Shape-Similarity Search of 3D Models by using Enhanced Shape Functions

Transcription

1 Shpe-Similrity Serch of 3D Models by using Enhnced Shpe Functions Ryutrou Ohbuchi, Tkhiro Minmitni, Tsuyoshi Tkei Grdute School of Medicl nd Engineering Science, University of Ymnshi, Tked, Kofu-shi, Ymnshi-ken, , Jpn Abstrct We propose pir of shpe fetures for serching surfce-bsed 3D shpe models bsed on their shpe similrity. Either of the fetures is computed by first converting n input surfce bsed model into n oriented point set model nd then computing joint 2D histogrm of distnce nd orienttion of pirs of points. Advntges of the shpe fetures re (1) they cn be computed for non-solid or non-mnifold models, (2) they re invrint to similrity trnsformtion, nd (3) they re tolernt of topologicl nd geometricl errors nd degenercies. Experiments showed tht, with only modest increse in computtionl cost, our shpe fetures chieved significnt performnce improvement over Osd s D2, on which our fetures re bsed. Keywords: content-bsed serch nd retrievl, geometric modeling, polygonl mesh. 1. Introduction Prolifertion of 3D models on the Internet nd in in-house dtbses prompted development of the technology for effective content-bsed serch nd retrievl of three-dimensionl (3D) models. A 3D model could be serched by its textul nnottion by using conventionl text-bsed serch engine. This pproch wouldn t work in mny of the ppliction scenrios for the 3D shpe model, however. The nnottions dded by humn beings depend on culture, lnguge, ge, nd other fctors. It is lso extremely difficult to describe by words shpe tht is not in n wellknown shpe or semntic ctegory. It is thus necessry to develop content-bsed serch nd retrievl systems for 3D models tht re bsed on the fetures intrinsic to the 3D models, one of the most importnt of which is shpe. In the study of shpe similrity serch of 3D models, current focuses re on the development of robust, concise, yet expressive shpe fetures, nd on the development of similrity (or, dissimilrity) comprison methods tht conform well to the humn notion of shpe similrity. In developing shpe feture for 3D models, we first hve to decide which clss of 3D shpe representtion we re trgeting. A 3D shpe my be defined by using ny of number of shpe representtions, mny of which re not mutully comptible. Some of the shpe representtions re mthemticlly well founded, llowing for computtions of such well-defined properties s volume, surfce curvture, or surfce (or volume) topology. Other shpe representtions re less nicer. For exmple, polygon-soup model is topologiclly disconnected collection of independent polygons nd/or polygonl meshes. Neither volume nor surfce curvture cn be computed for the model. As mny of the VRML models nd 3D models generted by using 3D 1

2 nimtion softwre re defined s polygon soup models, it is quite importnt to develop effective shpe similrity comprison methods for this clss of models. Another importnt requirement for 3D shpe similrity comprison method is invrince of the method to required clss of geometricl trnsformtions. Most of the time, n invrince to similrity trnsformtion, tht is, combintion of trnsltion, rottion, nd uniform scling, is required for 3D shpe similrity comprison. A 3D model hs higher degrees-of-freedom (DOF) for their pose thn 2D shpe model; description of similrity trnsformtion requires 7 DOF. On the other hnd, 2D shpe needs only 4 DOF to define similrity trnsformtion. A previous shpe similrity comprison method either: employed n orienttion insensitive shpe feture performed pose normliztion prior to pplying pose orienttion sensitive shpe feture. Osd et l [1] proposed wht they cll shpe distributions. Osd s shpe distributions, set of shpe fetures, hve the dvntge of being invrint, without pose normliztion, to similrity trnsformtions. Furthermore, they re designed to be pplicble to not-so-well-defined meshbsed model, i.e., polygon soup defining non-solid object consisting of non-mnifold surfces, multiple connected components, nd such degenerte surfces s zero-re polygons. All of their shpe fetures first converts the surfce bsed 3D models into unoriented point set models. Then, vrious sttistics re computed from the point set model. Among the proposed shpe distributions, the D2 showed the best retrievl performnce despite its low computtionl cost. The D2 is lso esy to implement, for it is 1D histogrm of distnce between pirs of points in the point set model. However, the retrievl performnce of the D2 is not sufficient, filing to distinguish shpes tht re quite different. In this pper, we propose pir of shpe fetures for compring polygon soup models. Our shpe fetures re bsed on the Osd s D2 shpe function [1]. Both our shpe fetures try to sttisticlly cpture surfce orienttion s well s surfce distnce of the model. Our method first convert given surfce model into oriented point set model, i.e., set of points hving 3D position s well s orienttion norml vector. Then, the methods compute, s feture, joint 2D histogrm of the distnces nd mutul orienttions of the pirs of oriented points. Experimentl evlution showed tht our shpe fetures hve significntly better retrievl performnce thn Osd s D2 shpe function while hving only slightly incresed computtionl cost. The pper is orgnized s follows. In the next section, we will review the previous work on 3D shpe similrity serch nd retrievl, focusing on those methods tht trget polygon soup models. Our shpe-mtching lgorithms re described in Section 3, nd the method nd results for the experimentl evlution of our lgorithm re presented in Section 4. We conclude the pper in Section Previous Work A method for shpe similrity comprison of 3D models cn be clssified by the shpe representtion it is trgeting. Some of the shpe comprison lgorithms ssume well-defined shpe representtion, tht re, 3D solid represented by using voxels, boundry representtion, or constructive solid geometry [2, 3, 4]. Others ssume topologiclly well-defined 2-mnifold surfces [5, 6]. However these methods cn t be used to compre polygon soup models. In this section, we review shpe similrity comprison methods for not-so-well-defined shpe 2

