Compact Vectors of Locally Aggregated Tensors for 3D shape retrieval

Transcription

1 Euogaphics Wokshop on 3D Object Retieval (2013) S. Biasotti, I. Patikakis, U. Castellani, and T. Scheck (Editos) Compact Vectos of Locally Aggegated Tensos fo 3D shape etieval Hedi Tabia 1, David Picad 1, Hamid Laga 2,3 and Philippe-Heni Gosselin 1,4 1 ETIS/ENSEA, Univesity of Cegy-Pontoise, CNRS, UMR 8051, Fance 2 Phenomics and Bioinfomatics Reseach Cente, Univesity of South Austalia, Austalia 3 Austalian Cente of Plant Functional Genonomics (ACPFG), Austalia 4 INRIA Rennes Betagne Atlantique, Fance Abstact Duing the last decade, a significant attention has been paid, by the compute vision and the compute gaphics communities, to thee dimensional (3D) object etieval. Shape etieval methods can be divided into thee main steps: the shape desciptos extaction, the shape signatues and thei associated similaity measues, and the machine leaning elevance functions. While the fist and the last points have vastly been addessed in ecent yeas, in this pape, we focus on the second point; pesenting a new 3D object etieval method using a new coding/pooling technique and poweful 3D shape desciptos extacted fom 2D views. Fo a given 3D shape, the appoach extacts a vey lage and dense set of local desciptos. Fom these desciptos, we build a new shape signatue by aggegating tenso poducts of visual desciptos. The similaity between 3D models can then be efficiently computed with a simple dot poduct. We futhe impove the compactness and discimination powe of the descipto using local Pincipal Component Analysis on each cluste of desciptos. Expeiments on the SHREC 2012 and the McGill benchmaks show that ou appoach outpefoms the state-of-the-at techniques, including othe BoF methods, both in compactness of the epesentation and in the etieval pefomance. Categoies and Subject Desciptos (accoding to ACM CCS): H.3.3 [ Infomation stoage and etieval]: Infomation Seach and Retieval I.5.4 [Patten ecognition]: Applications 1. Intoduction Databases of 3D models available in the public domain have ceated the need fo shape analysis and etieval algoithms capable of finding simila shapes in the same way a seach engine esponds to text and image queies. State-of-the-at 3D model etieval methods [TV08] often stat by extacting local featues and desciptos, then aggegating the featues into compact signatues, and finally compae the signatues with standad distances. Recently, Bag of Featues (BoF) epesentations, that have been widely adopted by the compute vision community fo image etieval and scene undestanding, ae gaining populaity in 3D shape analysis and etieval [BBGO11, LGS10, TDVC10]. One of the advantages of using BoF epesentations [JPD 12] is that one can benefit fom the ich liteatue of poweful local desciptos, such as SIFT [Low04], Histogam of Oiented Gadients (HoG) [DT05], and many othes [MS05]. Second, BoF epesentations can be compaed with standad distance measues. Thid, although BoF vectos can have lage dimensions when used fo etieval in lage databases [JPD 12], they ae often spase and inveted lists can be used to implement efficient seach [JPD 12, SZ03]. BoF-based appoaches stat by building a dictionay of K visual wods fom a set of taining samples. The visual wods ae usually obtained by k-means clusteing of the local featues extacted fom all the 3D models in the taining set. Each 3D model is then epesented with the statistics of the distibution of the visual wods in the 3D model. These vecto epesentations can be compaed with standad distances, and be subsequently used by obust classification methods such as Suppot Vecto Machines. Most of BoF-based 3D etieval techniques poposed so fa in the liteatue epesent a 3D model with a vecto of fequencies of occuences of visual wods [LG09, OOFB08, LGS10, TCF10, BBGO11, Lav12]. This coesponds to 0-ode statistics of the distibution of

2 the visual wods. In this pape, we exploe the usage of highode statistics and show that this enables the computation of compact desciptos while outpefoming 0-ode statistics in tems of etieval and classification pefomance. Ou appoach stats with the extaction of dense featues fom each shape. The advantage of using dense desciptos, in contast to few desciptos computed at spase locations, is well acknowledged by the compute vision community as it enables efficient 3D model etieval and classification. The gain in pefomance comes at the expense of significant incease in computation time and memoy equiement. In this pape we study and evaluate fou descipto aggegation pocedues, namely: (1) the standad BoF appoach based on 0- ode statistics, (2) the Vectos of Locally Aggegated Tensos (VLAT) method [NPG12b], which sums tenso poducts of the local desciptos, (3) Pincipal Component