25] B Li and H Johan ldquo3D model retrieval using hybrid featuresand class informationrdquoMultimedia Tools and Applications vol62 no 3 pp 821ndash846 2013

[26] Z Wu S Song A Khosla et al ldquo3D ShapeNets a deep rep-resentation for volumetric shapesrdquo in Proceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 1912ndash1920 Boston Mass USA June 2015

[27] K Lu Q Wang J Xue and W Pan ldquo3D model retrieval andclassification by semi-supervised learning with content-basedsimilarityrdquo Information Sciences vol 281 pp 703ndash713 2014

[28] B Leng X Zhang M Yao and Z Xiong ldquoA 3Dmodel recogni-tion mechanism based on deep Boltzmann machinesrdquo Neuro-computing vol 151 no 2 pp 593ndash602 2015

[29] F-W Qin L-Y Li S-M Gao X-L Yang and X Chen ldquoAdeep learning approach to the classification of 3DCADmodelsrdquoJournal of Zhejiang University Science C vol 15 no 2 pp 91ndash106 2014

[30] H S Kim H K Choi and K H Lee ldquoFeature detection oftriangular meshes based on tensor voting theoryrdquo Computer-Aided Design vol 41 no 1 pp 47ndash58 2009

[31] T Shimizu H Date S Kanai and T Kishinami ldquoA new bilateralmesh smoothing method by recognizing featuresrdquo in Proceed-ings of the 9th International Conference on Computer AidedDesign and Computer Graphics (CADCG rsquo05) pp 281ndash286Hong Kong China December 2005

[32] R C Eberhart and Y Shi ldquoParticle swarm optimization devel-opments applications and resourcesrdquo in Proceedings of the Con-ference on Evolutionary Computation pp 81ndash86 Soul KoreaMay 2001

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