3 representtions, especilly those for polygon-soup models. Another possible clssifiction is by the method used to chieve invrince of the shpe comprison method to clss of geometricl trnsformtions. To chieve geometricl trnsformtion invrince, some methods employ pose normliztion. To fully invert similrity trnsformtion, 7 DOF must be fixed; 3 for position, 3 for rottion, nd 1 for scling. Some methods normlize ll the 7 DOF, while the other normlize subset of the 7 DOF. A pioneering work in 3D shpe similrity serch by Pquet et l [7] computed, fter pose normliztion, set of geometricl fetures. Their method lso employed ttributes, such s color of the model, for their shpe similrity serch. Another pioneering work by Suzuki et l [8] computed, fter pose normliztion, distribution of vertices in the uniformly subdivided xisligned grid. Suzuki et l tried to relte impression words nd 3D shpe by mens of multidimensionl scling. Zhri [9] employed 3D Hough trnsformtion computed from distribution of vertices of model s its shpe feture, while Eld et l [10] computed vrious moments from the points generted rndomly on the surfce. Eld s tried to mtch the humn notion of shpe similrity with tht of mechnicl distnce by employing lerning clssifier support vector mchine. The method by Ohbuchi et l [11] normlizes pose, then computes moment of inerti, verge nd vrince of distnce from the three principl xes to the model surfce. These methods employed principl component nlysis of the covrince mtrix of the point distribution of the model for their pose normliztion. The points used to generte the covrince mtrix my be the originl vertices or they my be generted uniformly on the surfces for the purpose of pose normliztion. Ankerst [12] proposed one of the first 3D shpe similrity mtching lgorithms tht trgeted 3D moleculr dtbses. One of their shpe descriptor prmeterizes the 3D spce using concentric sphericl shells, mking the feture invrint to rottion. The sphericl hrmonics descriptor by Funkhouser, et l. [13] lso employs the concentric-shell prmeteriztion of the 3D spce, fter normlizing the position nd scle. Funkhouser et l used the sphericl hrmonics descriptor to cpture distribution of polygon for ech shell in the frequency domin. The sphericl hrmonics shpe feture is combined with the normliztion of trnsltion nd uniform scling to chieve invrince to ll the 7 DOF of similrity trnsformtion. The methods by Chen et l [14] nd Ohbuchi et l [15] compres 3D shpe by using set of 2D imges tken from multiple orienttions. These two methods first normlize position nd scle DOF to plce normlized model t the coordinte origin. Then, 2 out of 3 rottionl DOF re pproximted by few dozen discrete viewing loctions. The remining 1 DOF of rottion is removed by using rottion invrint shpe feture for 2D imges. These methods tht require pose normliztion could run into trouble if pose normliztion fils. For exmple, identifying nd inverting trnsltion typiclly employs computing brycenter of the model nd trnslting the brycenter to the origin of the coordinte system. Such normliztion of position my not work well if the shpe contins geometricl outliers. Similr problem could rise when normlizing rottion nd scle. A set of shpe fetures proposed by Osd et l [1] is inherently invrint to similrity trnsformtion so tht they don t require ny normliztion. Among the fetures, the D2 shpe distribution, one of the simplest to compute, performed the best in terms of computtionl cost nd retrievl performnce. Beside its geometricl trnsformtion invrince, their shpe fetures re quite robust ginst noise in, or even totl lck of, topology of polygons nd meshes. Their 3

4 shpe fetures re lso robust ginst geometricl noise nd degenercies such s zero-re polygons. The method proposed in this pper is n extension of Osd s method. We will describe the D2 shpe feture nd our shpe fetures in the next section. 2. Proposed Algorithm Figure 1 shows the structure of our proof-of-concept 3D model dtbse system. We dopted the query-by-3d-shpe-exmple pproch, in which user presents the system with n exmple 3D shpe nd sks for k most similr shpe models in the dtbse. The shpe feture extrction lgorithm ccepts 3D shpe defined s collection of polygons nd polygonl meshes. It my contin non-mnifolds or geometriclly degenerte polygons. If the surfce of the input model is known to be orientble, we employ mutul Angle-Distnce histogrm (AD) shpe feture. If we cn t ssume the surfces of the models to be properly nd consistently oriented, we employ mutul Absolute-Angle Distnce histogrm (AAD) shpe feture. Both re the extensions of the D2 [1] shpe feture. Unlike the D2, which is 1D histogrm, both our AD nd AAD re 2D histogrm. For the dissimilrity computtion mong shpe fetures, we compred few different methods bsed on L1 norm, L2 norm nd n elstic mtching lgorithm. The dtbse itself is orgnized s one-dimensionl rry, nd no ttempt hs yet been mde to speed up the dtbse ccess by indexing nd other methods. In the following, we first explin the D2 shpe function (Osd, et l. [Osd02]) on which our proposed shpe fetures re bsed. Then, our proposed shpe fetures, the AD nd the AAD will be described. 3D models Feture extrction Feture Feture 3D model dtbse Query Feture extrction Feture Distnce computtion 2.1. Osd s D2 User Results Retrieved results Figure 1. Structure of shpe similrity serch system for 3D models. Osd et l proposed severl different shpe fetures in [1]. Advntges of these shpe fetures 4