Analysis (PCA)-based VLAT desciptos, which educe the size of the dictionay while impoving futhe thei discimination powe, and (4) we popose to futhe incease the compactness of the educed-size signatue by pojection in a well chosen sub-dimensional space. Ou evaluation of these appoaches on 3D geneic shapes and on 3D non-igid shapes show that aggegation with high ode statistics significantly outpefoms standad BoF appoaches both in tems of etieval pefomance and in tems of compactness of the epesentation. The emainde of the pape is oganized as follows. In section 2, the elated woks ae pesented. In section 3, the method is detailed. Then, in Section 4 the expeiments ae pesented. Conclusions and futue developments (Section 5) end the pape. 2. Related wok While Bag of Featues appoaches ae vey popula in 2D image analysis, few woks have been intoduced fo 3D object ecognition. Most of them ae staightfowad extension of the 2D appoaches to 3D data. Existing appoaches diffe in the type of local featues used in the constuction of the visual vocabulay, and in the way these featues ae aggegated into signatues. Some of the appoaches epesent a 3D model by a set of 2D views which ae indexed using bags of 2D SIFT featues [OOFB08, LGS10]. Othe techniques use featues extacted diectly on the suface of 3D shapes. Liu et al. [LZQ06] and Li and Godil [LG09] use Spin Image desciptos computed on a dense set of featue points unifomly sampled on the suface of the 3D model. Toldo et al. [TCF10] segment the shape into egions and descibe each egion with seveal desciptos. A 3D shape is then modeled as a histogam of sub-pats occuences. Segmentation-based BoF have the advantage of captuing some semantics of the shapes. Tabia et al. [TDCV12] epesent each 3D object as a vecto of weighted occuences of featues. In all these woks, spatial elationships between featues ae lost in the constuction of the signatues. Lavoue [Lav12] and Bonstein et al. [BBGO11] add spatial elationships to take into account the spatial infomation between featue points. Most of BoF-based 3D etieval appoaches focus on the type of featues and little attention is given to the way these featues ae aggegated into signatues. Consequently the BoF vecto that esults fom the epesentation is of vey high dimension, paticulaly when dealing with lage-scale databases. Ou main contibution in this pape is a new step fo descipto aggegation into compact signatues, while impoving the high discimination powe of the descipto. The challenge in using dictionay-based appoaches is to find a good tadeoff between discimination powe, the size of the descipto, and the scalability to lage object databases. Fist, most of the methods used in 3D etieval epesent a 3D model with a histogam of fequency of occuences of the visual wods in the model. This is a diect adaptation fom the basic BoF appoach used by the compute vision and patten ecognition communities fo image etieval and categoization [SZ03]. To the best of ou knowledge, othe vaiants of BoF appoaches have neve been applied to the poblem of 3D model etieval. Fo example, these methods have been genealized using coding/pooling schemes, and achieve good pefomances in image categoization with the same featue size, using Locality-constained Linea Coding (LLC) [WYY 10]. Repesenting a 3D model with a vecto of fequencies of occuences of the visual wod coesponds to 0-ode statistics of the distibution of the visual wods. In this pape, we exploe the usage of high ode statistics. The most popula method is the Fishe Vectos (FV) [PSM10]. FV descibes how the set of desciptos deviates fom an aveage distibution of desciptos, modeled by a paametic geneative model, usually a Gaussian Mixtue Model. The FV achieves bette esults than BoF appoaches [CLVZ11], but at the cost of a lage index and highe seach time. Jege et al. [JPD 12] poposed a set of appoximation and compession techniques so that FV can be used on lage image datasets. Negel et al. [NPG12b] build efficient signatues by lineaizing the kenel function on bags. A Kenel function on bags aims at computing a similaity analog to votebased systems, but with espect to Mece s condition. Thei huge computational complexity can be highly educed using lineaization techniques based on tensos, called Vectos of Locally Aggegated Tensos (VLAT). VLAT pefomances ae compaable to Fishe Vectos [NPG12b]. The size of the dictionay poduced by VLAT can be significantly educed using Compact VLAT [NPG12a]. In this pape, we study fo the fist time the pefomance of these diffeent vaiants of BoF techniques on 3D model etieval tasks. We will show that Vectos of Locally Aggegated Tensos (VLAT) and Pincipal Component Analysisbased VLAT outpefom significantly the standad BoF techniques in tems of etieval pefomance. We popose also Compact VLAT, which educes significantly the size of the shape descipto, and show that it achieves elatively good pefomance which makes it suitable fo boad classification

