Combining Multiresolution Shape Descriptors for 3D Model Retrieval

Transcription

1 Ryutarou Ohbuch, Yushn Hata, Combnng Multresoluton Shape Descrptors for 3D Model Retreval, Accepted, Proc. WSCG 2006, Plzen, Czech Republc, Jan. 30~Feb. 3, Combnng Multresoluton Shape Descrptors for 3D Model Retreval Ryuatrou Ohbuch Unversty of Yamanash Takeda, Kofu-sh , Yamanash-ken, Japan Yushn Hata Unversty of Yamanash Takeda, Kofu-sh , Yamanash-ken, Japan ABSTRACT In ths paper, we propose and evaluate a systematc approach for mprovng performance of 3D model retreval by combnng multple shape descrptors. We explored two approaches for generatng multple, mutually ndependent, shape descrptors; (1) applcaton of a (sngle-resoluton) shape descrptor on a set of multresoluton shape models generated from a query 3D shape model, and (2) applcaton of multple, heterogeneous shape descrptors on the query 3D shape model. The shape descrptors are ntegrated va the lnear combnaton of the dstance values they produce, usng ether fxed or adaptve weghts. Our experment showed that both multresoluton and heterogeneous sets of shape descrptors are effectve n mprovng retreval performance. For example, by usng the multresoluton approach, the R-precson of the SPRH shape descrptor by Wahl, et al, mproved by 8%, from 29% to 37%. A combnaton of three heterogeneous shape descrptors acheved the R-precson of about 42%; ths fgure s about 5% better than the R-precson of 38% acheved by the Lght Feld Descrptor by Chen, et al., whch s arguably the best sngle shape descrptor reported to date. Keywords 3D model database, content-based retreval, geometrc modelng, multresoluton analyss, feature combnaton. 1. INTRODUCTION 3D shape models are ncreasngly popular n many applcaton domans, rangng from move specal effects, 3D games on cellular phones or on game consoles, to 3D mechancal CAD/CAE systems. The popularty has prompted research nto effectve management and reuse of 3D shape models by means of shape-based retreval of 3D models. An example of such database s the 3D Search engne at the Prnceton Unversty [Funkhouser03]. Typcal steps for shape smlarty based retreval of 3D models starts wth query specfcaton (See Fgure 1.) As queres, texts, 2D sketches, 2D mages, 3D sketches, and 3D shapes have been used n the past. Multple query specfcaton methods may be Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Conference proceedngs ISBN WSCG 2006, January 30-February 3, 2006 Plzen, Czech Republc. Copyrght UNION Agency Scence Press combned, as n the work of Funkhouser et al [Funkhouser03]. The next step s to extract feature, or shape descrptor (SD), from the query. Also, as a pre-computng step, SDs for 3D shape models n the database have been computed. The queryng method and the 3D shape representaton used for the database nfluences the shape descrptor and the smlarty (or more often, dstance) computaton method. Durng the dstance computaton, t s often desrable to reflect human judgment. The database retreves and presents the models most smlar to the query based on the dstances computed.... 3D models... Query Response Preprocessng Compute shape descrptor Compute shape descrptor Compute dstance Fgure 1. A typcal 3D model database system. Features vector 3D models database

2 Ryutarou Ohbuch, Yushn Hata, Combnng Multresoluton Shape Descrptors for 3D Model Retreval, Accepted, Proc. WSCG 2006, Plzen, Czech Republc, Jan. 30~Feb. 3, In ths paper, we explore an approach to boost shape smlarty retreval performance of a 3D model database by extractng more features from shape models. Our method uses two method to try to extract more shape feature from a query 3D model; (1) extracton of a multresoluton set of features by applyng a SD to the models havng multple resoluton levels generated by usng the method by Ohbuch, et al [Ohbuch03], and (2) extracton of a heterogeneous set of features by applyng