5 re tht (1) they cn be computed for 3D polygon soup model tht contin geometricl nd topologicl noise, errors, nd degenercies, nd tht (2) the shpe fetures is invrint, without pose normliztion, to similrity trnsformtion of the 3D model. Among the proposed shpe descriptors, the D2 performed the best in terms of combined computtionl cost nd retrievl performnce. To compute the D2 shpe feture for surfce-bsed 3D model, the model is first converted into n unoriented point set representtion by stochsticlly smpling the geometry of the model by generting points t rndom loction on every surfce of the model (See Figure 2). Distnce is N N 1 2 pirs for the then computed for every possible pir of points generted, tht is, p( p ) N p points generted. The D2 shpe function is 1D histogrm generted by counting the popultion of the point-pir distnces tht flls within certin distnce intervl. The D2 shpe feture is insensitive to the vrition in (or, totl lck of) connectivity of polygons. As it is bsed on the unoriented point set representtion, it is insensitive to the orienttion of the surfces in the originl model. p 1 p 2 d = ( p p ) () Surfce bsed model. (b) Generte point set model (c) Compute distnce d between the pir of points. (c) The D2 shpe function Figure 2. Computing the D2 shpe function from the surfce bsed 3D shpe model. We used Osd s method [1] to generte point t rndom loction P on the surfce of tringle. ( 1 r ) ( r1 r2) 2 r1( r2 3) P= t + t + t. (1) In the formul, t 1, t 2, nd t 3 re vertices of the tringle, nd r 1 nd r 2 re pseudo-rndom number sequences (PRNS) hving the rnge [0,1]. If the model contined non-tringulr polygons, they re tringulted prior to the point genertion. The number n i of points per ith tringle is determined in proportion to the re of the ith tringle by the following formul; M h = N S S i p i i i= 1, ( ) ni = hi + hi hi r3. Here, S i is the re of the ith tringle, M is the number of tringles for the model, nd r 3 [0,1] is PRNS. The integrl number n i is probbilisticlly generted from the expected vlue h i. Our implementtion of the D2 is somewht different from the originl D2, nd we cll our (2) 5

6 version the modified D2 (md2). Insted of PRNS used by Osd, we used Qusi-Rndom Number Sequence (QRNS) by Sobol [16] for the r 1, r 2 bove. The sptilly uniformity of distribution of points on polygons re better if QRNS is used, compred to the cse in which PRNS is used (See Figure 3.) Tht is, feture vectors generted by using QRNS [16] tend to be more consistent, tht is, low-vrince, thn those generted by using PRNS. In the preliminry experiment, we experimentlly compred the retrievl performnce of the D2 shpe feture computed by using the Sobol s QRNS [16] with tht of the feture generted by using the PRNS drnd48() function vilble in the stndrd C librry. The QRNS version performed better thn the PRNS version given number of point N p, especilly if the N p is smll. We lso performed the sme experiment for the AD nd the AAD shpe fetures, nd the QRNS versions of the AD nd the AAD performed better thn their PRNS counterprts. We thus chose the Sobol s QRNS for generting points for the md2 shpe feture s well s for the AD nd the AAD shpe fetures in the experiment described in Section 4. () Sobol s QRNS. (b) drnd48() PRNS. Figure 3. Plots of 2D points tht used the Sobol s qusi-rndom number sequence () compred with the plot tht used the stndrd pseudo-rndom number genertor function drnd48()(b). The Sobol s QRNS smples the rectngle more uniformly thn the PRNS drnd48(). For proper comprison mong the models hving different size, the distnce xis of the histogrm needs to be normlized. We normlize the histogrm by using the mximum, minimum, nd the verge of the point pir distnce. Of the totl I d intervls of the histogrm, I d / 2 eqully spced intervls (bins) re llocted for distnce vlues tht rnge from the minimum to the verge. The remining I d / 2 eqully spced intervls re llocted to the vlues from the verge to the mximum. Consequently, the size of intervls in the upper hlf (i.e., bove verge) of the histogrm is in generl different from tht of the intervls in the lower hlf (i.e., below verge) of the histogrm. L1 The dissimilrity D (, ) md2 x y of pir of models A nd B, whose fetures re 1D vectors x nd y, respectively, cn be computed s follows; I d L1 md2 (, ) = i i i= 1 D xy x y. (3) 6

7 Here, Id is the number of histogrm intervls for the distnce xis Shpe Fetures AD nd AAD The AD nd the AAD shpe feture re 2D histogrms of distnces nd ngles formed by pirs of oriented points tht re generted on the surfces of the given 3D shpe model. In computing the AD or the AAD shpe feture, n oriented point set representtion of the originl (surfcebsed) 3D shpe model is computed first. The orienttions of the point re inherited from the surfce norml vectors of the polygons on which the points re generted. The ngle between pir of points is ctully represented s n inner product of the orienttion vectors. The difference between the AD nd the AAD is tht the AAD ignores the sign of the inner product. Consequently, the AAD is more robust ginst models hving inconsistent surfce orienttions thn the AD AD The AD shpe feture mesures, for ech pir of points p 1 nd p 2, the 3D Euclidin distnce d = ( p ) 2 1 p 2 between the points nd the inner product = n1i n2 of the orienttion vectors n 1 nd n 2 of the points (Figure 4.) The points re generted in the mnner identicl to tht of md2, using the eqution (1) nd the Sobol s QRNS. Unlike the md2, ech point is oriented. Orienttion of point is inherited from the surfce norml vector of polygon on which the point is generted. Given the distnce nd the inner product for every pirs of the points, the AD is joint 2D histogrm of the distnce d nd the inner product. n 1 P 1 p 2 n2 () Surfce bsed model. (b) Convert the model into n oriented point set. d = = n1in2 ( p p ) (c) Compute distnce d nd the inner product. d (d) The AD shpe feture. Figure 4. Computing the AD nd AAD shpe function from the surfce bsed 3D shpe model. Similr to the D2 nd md2 shpe fetures, proper comprison of models hving different sizes requires normliztion of the distnce xis of the AD histogrm. We experimented with the four different normliztion methods; (1) by mximum, (2) by verge, (3) by medin, nd (4) by mode, of the distnce vlues. Normliztion by Mximum: The mximum nd the minimum of the distnce vlues re found. Then, the intervl the minimum distnce nd the mximum distnces is subdivided into d I equl width bins. Normliztion by Averge: An verge distnce is computed. Then, given the totl number of 7