3 Figue 1: Method oveview: Fist we compute depth images of the model captued fom cameas localized on the unit sphee. Then, local featues (e.g. HoG) ae extacted fom the images. Finally, we compute the VLAT signatue using deviations between covaiance matices of the codebook and covaiance matices of the desciptos. of shapes. We evaluate the pefomance of these desciptos on two diffeent types of databases: the SHREC 2012 Geneic 3D Shape Retieval benchmak [LGA 12] and the Mcgill dataset [SZM 08] fo aticulated 3-D models. 3. Method oveview Figue 1 gives an oveview of the poposed method. Fist in a pe-pocessing step, we nomalize the models to ensue that the extacted desciptos ae invaiant to tanslation and scale. Then, we ende depth maps of the object fom n views unifomly sampled on the suface of a bounding unit sphee. We epesent each depth map as a collection of dense Histogam of Oiented Gadients (HoG) desciptos. HoG have the advantage of being compact and easy to compute. We then build a shape signatue by aggegating the desciptos using the BoF paadigm. In this pape we exploe and evaluate vaious BoF schemes, namely: (1) the standad BoF appoaches, which descibe a 3D shape as a vecto of fequencies of occuences of the visual wods, (2) the Vecto of Locally Aggegated Tensos (VLAT), which have been oiginally poposed fo image analysis [NPG12a]. They have the advantage of being compact and efficient fo lage-scale image etieval, (3) we futhe educe the size of desciptos constucted with VLAT, while impoving futhe thei discimination powe, by using Pincipal Component Analysis (PCA) on the VLAT vectos, and (4) finally, we poposed a new vesion called compact VLAT, which builds a compact vecto and show that it is suitable fo ough classification of 3D shapes. The methods poposed in this pape have seveal advantages. They ae invaiant to igid tansfomations and some aticulated defomations and ae obust to geometical and topological noise. They ae obust to the level of tessellation of 3D models. They handle any type of 3D shape epesentations as long as depth images can be endeed. The impotant deviation fom the state-of-the-at is that they ely on compact dictionaies (256 o 512 visual wods) but achieve significantly bette pefomance than classical BoF methods Dense featue extaction Pio to featue extaction, we need to poceed to a obust nomalization of pose and scale of the 3D objects in ode to emain invaiant to two geometical tansfomations (tanslation, scaling). Hee, we do not pefom the pose nomalization fo otation because the locations of local featues ae completely ignoed in ou method. Fo the cente and the scale, we use the smallest enclosing sphee [LGS10]. The use of the smallest enclosing sphee has seveal advantages: it is fast to compute, it allows the maximization of the model size inside the unit sphee. We epesent a 3D object with a set of depth maps captued by vitual cameas distibuted unifomly aound the object, see Figue 2. In ode to captue all the impotant featues of the object, we use a lage numbe of views (80 in ou implementation) and captue depth images of size Fom the depth images, we extact a dense set of HoG desciptos on a dense egula gid. Evey two pixels we compute one HoG descipto at fou diffeent scales (16 16, 24 24, and pixels). We then obtain a lage unodeed set of local desciptos. Let B i = {b i, = 1...n} be the set of desciptos extacted fom the depth images of model i. Unlike [LGS10], we do not maintain a sepaate set of desciptos fo each view. Instead, we put all the desciptos in a single bag and use them to build ou compact dictionay. This makes the descipto invaiant to otations of the 3D model. Also, given a sufficiently lage numbe of views and a densely sampled HoG desciptos, we insue that the most impotant featues of an object ae captued by the epesentation. One method to map the set of desciptos into a single vecto that can be used fo indexation is the Bag of Featues (BoF) [SZ03]. It involves the constuction of a visual codebook (visual wods) and the count of occuences of these wods in the 3D model. Applying in a staightfowad manne the taditional BoF paadigm to 3D models that ae epesented with a lage and dense set of desciptos will esult in lage dictionaies. We popose to use Vectos of Locally Aggegated Tensos (VLAT) ecently poposed in [NPG12a] fo image