multple, heterogeneous SDs to the model. Multple SDs are ntegrated va dstance value they produce by usng lnear combnaton, whch allows ntegraton of features that does not explctly produce feature vectors, namely, the Lght Feld Descrptor (LFD) [Chen03] by Chen, et al. Integraton of a set of dstances are done through ether fxed weght lnear combnaton of dstances, or knowledge based adaptve weght lnear combnaton of dstances. The latter s a modfcaton of the purty based method of Bustos, et al. [Bustos04a, Bustos04b]. Our experments showed that the multresoluton approach boosted performances of many, but not all, of the SDs we tested. Combnatons of multresoluton, heterogeneous shape descrptors produced the best performance of the combnatons we tested. For example, one of the combnatons produced R-precson of 42%, whch s about 5% hgher than the 38% acheved by the LFD by Chen, et al., whch s one of the best performng SDs accordng to Tangelder, et al. [Tangelder04]. Ths paper s organzed as follows. In the followng secton, we revew the prevous ntegrate features for 3D model retreval. In Secton 3, we descrbe the method to compute multresoluton SDs, and the method to ntegrate multple SDs by ther dstance values. A set of performance evaluaton experments and ther results are descrbed n Secton 4. We conclude the paper n Secton Prevous Work In the feld of content based search and retreval of 2D mages, t s typcal to extract more than one feature from an mage, and combne these features for an overall smlarty comparson. Thus t s natural to thnk of such an approach for the 3D model retreval. Ths approach has been taken by several groups; Iyer, et al. [Iyer03], Bustos et al [Bustos04a, Bustos04b], and Atmosukarto, et al. [Atmosukarto05]. Iyer et al. weghted and combned multple heterogeneous feature vectors, lettng the user control the weghts explctly through a user nterface or mplctly through a relevance feedback mechansm that employs nteractve learnng. The method by Atmosukarto et al. also weghted and combned heterogeneous feature vectors. Ther experments showed that combnatons of descrptors have better performance than any sngle shape descrptor they evaluated. Unlke the former two, the method by Bustos et al. ntegrated multple, heterogeneous features va dstance. Bustos computed the overall dstance between a par of models as a lnear combnaton of the dstances usng ether fxed or adaptve weghts. In the fxed weght combnaton, weghts are the same regardless of the model (or the model category.) In the adaptve weghts combnaton, weghts are computed by usng purty, whch s an estmate of the performance of the SD determned by usng a pre-classfed tranng database. Bustos et al. reported sgnfcant performance gan usng both fxed weght and adaptve weght lnear combnaton of dstances, although the adaptve one performed better. Integraton of multple shape descrptors can be acheved usng ether (1) feature vectors of the descrptors, f avalable (the approach by Iyer et al. and Atmosukarto, et al.), or (2) dstance values computed usng the descrptors (the approach by Bustos et al.). Integraton usng feature vectors potentally allows fne tunng of dstance computaton, e.g., by weghtng each element of the vectors. However, ths approach can t be used f a shape descrptor produces dstance but not feature vector. For example, one of the most powerful 3D shape descrptors, the LFD by Chen et al. [Chen03] produces dstance only, and s useful only for dstance based ntegraton. Fndngs by Atmosukarto et al. and by Bustos et al. are contradctory regardng whether the shape descrptor should be combned as feature vectors or as dstances. Atmosukarto reported that a combnaton of dstances does not produce any performance mprovement. Bustos et al., on the other hand, reported that combnatons of dstances are benefcal. Our fndng reported n ths paper agrees wth that of Bustos et al. 3. METHOD In ths secton, we descrbe the method to compute multresoluton shape descrptors and the method to combne multple shape descrptors by ther dstance values Computng multple shape descrptors To mprove retreval performance through feature ntegraton, features should capture as dfferent shape feature of the model as possble. Our method uses the