8 distnce bins I d, hlf ( I d / 2 equl sized bins) is llocted to the distnce vlues bove the verge the other hlf (nother I d / 2 equl sized bins) is llocted to the distnce vlues below the verge. Normliztion by Medin or Mode: These two normliztion methods re similr to the normliztion by verge bove, except tht, insted of the verge, either the medin or the mode of the distnce is used. Inner product of pir of orienttion vectors lwys fll within the rnge [ 1,1], regrdless of the size of the model. The structures of the bins for the ngulr xis re the sme for ll the normliztion methods. The ngulr xis divides the intervl [ 1,1] into I equl sized bins. The result is 2D histogrm hving I Id elements for ll the normliztion methods. Exmples of the AD shpe fetures computed by using the four normliztion methods bove re shown in Figure 5-5d. In these figures, the drker the imge, the higher the popultion is. These fetures re computed for the bunny model of Figure 2 by generting 2,000 points on the surfce of the model for the 2D histogrm hving I I = 64 8 elements. d 8

9 +1-1 Minimum Mximum d () Normlized by using the minimum nd the mximum Minimum Averge Mximum d (b) Normlized by using the verge Minimum Medin Mximum d (c) Normlized by using the medin Minimum Mode Mximum d (d) Normlized by using the mode. Figure 5. Exmples of the AD shpe feture, 2D histogrm, generted by using four distnce normliztion methods AAD The AD shpe feture described bove is sensitive to the sign of the orienttion vector of the point set model. If the models to be compred hve consistent surfce orienttion, e.g., consistent trversl order of the vertices mong polygons, the AD shpe feture performs well. If, however, the dtbse contins models hving surfces tht re inconsistently oriented, the 9

10 performnce of the AD shpe feture suffers. Models generted by using different shpe modeling tools might hve different rules in determining surfce orienttions. Some of the models do not hve coherent surfce orienttion t ll. The mutul Absolute Angle nd Distnce (AAD) histogrm is computed similrly to the AD, except tht the AAD ignores the sign of the inner product. This mkes the AAD more robust shpe feture thn the AD for the models hving unoriented or inconsistently oriented surfce orienttions. The nglulr xis of the 2D histogrm of the AD tkes the vlue in the rnge [0,1]. Figure 6 shows n exmple of the AAD shpe feture computed for the sme bunny model of Figure 2 using the mximum vlue normliztion, N p = 2,000 points nd the histogrm size of I I = d +1 0 Minimum Figure 6. The AAD normlized by using the mximum. Mximum d 2.3. Dissimilrity computtion for the AD nd AAD We hve implemented nd compred three distnce computtion methods, the L1 norm (Mnhttn distnce), the L2 norm (Euclidin distnce), nd the elstic mtching distnce for the AD nd the AAD shpe fetures Dissimilrity mesures using L1-norm nd L2-norm Assume tht X = ( x i, j ) (1 i Id,1 j I ) nd Y = ( y i, j ) (1 i Id,1 j I ) re the feture vectors for the models A nd B, respectively. Note tht feture vector for the AD nd the AAD shpe fetures re in fct 2D mtrix of the dimension I d I, in which I d is the number of distnce intervls nd I is the number of ngulr (or, inner product) intervls. The L1 norm-bsed dissimilrity mesure D (, ) L1 XY nd the L2 norm-bsed distnce D (, ) L2 XY for the AD nd AAD shpe fetures re defined s follows; Id I L1 = i, j i, j i= 1 j= 1 D ( XY, ) ( x y ). (4) Id I 2 L2 ( XY, ) = ( i, j i, j). (5) i= 1 j= 1 D x y The distnce xis is treted differently from the ngulr xis. The L1 or L2 distnce mong pir of column vectors, ech of which consisting of vlues from ngulr bins t the distnce bin i, 10

11 is computed first. Then, simple sum of these distnce vlues over ll the I d intervls is computed Dissimilrity mesure using elstic mtching We lso computed elstic mtching distnce long the distnce xis to compute the distnce D (, ) E XY between the shpe fetures X nd Y. It loclly stretches nd shrinks the distnce xis of the histogrm in order to find miniml distnce mtch. The elstic mtching lgorithm employs dynmic progrmming technique for n efficient implementtion. In the pst, elstic mtching in the temporl xis hd been used extensively in speech recognition in order to bsorb vrition in the speed of utternce. If the mtching is too elstic, pir of shpes tht re different could produce smll distnce vlue. We compred the performnce of liner nd qudrtic penlty functions, nd chose the better performing qudrtic penlty function, s shown in eqution (8). The elstic distnce D (, ) E XY mong pir of fetures X nd Y is computed s follows. 3. Experiments nd results D (, ) = g(, ) XY X Y, (6) E n n ( Xn, Yn 1) + g( Xn, Yn) ( 1 1) ( ) ( X, Y ) + g( X, Y ) g g( X, Y ) = min g X, Y + 2 g X, Y g n 1 n n n n n n n n n ( ) ( ) 2 i, j i, k j, k I k = 1, (7) g X Y = i j x y. (8) To evlute the proposed shpe fetures, we implemented the proof-of-concept 3D shpe similrity dtbse system using C++ on Linux operting system Evlution method For the experiment, we used three different dtbses, ech with its own correct nswer ctegories. Dtbse A: The dtbse A consisted of 215 VRML models provided by Ptrick Min nd Prof. Funkhouser t the Princeton University. The model dtbse is ctegorized priori into 42, which re listed in Tble 1. Dtbse B: The dtbse B consisted of 1,213 VRML models we hve collected, modified, or creted. We combined the dtbse A bove with the models from John Tngelder et l t the University of Utrecht [17]. We lso collected more thn 300 copyright free models from the Internet. To generte similr but different models, we modified some of the models (e.g., bunny model) by using our shpe morphing lgorithm, mesh simplifiction lgorithm, or by dding geometricl noise. We lso creted dozen or so new models, e.g., bunny house model tht contins the bunny model in cube. Bsed on consensus mong 11