4 Figue 2: Example of 3D models fom the McGill (models in the two fist ows) and the Shec12 (models in the two othe ows) data-sets. The fist column shows endeings of the models, while the othe columns show depth images extacted andomly fom the coesponding models. analysis to build compact yet vey disciminative shape signatues using a small dictionay size VLAT: Tenso based aggegation Let B i = {b i } and B j = {b j }, = 1...n, be two bags of featues epesenting the sets of desciptos in two 3D shapes i and j. An effective method to compute the similaity between two bags is based on kenel functions. Thanks to mathematical popeties like Mece conditions, these kenel functions on bags can be used with many poweful kenelbased leaning techniques. The novelty in this pape is that we conside a kenel function on bags fo each cluste c: K(B i,b j ) = c K B (B ic,b jc ) (1) with B ic = {b ic } the desciptos of model i that belong to cluste c. K B is a kenel on the bags B ic and B jc, which we define as the sum of Gaussian kenels on each pai of desciptos belonging to B ic and B jc. K B (B ic,b jc ) =,s e 1 σ 2 bic b jcs 2 (2) Computing a Gaussian kenel on each pai of desciptos is computationally pohibitive, especially fo lage bags (O(n 2 ) fo a dictionay of size n). Thus ou second idea is to lineaize the Kenel functions on bags. To do so, we fist nomalize all the desciptos to have a unit length. Then we expand it using Taylo seies, and finally lineaize the kenel function using tenso poducts: K B (B ic,b jc ) = e 2 2 b ic,b jcs σ 2 e σ 2,s = e 2 σ 2,s p = e 2 σ 2 p α p σ 2p b ic,b jcs p α p σ 2p p b ic, s p b jcs with p x the tenso poduct of ode p of the vecto x. When stopped at p = 2, the expansion coesponds to the second ode statistics of the set B ic. To extends the ange of the similaity measue, we popose to cente the obtained tensos by the mean tensos of each cluste c. The mean and tensos of the cluste have been leaned duing the dictionay constuction. We poceed as follows; Fist, we compute the mean descipto µ c and mean tenso matix T c of each cluste c: T c = 1 c i µ c = 1 c i b ic (3) (b ic µ c)(b ic µ c) T (4) whee c is the numbe of desciptos in cluste c and b ic ae the desciptos of model i that belong to c. The quantities µ c and T c povide statistical summaies of the shape cluste and captue the main vaiabilities. Next, fo evey 3D model i we compute one featue pe cluste c. We compute a tenso T ic as an aggegation of

5 the centeed tensos of centeed desciptos: T ic = (b ic µ c )(b ic µ c ) T c (5) whee µ c is the cente of cluste c. We flatten the matix T ic into a featue vecto x ic, the descipto of the 3D shape i with espect to cluste c. Finally, we concatenate all the vectos x ic of object i to fom a single featue X i = {x ic },c = 1...n, whee n is the size of the dictionay. X i is called the VLAT signatue of model i. Fo best pefomance, we pefom a nomalization step of the X i signatue. j,x i [ j] = sign(x i [ j]) X i [ j] α, (6) V i = X i X i (7) With α = 0.05 typically. VLAT pefoms vey good esults in similaity seach and automatic indexing of 3D objects with linea metic, but leads to lage featue vectos. The size of VLAT featues is n d d, whee n is the numbe of clustes and d is the dimension of the descipto PCA-based VLAT (PVLAT) To educe the dimension of the VLAT descipto, we pefom a tenso decomposition using Takagi s factoization: T c = A cd ca c (8) Whee D c is a eal non-negative diagonal matix containing the eigenvalues of T c and A c is unitay composed of eigenvectos of T c. Then we poject the centeed desciptos belonging to c on the eigenvectos: b ic = A c (b ic µ c ). (9) We can then deduce fom equation 5 the new signatue called PVLAT of the model i in cluste c as the sum of tensos of pojected desciptos b ic belonging to cluste c, centeed by D c: T ic = b icb ic D c. (10) Simila to VLAT, we concatenate and nomalize each cluste signatue. The size of the final PVLAT signatue depends on the numbe of eigenvecto selected in each cluste Compact PCA-based VLAT (CPVLAT) We popose to futhe educe the size of the PVLAT signatue while etaining its disciminative powe. Given a set S of N 3D models, we compute the Gam matix G of PVLAT signatues: G i, j = V i V j, i, j S (11) Then, we compute the eigenvalues and eigenvectos of the Gam matix G. Then, we compute a low ank appoximation of G. We denote by L t the matix with the t lagest eigenvalues on the diagonal: L t = diag(λ 1...λ t), (12) and we denote by Λ t the matix of the fist t eigenvectos: U t = [Λ 1...Λ t], (13) We can then define G t as an appoximation of G: G t = U tl tu t (14) Then, we compute the pojectos of PVLAT signatues in appoximated subspace: P t = VU tl 1/2 t (15) with V = [V 1...V N ] is the matix of PVLAT signatues. This method is analogous to Kenel-PCA using a dot poduct Kenel. Fo each 3D model, we compute the pojection of PVLAT in the sub-space as: Y i = P t V i (16) Y i contains an appoximate and low dimensional vesion of V i. The subspace defined by the pojectos peseves most of the similaity even fo vey small dimension because the PVLAT optimization has concentated the infomation in a small numbe of dimensions. One can notice that this pocedue is analogous to that of a kenel PCA with a linea kenel. 