3 Ryutarou Ohbuch, Yushn Hata, Combnng Multresoluton Shape Descrptors for 3D Model Retreval, Accepted, Proc. WSCG 2006, Plzen, Czech Republc, Jan. 30~Feb. 3, followng two dfferent approaches to extract mutually ndependent shape descrptors. 1. Multresoluton approach: Generate a set of multresoluton (MR) shape models from a model to be compared. A (sngle-resoluton) shape descrptor s appled to the MR shape models to produce multple SDs. 2. Heterogeneous shape descrptor approach: Apply multple shape descrptors to the model to be compared. A combnaton of the two approaches above s also possble. For example, an MR representaton havng m levels may be combned wth n heterogeneous shape descrptors to produce m n shape descrptors Multresoluton shape descrptors The method by Ohbuch, et al. [Ohbuch03] compares 3D models at multple scales followng the steps below: 1. Compute a multresoluton representaton: The surface-based nput model s converted nto a pont-based model by Monte-Carlo samplng of the surfaces of the model. A set of L-1 scale values α ι, =1 L-1 s computed based on the sze of the model. The set of scale values are used to normalze sze among shape models. Then, compute L-1 3D alpha shapes from the pont set model by usng the L-1 scale values α. ι Of L shape features, those of the coarser (L-1) levels are computed by usng the 3D alpha shapes [Edelsbrunner94]. For the fnest resoluton level L, however, the orgnal, polygon soup model s used. Ohbuch et al calls ths set of multresoluton models Alpha- Multresoluton Representaton (AMR). 2. Compute multresoluton shape descrptors: Apply a shape descrptor x to a model at each resoluton level of the AMR, creatng a set of multresoluton shape descrptors AMR-x. 3. Compute dstance between a par of features: Compute a dstance between AMR-x shape descrptors of a par of models to be compared, usng a statcally weghted lnear combnaton of L dstances. An advantage of the AMR above s that t can be computed for polygon soup models or even for pont set models (wthout the step 1 above.) Fgure 3 shows an example of AMR representaton for a surface-based 3D model of rabbt. The AMR s remnscent of morphologcal multresoluton representatons for 2D mages. Orgnal model (surface-based) Generate pont-set Pot set model Compute α-shapes α values α ι at level computed based on the dameter of the model. Level 6 model = orgnal AMR Level = Compute feature x AMR-x shape-descrptor Level = Fgure 2. Computng the AMR-x multresoluton shape feature. A surface based model and ts pont set representaton usng 2,048ponts. Level 1(α 1 ) Level 2(α 2 ) Level 3(α 3 ) Level 4(α 4 ) Level 5(α 5 ) Level 6( org.) Fgure 3. An example of AMR representaton for the surface-based model of a rabbt Heterogeneous shape descrptors As the heterogeneous shape descrptors, we used the D2 by Osada, et al.[osada02], the AAD by Ohbuch et al. [Ohbuch05], the SPRH by Wahl, et al. [Wahl03], and the LFD by Chen, et al. [Chen03]. The D2 s a 1-dmensonal (1D) hstogram of dstances between every par of ponts generated on surfaces of a model. The AAD and the SPRH are both extensons to the D2 above. In addton to the