12 few grdute students, we clssified the models into 35 ctegories listed in Tble 2. The 35 ctegories included the other ctegory contining s mny s 352 difficult to clssify models. The dtbse B contins disproportiontely lrge number of irplne models from the Utrecht dtbse [17]. Dtbse C: The dtbse C is the Princeton Shpe Benchmrk (PSB), publicly vilble 3D model dtbse [18]. The PSB version 1.0 consists of 1,814 models, which is divided into two groups; the trining dtbse (907 models) nd the test dtbse (907 models). We used the test dtbse only for the experiment described in this pper. The 92 ctegories in the dtbse C re listed in Tble 3. The retrievl experiment is s follows. We pick query model q from ctegory C q, nd sk the system to find, in dtbse, models similr to q. If retrieved model r Cq, it is success retrievl. If r Cq, then it is filure. (In the cse of the dtbse B, we drew query models from ctegories other thn the other ctegory. Consequently, the lrger the size of the other clss, the lower vrious performnce figures become.) Note lso tht mny of the ctegories contined smll number (2, 3, 4, or 5) of models. As the objective mesures of retrievl performnce, we used the First Tier (FT), Second Tier (ST), nd Nerest Neighbor (NN), s well s the recll-precision plot. First Tier (FT): Assume tht the query belongs to the clss C q contining k = Cq models. The FT figure is the percentge of the models from the clss C q in the top ( k 1) mtches. As the query model is excluded, ( k 1) models from the clss C q in the top ( k 1) results produces the figure 100%. Second Tier (ST): The ST figure is the percentge of the models from the clss C q in the top 2( k 1) mtches. Nerest Neighbor (NN): The percentge of the cses in which the top mtch is drwn from the query s clss C. q Recll nd precision re well known in the literture of content-bsed serch nd retrievl. Precision is the number of retrieved models tht re in the clss C q divided by the number of ll the retrieved models. Recll is the number of retrieved models tht re in the clss C q divided by the number of models in the clss C q. In generl, recll nd precision re in trde-off reltionship. If one goes up, the other usully comes down. As the objective of this dtbse is similrity bsed serch, if the similrity mtching criteri is rther strict, the precision vlue goes up, while the recll vlue goes down. On the other hnd, if the mtching criterion is too loose, most of wht hs been retrieved is useless. The FT, ST, NN figures s well s the recll-precision plots shown lter re the verges produced by querying every model in the dtbse once. As n exception, in the cse of the dtbse B, models in the other ctegory re not queried. 12

13 Tble 1. The 42 ctegories creted for the test dtbse A, which contins 215 models. Clss nme C q Clss nme C q Clss nme C q niml 5 hed 4 plnt 3 bll 6 helicopter 6 rifle 4 bed 2 humn 17 shrk 2 belt 3 lmp 9 sktebord 3 bicycle 2 leg 2 sof 4 blimp 3 lightning 3 spceship 6 bot 4 mechnoids 2 sub 3 bookshelf 2 misc 3 tble 4 box 6 missile 7 tnk 5 building 2 mug 5 tiefighter 2 cr 7 openbook 4 tools 5 chir 9 pen 5 torus 2 clw 3 phone 4 tower 3 glove 2 plne 37 vse 4 Tble 2. The 35 ctegories creted for the test dtbse B, which contins 1,213 models. Clss nme C q Clss nme C q Clss nme C q irship 5 delt-jet 74 plne2 49 niml4legs 22 dolphin 11 plne3 90 bll 33 hed 23 plne4 54 biplne 24 helicopter 17 plne5 55 bord-circulr 6 holes 9 plne6 55 bord-thick 16 humnoid 26 plne7 53 bord 16 lmp 12 sof 9 bot 10 missile 11 submrine 8 bunny 23 mug 4 sword 9 cr 19 multi-fuselge 54 tnk 7 chir 6 office-chir 8 other 352 cube 8 plne

14 Tble 3. The 92 ctegories used in the dtbse C, which is the test dtset contining 907 models of the Princeton Shpe Benchmrk [PSB] dtbse. Clss nme C q Clss nme Cq Clss nme C q Biplne 14 Book 4 Vse 11 Commercil 11 Brn 5 Milbox 7 Fighter_jet 50 Church 4 Electricl_guitr 13 Glider 19 Gzebo 5 Newtonin_toy 4 Stelth_bomber 5 One_story_home 14 Bush 9 Hot_ir_blloon 9 Skyscrper 5 Flowers 4 Helicopter 18 One_pek_tent 4 Potted_plnt 26 Enterprise_like 11 Two_story_home 10 Brren 11 Flying_sucer 13 Chess_set 9 Conicl 10 Stellite 7 City 10 Stellite_dish 4 Tie_fighter 5 Desktop 11 Lrge_sil_bot 6 Ant 5 Computer_monitor 13 Ship 11 Butterfly 7 Door 18 Submrine 9 Humn 50 Eyeglsses 7 Billbord 4 Humn_rms_out 20 Fireplce 6 Sink 4 Wlking 8 Cbinet 9 Slot_mchine 4 Flying_bird 14 School_desk 4 Stircse 7 Stnding_bird 7 Bench 11 Hmmer 4 Dog 7 Dining_chir 11 Shovel 6 Horse 6 Desk_chir 15 Umbrell 6 Rbbit 4 Shelves 13 Rce_cr 14 Snke 4 Rectngulr 25 Sedn 10 Fish 17 Single_leg 6 Covered_wgon 5 Se_turtle 6 Geogrphic_mp 12 Motorcycle 6 Axe 4 Hndgun 10 Monster_tuck 5 Knife 7 Ht 6 Semi 7 Sword 16 Hourglss 6 Jeep 5 Fce 16 Ldder 4 Trin_cr 5 Hnd 17 Streetlight 8 Wheel 4 Hed 16 Glss_with_stem 9 Ger 9 Skull 6 Pil 4 14

15 3.2. Selecting Prmeters for the Proposed Shpe Fetures The AD nd the AAD contins three prmeters tht ffect their performnce s well s computtionl cost; the number of points generted on ech model N p nd the number of distnce intervls I d nd ngulr intervls I. In this experiment, we vried these three prmeters to find best performing combintion of prmeters without too much in computtion. For the AD, we tested set of 27 prmeter combintions; N p = { 512,1024, 2048}, I d = { 32,64,128}, I = { 4,8,16}. For the AAD, we evluted nother set of 27 prmeter combintions; N p = { 512,1024, 2048}, I d = { 32,64,128}, I = { 4,8,16}. In both of the cses, histogrms re normlized by using the verge, nd the distnce mong fetures re computed by using the L2 norm. We compred the results using the retrievl performnce in terms of FT, ST, NN, nd the computtionl cost. If performnces of two prmeter combintions re equl, we chose the one hving lower computtionl cost. In terms of the number of points, N p = 1024 performed better thn N p = 512. But the performnces of N p = 1024 nd N p = 2048 re indistinguishble. We chose N p = 1024 for its lower computtionl costs. The numbers of distnce intervls I d nd I ffected retrievl performnce. If they re too smll, the feture becomes insensitive to shpe differences. If they re too lrge, the shpe feture my be overly sensitive to minute shpe differences, decresing the overll similrity serch nd retrievl performnce. And, the lrger the number of intervls, the higher the storge cost nd the distnce computtion cost. For exmple, when we incresed the I from 8 to 16, the retrievl performnce of the AD nd the AAD shpe feture decresed. Wht hppened ws tht the fetures hve become too sensitive. For exmple, two polygonl pproximtions of sphere, one hving 20 fcets nd the other hving 80 fcets re determined to be different, lthough they re in the sme bll ctegory. Overll, the combintion of prmeters we found to be the best re shown in Tble 4. Tble 4. Prmeters selected for the shpe feture vectors. Fetures N p AD AAD I d I 3.3. Performnce Comprison mong the proposed methods We compred the performnce of vrious vritions of our proposed methods using the dtbse A. We performed three sets of comprisons; (1) Comprison mong the two shpe fetures AD nd AAD, (2) Comprison mong the four histogrm normliztion methods (by mximum, by verge, by medin, nd by mode), nd (3) Comprison mong the three distnce computtion methods, the L1-norm, the L2-norm, nd the elstic mtching distnce. 15

16 Tble 5 shows the results of the comprison (1) bove. In this experiment, we used the verge normliztion method for the histogrm normliztion nd the L2 norm for the distnce computtion. The figure shows tht the AAD hs the higher NN vlue, while its FT vlue is slightly lower, thn the AD. Due to its smller feture vector size (64 4 insted of 64 8), the AAD is somewht fster thn the AD t the dtbse serch step. Tble 6 shows the results of the comprison (2) bove tht compred mong the histogrm normliztion methods. For this experiment, the AAD shpe feture nd the L2 norm re used. The result showed tht the verge bsed normliztion performed the best. In terms of computtionl cost, mximum normliztion ws the lest expensive, followed closely by the verge normliztion method. Tble 7 shows the results the comprison mong the three distnce computtion methods (comprison (3) bove.) In this experiment, we used the AAD shpe feture nd the verge bsed normliztion method. The figures in the tble show tht the L2 norm performed the best. The elstic mtching did improve the performnce for certin clsses. But overll, the simple L2 norm performed better thn the elstic mtching. Tble 5. Performnce comprison mong the proposed shpe fetures. (Dtbse A). Fetures FT ST NN Retrievl time AD 39% 51% 56% 0.84s AAD 38% 51% 60% 0.70s Tble 6. Performnce comprison mong the histogrm normliztion methods. (Dtbse A.) Feture Normliztion FT ST NN computtion methods time Mx-Min 36% 49% 58% 0.52s Averge 38% 51% 60% 0.54s Medin 36% 48% 58% 0.60s Mode 33% 47% 54% 0.60s Tble 7. Performnce comprison mong the distnce computtion methods. (Dtbse A.) Distnce Retrievl computtion FT ST NN time methods L1 norm 38% 49% 58% 0.68s L2 norm 38% 51% 60% 0.70s Elstic mtching 37% 50% 54% 0.77s 16