4 Ryutarou Ohbuch, Yushn Hata, Combnng Multresoluton Shape Descrptors for 3D Model Retreval, Accepted, Proc. WSCG 2006, Plzen, Czech Republc, Jan. 30~Feb. 3, dstance used n the D2, the AAD and the SPRH extract such features as the mutual orentaton of the par of ponts, resultng n a 2D hstogram for the AAD and a 4D hstogram for the SPRH. The LFD by Chen, et al. s dfferent from all of the above. It compares smlartes of a set of mages generated from multple vewponts about the 3D model. Accordng to the survey paper by Tangelder et al. [Tangelder04], the LFD s by far the most powerful shape descrptor. 3.2 Integratng multple shape descrptors In combnng dstances produced by shape descrptors, our method normalzes the dstances frst, and then computes a weghted lnear combnaton of the dstances Normalzaton Pror to ntegratng dstances, the dstances are normalzed by ther standard devatons. Let U be the set of 3D models n the database, and o U be the model from the database, and SD, 1 N be the shape descrptors. For the shape descrptor SD, rhe average μ ( d ) and the standard devaton σ ( d ) are computed for the databaseu. Let d ( q, o ) be the dstance pror to normalzaton computed usng the SD for the model par q and o. Then the normalzed dstance dˆ ( q, o ) for the SD s computed as below. ˆ 1 d( q, o) μ ( d ) d ( q, o) = σ ( d ) (1) Fgure 4 shows examples of hstograms of dstances for the four SDs we have used. The dstance axs s normalzed to ts maxmum dstance value (=100%). It can be seen that the normalzaton usng standard devaton would perform better than the normalzaton usng maxmum dstance. Frequency D2 AAD SPRH LFD Normalzed dstance [%] (max.=100%) Fgure 4. Dstrbuton of dstances for the SDs. Integratng shape descrptors usng normalzed dstance usng the formula (1) above should perform better than our prevous method descrbed n [Ohbuch03], whch employed no dstance normalzaton at all n ntegratng SDs generated by usng the AMR. Also, we expect our normalzaton method usng standard devaton of dstances to be more robust aganst outlers than the normalzaton method usng maxmum dstance employed by Bustos, et al. [Bustos04a] Weghted Lnear Combnaton Our method computes the overall dstance dqo (, ) as the lnear combnaton of N normalzed dstances d ˆ ( q, o ) usng the followng formula. N (, ) ˆ (, ) d q o = wd q o (2) = 1 We compared two dfferent methods to determne the weghts w. fx. 1. Fxed weghtng: weghts w are predetermned and fxed across the query. 2. Adaptve weghtng: weghts w are adaptvely vared accordng to the query and ts (estmated) category. Usng fxed weghts has two drawbacks. One s that the fxed weghts won t produce the best performance across all the classes. Assumng a precategorzed database, the performance of a SD vares dependng on the class the query model (s supposed to) belongs to. That s, for example, one SD s good at queryng human fgures whle another SD s good at queryng arplanes. The other s that expermentally fndng a best set of weghts for a gven set of SDs and a database can be computatonally expensve for a nontrval number of SDs and models. An adaptve weghtng method that adapts to the query model and/or to the class n whch the query model s lkely to belong s qute mportant. For the adaptve weghtng, we adopted Bustos s purty based weghtng scheme [Bustos04a, Bustos04b] wth a few mnor modfcatons. The purty s an estmate of goodness of a SD n characterzng a category n a gven database. The dea behnd the purty s the maxmum nformaton gan crtera for selectng an attrbute n splttng a set n the decson tree learnng algorthm. The purty assumes that the database of the 3D models s preclassfed nto M classes. Let S be the number of k models retreved from a class C k, 1 k M, usng the shape descrptor SD. The purty purty( SD, q, t ) s computed as below for a shape descrptor SD, query q, and a postve nteger constant t.

5 Ryutarou Ohbuch, Yushn Hata, Combnng Multresoluton Shape Descrptors for 3D Model Retreval, Accepted, Proc. WSCG 2006, Plzen, Czech Republc, Jan. 30~Feb. 3, k (,, ) max( S ) purty SD q t = (3) 1 k M In other words, the purty for the SD s hgher f the SD returned more model from the category C k n the top t retrevals, regardless of the class. Usng the purty( SD, q, t ), the adaptve weght w between the query q and the 3D model o U s computed as follows. n w = purty( SD, q, t) (4) Bustos, et al. used a slghtly dfferent w. Our verson could have a larger dfference n weghts, dependng on the selecton of the purty power parameter n. w = purty( SD, q, t) 1 (5) In the followng, we call our weghtng method purty* and Bustos s orgnal adaptve weghtng method purty. 4. EXPERIMENTS AND RESULTS We mplemented our own versons of the D2 [Osada02] and the SPRH [Wahl03] shape descrptors. We used our orgnal mplementaton of the AAD [Ohbuch05] shape descrptor. For the LFD [Chan03], we used the executable provded by the orgnal authors of the LFD paper [Chan03] found at ther web ste. In our prevous work [Ohbuch03], we combned the AMR only wth the AAD shape descrptor. In ths work, we combne the AMR approach wth all the four shape descrptors lsted above. For the performance evaluaton experments, we used the Prnceton Shape Benchmark (PSB) [Funkhouser03] database. It contans the total of 1914 models, dvded nto the 907 model tranng set and the 907 model test set. Each set s classfed nto 90+ base classes. We used the most detaled base classfcatons for the experment. As quanttatve measures of performance, we used the R-precson (1R), the 2R-precson (2R), the 11 pont average precson (11P) fgures, and the precson-recall plot [Baeza-Yates99]. The R- precson s the rato, n percentle, of the models retreved from the desred class C k (.e., the same class as the query) n the top R retrevals, n whch R s the sze of the class C k. The 2R-precson s smlar to the R-precson, except that the fgure s computed usng the top 2R retrevals. In computng the R- and 2R-precson values, the query q s not counted as the retreved model. That s, the q s drawn from the database U, the 1R and 2R values are computed by usng Ck 1. The 11-pont average 11P s the average of precson values taken at 11 equally spaced recall values {0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}. A 11P average precson value can be consdered as a summary of the recallprecson plot, whch emphaszes overall performance. The 1R and the 2R values favor methods havng hgher precson for the near the top retrevals Multresoluton shape descrptors In ths experment, we evaluated the effectveness of ntegratng multresoluton shape descrptors. We frst compared the retreval performances of the shape descrptors at dfferent resoluton levels. Interestngly enough, as shown n Table 1, the most detaled models (the orgnal models) may not acheve the hghest retreval scores. In the cases of the D2 (not shown), the AAD, and the SPRH, retreval usng coarser scales (the level 5, and to a lesser degree, the level 4), produced better 1R score than usng level 6 (.e., most detaled) models. For the LFD, however, retreval usng the most detaled models produced the hghest 1R score. A possble explanaton for ths s that the LFD favors shape detals of the models and that the corners and edges of the convex hull models nterfere wth the performance of the shape descrptor. Note also that retreval usng the coarsest level (.e., convex hull) models performed surprsngly well. As shown n Table2 and n Fgure 5, for some of the SDs, combnatons of multresoluton (MR) shape descrptors usng ether the fxed or the adaptve weghts sgnfcantly outperformed ther sngleresoluton (SR) counterparts. In the case of the AAD, 1R fgure mproved by 10% from 24.4% for the SR verson to 34.9% for the MR verson usng the purty* weghtng. The performance of the SPRH ncreased from 28.6% (SR) to 36.6% (MR), approachng the 38.0% of the LFD. The performance of the LFD, however, dd not mprove due to the MR combnaton. Also, adaptve weghtng usng the purty* showed small but consstent advantage over ther fxed weghtng counterparts for all of the four SDs tested. Table 1. R-Precson vares across resoluton levels. As the numbers show, the most detaled model may not be the best choce for retreval. Resoluton R-precson [%] levels D2 AAD SPRH LFD Org=