17 3.4. Comprison with the other methods We compred the performnce of the AAD nd the md2 shpe fetures by using the dtbse A nd the dtbse B. The prmeters used for this experiment re s follows; md2: The number of points per model N p = 1024 nd the number of distnce intervls I d = 512. The normliztion is performed by using the verge bsed method [1]. Dissimilrity is computed by using the L1 norm. AAD: The number of points per model N p = 1024 nd the number of distnce intervls I d = 64, nd the number of ngle intervls I = 4. The normliztion is performed by the verge-bsed method. Dissimilrity is computed by using the L2 norm-bsed method. The prmeters for the md2 re chosen so tht its performnce is the highest in our preliminry experiment. The choice of the AAD shpe feture nd the choice of vrious prmeters of the AAD shpe feture re due to the experiments described in the Section 3.3. The size, counted in number of numericl vlues to be stored, of feture for the md2 is 512, while tht for the AAD is 64 4 = 256. The FT, ST nd NN figures resulted re shown in Tble 8, Tble 9, nd Tble 10, respectively, for the dtbse A, the dtbse B, nd the dtbse C. The recll-precision plots re shown in Figure 7, Figure 8, Figure 9, respectively, for the dtbse A, the dtbse B, nd the dtbse C. Tble 8 nd Tble 9 lso show computtionl costs in two prts; (1) the feture computtion time, nd (2) the totl retrievl time, which is the sum of the feture computtion time nd distnce computtion time. Tble 8 shows tht, using the dtbse A, the AAD methods outperformed the md2 in ll of the FT, ST, nd NN figures by the mrgin of 6% to 7%. Recll tht these numbers re computed s verges over ll the models nd ctegories in the dtbse. The AAD lso outperformed the md2 in the experiments tht used the dtbse B nd the dtbse C tht re shown in Tble 9 nd Tble 10. The performnce dvntge of the AAD is pprent in the recll-precision plots shown in Figure 7, Figure 8, nd Figure 9 for the dtbse A, B, nd C, respectively. (In recllprecision plot, curve closer to the upper right corner mens better retrievl performnce. Idel retrievl performnce would be the precision vlue of 1.0 for ll the recll vlues.) The plots re smoother for the Figure 8 nd Figure 9 thn the Figure 7. This is becuse the dtbse A hs much smller size thn the dtbse B nd C. In terms of computtionl cost, the feture extrction costs more for the AAD thn the md2. This modest increse in cost is well justified considering the performnce dvntge of the AAD. Furthermore, when it comes to the cost of distnce computtion, the AAD cost less to compre thn the md2 due to its smller feture size. Figure show query exmples using the md2 (Figure 10, 11, 12, nd 13) nd the AAD (Figure 10b, 11b, 12b, nd 13b) using the dtbse B. In ech figure, the upper left entry is the query, nd the 5 by 4 mtrix to the right shows the top 20 mtches. Of the top 20 retrievls, the upper left corner shows the best mtch, which in every cse is the query model itself. In these exmples, the AAD ppers to perform better thn the md2. In Figure 10, for exmple, the AAD ppers to retrieve more chir-like models thn the md2. In the exmple of Figure 11 nd Figure 12, compred to the md2, the AAD retrieved models tht pper to hve smooth nd continuous surfces orienttions.. 17

18 Note tht, in these figures, some of the imges of the retrieved models contined blck surfces. Exmples of blck surfces cn be found in irplne models in Figure 11. These re bckwrdfcing polygons hving flipped surfce norml vectors. In these irplne models, one of the wings is mde of bckwrd-fcing polygons presumbly becuse of the mirror nd copy opertion used during the modeling process. Tble 8. Comprison mong the md2 nd the AAD shpe fetures using the dtbse A (215 models). Performnce Computtionl cost Fetures Feture Retrievl FT ST NN computtion totl md2 33% 44% 47% 0.41s 0.70s AAD 38% 51% 60% 0.54s 0.70s Tble 9. Comprison mong the md2 nd the AAD shpe fetures using the dtbse B (1215 models). Performnce Computtionl cost Fetures Feture Retrievl FT ST NN computtion totl md2 20% 31% 37% 0.40s 2.00s AAD 24% 35% 43% 0.52s 1.37s Tble 10. Comprison mong the md2 nd the AAD shpe fetures using the dtbse C (907 models) (Computtionl costs re not vilble.) Performnce Fetures FT ST NN md2 19% 27% 36% AAD 25% 35% 47% 18

19 1 0.8 md2 AAD Precision Recll Figure 7. The recll-precision plot of the md2 nd the AAD shpe fetures using the dtbse A (215 models) md2 AAD 0.6 precision recll Figure 8. The recll-precision plot of the md2 nd the AAD shpe fetures using the dtbse B (1213 models). 19

20 1 0.8 md2 AAD 0.6 precision recll Figure 9. The recll-precision plot of the md2 nd the AAD shpe fetures using the dtbse C (907 models). 20

21 () md2 (b) AAD Figure 10. The retrievl exmple using the md2 nd the AAD shpe fetures for the dtbse B. 21

22 () md2 (b) AAD Figure 11. The retrievl exmple using the md2 nd the AAD shpe fetures for the dtbse B. 22

23 () md2 (b) AAD Figure 12. The retrievl exmple using the md2 nd the AAD shpe fetures for the dtbse B. 23

24 () md2 (b) AAD Figure 13. The retrievl exmple using the md2 nd the AAD shpe fetures for the dtbse B. 24