6 Ryutarou Ohbuch, Yushn Hata, Combnng Multresoluton Shape Descrptors for 3D Model Retreval, Accepted, Proc. WSCG 2006, Plzen, Czech Republc, Jan. 30~Feb. 3, Table 2. The multresoluton (MR) versons of D2, AAD, and SPRH outperform ther sngle resoluton (SR) counterparts. Such s not the case for the LFD. SDs Weghts 1R 2R 11P D Fxed, MRD2 Fxed, Adaptve, purty* AAD Fxed, MRAAD Fxed, Adaptve, purty* SPRH Fxed, MRSPRH Fxed, Adaptve, purty* LFD Fxed, MRLFD Fxed, Adaptve, purty* Weghtng methods We compared the performance of the purty* based weghtng for dfferent values of the parameter purty power n n the equaton (2). We also compared the performance of our purty* and the orgnal purty by Bustos, et al. [Bustos04a]. Table 3 shows the performance of the MRLFD- MRAAD-MRSPRH combnaton usng dfferent values of purty power n. In terms of 1R scores, n = 3.0 produced the best 1R performance, and n = 9 produced the best 11P performance. Table 4 compares the Bustos s purty wth our purty* for ther retreval performance. In all the combnatons tested, our purty* usng the selected parameter n performed better than the orgnal purty of Bustos et al. Our purty* performed better probably because the weght can have a larger dynamc range. As a dsadvantage, our scheme requres a search for the bester value n, although the search should be relatvely easy for t s a 1D search space. Table 3. Effects of purty power n on the retreval performance for the MRLFD-MRAAD-MRSPRH combnaton. N 1R 2R 11A Bustos s purty Purty* Purty* Purty* Purty* Purty* Table 4. Comparson of retreval performance between Bustos s purty v.s. our purty*. Descrptors Weghts 1R 2R 11P MRAAD purty purty* n= MRSPRH purty purty* n= SPRH-AAD purty purty* n= LFD-AAD purty purty* n= LFD-SPRH purty purty* n= LFD-SPRH- purty AAD purty* n= Combnaton of heterogeneous shape descrptors In ths experment, we compared the performance of ntegrated SDs usng both multresoluton and heterogeneous combnatons of varous SDs. Table 5 summarzes the experment. In all the results lsted, we used the purty* adaptve weghtngs. In each combnaton, a best performng parameter n s chosen out of the 10 canddates values of n={1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0}. The Fgure 6 shows the recall-precson plot of fve combnatons selected from the Table 5. Table 5 and Fgure 6 clearly show that combnng multple, heterogeneous shape descrptors va dstances could produce sgnfcant performance gan compared to any one of the sngle resoluton SDs. As a reference, the LFD has the 1R precson of 38%. The combnaton of LFD, AAD, and SPRH produced nearly 5% performance gan over that of the LFD, resultng n the 1R precson of 42.5%. Table 5. Performance comparson of some of the combnatons tested usng both heterogeneous and multresoluton shape descrptors. All the combnatons used the purty* adaptve weghtng. Shape descrptors 1R 2R 11P LFD SPRH MRSPRH SPRH+AAD MRSPRH+MRAAD LFD+AAD LFD+MRAAD LFD+SPRH LFD+MRSPRH LFD+AAD+SPRH LFD+MRAAD+MRSPRH MRLFD+MRAAD+MRSPRH

7 Ryutarou Ohbuch, Yushn Hata, Combnng Multresoluton Shape Descrptors for 3D Model Retreval, Accepted, Proc. WSCG 2006, Plzen, Czech Republc, Jan. 30~Feb. 3, On the other hand, combnatons of the LFD wth the MR shape descrptors dd not produce consstent and sgnfcant performance gan. The reason for ths may be attrbuted to the AMR model, the purty* adaptve weghtng scheme, or both. Further study s needed to determne the exact cause. 5. CONCLUSION AND FUTURE WORK In ths paper, we explored a systematc approach for mprovng shape-based retreval performance of 3D shape models. Our approach s to (1) extract as many (mutually ndependent) shape features as possble, and (2) combne the dstances computed usng these features by usng an adaptve weghtng scheme. To extract dfferent shape features, the method employed a combnaton of multresoluton shape descrptors and heterogeneous shape descrptors. We used the 3D alpha-shape [Edelsbrunner94] based method we have proposed prevously [Ohbuch03] to capture multresoluton shape features. To combne dstances computed usng these shape descrptors, we adopted Bustos s purty weghtng scheme wth slght modfcaton. Experments showed that the proposed method of ntegratng multresoluton and heterogeneous shape descrptors s effectve n mprovng retreval performance. Many combnatons of the shape descrptors we tested surpassed the performance of the arguably the best (sngle) shape descrptor, the Lght Feld Descrptor by Chen, et al. [Chen03]. We ntend to explore better adaptve weghtng schemes and better multresoluton shape feature extracton approaches. 6. ACKNOWLEDGMENTS Ths research s funded, n part, by the Grants-n-Ad for Scentfc Research from the Japan Socety for the Promoton of Scence (No ). 7. REFERENCES [Atmosukarto05] I. Atmosukarto, W.K. Leow, Z. Huang, Feature Combnaton and Relevance Feedback for 3D Model Retreval, Proc. MMM 2005, pp , (2005). [Bustos04a] B. Bustos, D. Kem, D. Saupe, T. Schreck, D. Vranć, Automatc Selecton and Combnaton of Descrptors for Effectve 3D Smlarty Search, Proc. IEEE MCBAR'04, pp , (2004). [Bustos04b] B. Bustos, D. Kem, D. Saupe, T. Schreck, D. Vranć, Usng Entropy Impurty for Improved 3D Object Smlarty Search, Proc. IEEE ICME 2004 (2004). [Chen03] D.-Y. Chen, X.-P. Tan, Y.-T. Shen, M. Ouhyoung, On Vsual Smlarty Based 3D Model Retreval, Computer Graphcs Forum, 22(3), pp , (2003). [Edelsbrunner94] H. Edelsbrunner, E. P. Mücke, Three-dmensonal Alpha Shapes, ACM TOG, 13(1), pp , (1994). [Funkhouser03] T. Funkhouser, P. Mn, M. Kazhdan, J. Chen, A. Halderman, D. Dobkn, D. Jacobs, A search engne for 3D models, ACM TOG, 22(1), pp , (2003). [Funkhouser04] Thomas Funkhouser, Mchael Kazhdan, Phlp Shlane, Patrck Mn, Wllam Kefer, Ayellet Tal, Szymon Rusnkewcz, Davd Dobkn, Modelng by ex-ample, ACM TOG, 23(3), pp (2004). [Iyer03] Iyer, N., Kalyanaraman, Y., Lou, K., Jayant, S., Raman, K., A Reconfgurable, Intellgent 3D Engneerng Shape Search System Part I: Shape Representaton, Proc. ASME DETC '03, 23rd CIE Conf. (2003). [Iyer05] M. Iyer, S. Jayant, K. Lou, Y. Kalyanaraman, K. Raman, Three Dmensonal Shape Searchng: State-of-the-art Revew and Future Trends, Computer Aded Desgn, 5(15), pp , [Lefman03] G. Lefman, S. Katz, A. Tal, R. Mer., Sgnatures of 3D Models for Retreval, The 4th Israel-Korea B-Natonal Conf. on Geom. Modelng and Comp. Graph., February 2003, (2003). [Ohbuch05] Ryutarou Ohbuch, Takahro Mnamtan, Tsuyosh Take, Shape-smlarty search of 3D models by usng enhanced shape functons, Internatonal Journal of Computer Applcatons n Technology (IJCAT), 23(3/4/5), pp , (2005). [Ohbuch03] R. Ohbuch, T. Take, Shape- Smlarty Comparson of 3D Shapes Usng Alpha Shapes, Proc. PG 2003, pp , IEEE Press, (2003). [Osada02] R. Osada, T. Funkhouser, B. Chazelle, D. Dobkn, Shape Dstrbutons, ACM TOG, 21(4), pp , (2002). [Shlane04] P. Shlane, P. Mn, M. Kazhdan, T. Funkhouser, The Prnceton Shape Benchmark, Proc. SMI 04, pp , (2004). [Tangelder04] J. Tangelder, R. C. Veltkamp, A Survey of Content Based 3D Shape Retreval Methods, Proc. SMI '04, pp [Wahl03] E. Wahl, U. Hllenbrand, G. Hrznger, Surflet-Par-Relaton Hstograms: A Statstcal 3D-Shape Representaton for Rapd Classfcaton, Proc. 3DIM 2003, pp , IEEE Press, (2003).

8 Ryutarou Ohbuch, Yushn Hata, Combnng Multresoluton Shape Descrptors for 3D Model Retreval, Accepted, Proc. WSCG 2006, Plzen, Czech Republc, Jan. 30~Feb. 3, D2 MRD2 AAD MRAAD SPRH MRSPRH LFD 0.6 Precson Recall Fgure 5. Performance gan due to the multresoluton approach. The multresoluton shape descrptors usng AMR ncreased performance of many but not all of the SDs LFD MRSPRH MRAAD-MRSPRH LFD-MRSPRH LFD-AAD-SPRH 0.6 Precson Recall Fgure 6. Precson-recall plots of some of the combnatons of descrptors. Some of the combnatons sgnfcantly outperform the LFD [Chen03], arguably the best sngle shape descrptor to date.