25 4. Summry nd conclusion In this pper, we proposed nd evluted pir of shpe fetures for shpe similrity serch of 3D models. The shpe fetures, clled the AD nd the AAD, re robust ginst topologicl nd geometricl irregulrities nd degenercies, which mke them pplicble to VRML nd other so clled polygon soup models. They re lso invrint to similrity trnsformtion, qulity vluble in compring 3D shpe models. While the AD nd the AAD hve computtionl cost somewht higher (bout 1.5 times) thn the D2, they significntly outperformed D2 in our retrievl experiments. Although direct comprison hs not been mde, the AD nd the AAD might hve the performnce lower thn tht of the more elborte methods, such s [13] nd [14]. However, the computtionl costs of AD nd AAD re significntly lower thn these methods. The AD nd AAD could thus be useful for quick pre-screening of 3D shpes. As future work, we would like to improve our shpe feture, for exmple by dding some form of multi-resolution pproch to mtching 3D shpes. We lso would like to explore hybrid shpe feture tht combines, possibly dptively, shpe fetures hving different chrcteristics. Acknowledgements We thnk Prof. Thoms Funkhouser nd Ptrick Min for providing us with the model dtbse. We thnk Prof. Shigeo Tkhshi for insightful discussion nd providing us with the softwre toolkit gmtools on which prt of our experimentl system is bsed. This reserch hs been funded by the Ministry of Eduction, Culture, Sports, Sciences, nd Technology of Jpn (No ), s well s by the grnts from Okw Foundtion for Informtion nd Telecommunictions, nd Artificil Intelligence Reserch Promotion Foundtion. 25

26 References [1] R. Osd, T. Funkhouser, Bernrd Chzelle, nd Dvid Dobkin Shpe Distributions, ACM TOG, 21(4), pp , (October 2002). [2] D. Keim, Efficient Geometry-bsed Similrity Serch of 3D Sptil Dtbses, Proc. ACM SIGMOD Int. Conf. On Mngement of Dt, pp , Phildelphi, PA., [3] D. McWherter, M. Pebody, W. Regli, A. Shokoufndeh, Trnsformtion Invrint Shpe Similrity Comprison of Solid Models, Proc. ASME DETC 2001, September 2002, Pittsburgh, Pennsylvni. [4] S. Muki, S. Furukw, M. Kurod, An Algorithm for Deciding Similrities of 3-D Objects, Proc. ACM Symposium on Solid Modeling nd Applictions 2002, Srbrücken, Germny, June [5] M. Hilg, Y. Shingw, T. Kohmur, nd T. Kunii, Topology Mtching for Fully Automtic Similrity Estimtion of 3D Shpes. Proc. SIGGRAPH 2001, pp , Los Angeles, USA [6] T. Zhri, F. Préteux, Three-dimensionl shpe-bsed retrievl within the MPEG-7 frmework, Proceedings SPIE Conference 4304 on Nonliner Imge Processing nd Pttern Anlysis XII, Sn Jose, CA, Jnury 2001, pp [7] E. Pquet nd M. Rioux, Nefertiti: Query by Content Softwre for Three-Dimensionl Dtbses Mngement, Proc. Int l Conf. on Recent Advnces in 3-D Digitl Imging nd Modeling, pp , Ottw, Cnd, My 12-15, [8] M. T. Suzuki, T. Kto, H. Tsukune, 3D Object Retrievl bsed on subject mesures, Proc. 9th Int l Conf. nd Workshop on Dtbse nd Expert Systems Applictions (DEXA98), pp , IEEE-PR08353, Vienn, Austri, Aug [9] T. Zhri, F. Préteux, Shpe-bsed retrievl of 3D mesh models, Proc. IEEE ICME 2002, Lusnne, Switzerlnd, August, [10] M. Eld, A. Tl, S. Ar., Content bsed retrievl of VRML objects: n itertive nd interctive pproch, Proc. sixth Eurogrphics workshop on Multimedi 2001, pp , [11] R. Ohbuchi, T. Otgiri, M. Ibto, T. Tkei, Shpe-Similrity Serch of Three-Dimensionl Models Using Prmeterized Sttistics, proc. Pcific Grphics 2002, pp , October 2002, Beijing, Chin. [12] M. Ankerst, G. Kstenmuller, H-P. Kriegel, T. Seidl, 3D Shpe Histogrm for Similrity Serch nd Clssifiction in Sptil Dtbses, Proc. Int l Symp. Sptil Dtbses (SSD 99), Hong Kong, Chin, July [13] T. Funkhouser, P. Min, M. Kzhdn, J. Chen, A. Hldermn, D. Dobkin, D. Jcobs, A serch engine for 3D models, ACM TOG, 22(1), pp , (Jnury, 2003). [14] Ding-Yun Chen, Xio-Pei Tin, Yu-Te Shen, Ming Ouhyoung, On Visul Similrity Bsed 3D Model Retrievl, Computer Grphics Forum, Vol. 22, No. 3, pp , (2003) (lso s the proc. of EUROGRAPHICS 2003.) [15] Ryutrou Ohbuchi, Mstoshi Nkzw, Tsuyoshi Tkei, Retrieving 3D Shpes Bsed On Their Appernce, Proc. 5th ACM SIGMM Workshop on Multimedi Informtion Retrievl (MIR 2003), pp , Berkeley, Clifornic, USA, November [16] W. H. Press et l., Numericl Recipes in C-The Art of Scientific Progrmming, 2nd Ed., Cmbridge University Press, Cmbridge, UK,

27 [17] Tngelder, et l. [18] Princeton Shpe Benchmrk, 27