4. Expeiments and esults In ou method implementation, fo each 3D shape in the dataset, we captue a set of depth maps fom diffeent view points. To geneate this set of depth maps fo a model, we ceate 2D pojections fom multiple viewpoints. These viewpoints ae equally spaced on the unit sphee. In ou cuent implementation, we use 80 depth maps. Actually, each model was nomalized fo size by escaling it so that the aveage Euclidean distance fom points on its suface to the cente of mass is 0.5. Then, all models wee nomalized

6 fo tanslation by moving thei cente of mass to the oigin. Then, fo each depth map, we extact HOG featues on a dense gid, one HOG featue evey two pixels, with fou diffeent scales 16 16, 24 24, and pixels. Fo dictionay constuction, we use the K-Means Algoithm. To evaluate ou method, we used two diffeent 3D model databases. The fist one is the SHREC 2012 Geneic 3D Shape Retieval benchmak [LGA 12] which is a standad shape benchmak widely used in shape etieval community. The dataset contains 1200 thee-dimensional models, classified into 60 object categoies based mainly on visual similaity. We also pesent ou esults on a non-igid database, the Mcgill dataset povided by Siddiqi et al. [SZM 08] fo aticulated 3-D models. It contains 255 objects divided into ten classes (Ant, Cabs, Hands, Humans, Octopuses, Plies, Snakes, spectacles, Spides and Teddy). Each class contains simila 3D shapes unde a vaiety of poses. We conducted fou diffeent expeiments. In the fist one (Table 1), we evaluate the pefomance of the thee poposed signatues and analyze the effect of the dictionay size on thei pefomances. In the second expeiment (Table 2), we compae the pefomance of the poposed signatues to standad bag of featues ones. In the Thid expeiment, we compae ou methods to state-of-the-at methods benchmaked in SHREC 2012 [LGA 12]. Finally, we investigate the obustness of the poposed PVLAT signatue against non-igid tansfomations. The fist thee expeiments wee conducted on the SHREC 2012 [LGA 12] dataset. In the last expeiment we descibe esults on the McGill dataset. To objectively evaluate ou method we use a statistical tool povided by the SHREC 2012 and the McGill datasets to compae 3-D etieval methods. Given a classification and a distance matix computed with any shape matching algoithm, a suite of tools poduces statistics and visualizations that facilitate the evaluation of the match esults. It includes five evaluation metics, namely: the Neaest Neighbo (NN), Fist-tie (1-Tie), Second-tie (2-Tie), E-Measues and Discounted Cumulative Gain (DCG). We also epot the pecision-ecall gaph Geneic shapes We fist show the contibution of the VLAT, PVLAT and CPVLAT signatues to etieval pefomance on geneic 3D shapes fom SHREC2012 [LGA 12]. Fom Table 1, we see that ou signatues have the same behaviou with dictionay size change. Poposed signatues ae faily stable when moving fom 256 to 512 visual wods. We can also see fom Table 1 that the PVLAT based method pefoms the best, followed by the PCVLAT and the VLAT one. We obseve a gain of 5.75% between VLAT and PVLAT which highlights the impovements bought by the signatue optimization. Note that the size of the VLAT featue is n d d, with n the numbe of clustes and d the size of desciptos. That means, when d = 128 and n = 512, the size of the VLAT signatue is 8M. When using the PVLAT, the size of the signatue is educed accoding to the numbe of eigenvecto selected in each cluste. Fo the CPVLAT, the size of the signatue depends on the low ank appoximation of the Gam matix. Hee, we dastically educe its dimension by keeping only 128 dimensions. So that, the final CPVLAT size is set to 128. Table 2 shows the pefomance of ou method compaed to standad BoF appoach. We can clealy see that ou method achieves significantly bette etieval pefomance. With only a dictionay of size 512 ou method achieves 0.83 in the Neaest Neighbou measue against 0.68 fo a standad BoF with a dictionay of visual wods. Table 3 shows a compaative evaluation of ou method and five othe methods pesented in [LGA 12]. As one can see ou method pefoms bette than LSD-sum, 3DSP and ZFDR. The DG1SIFT pefoms the best, followed by DVD+DB. Note that DG1SIFT combines thee diffeent methods fo descipto sampling: dense egula sampling, andom sampling, and one global SIFT descipto. DG1SIFT also uses the BoF paadigm, with a vocabulay size exceeding 13k, which is much highe than the appoach we popose in this pape. These expeiments show that ou appoach achieves good etieval pefomances on standad 3D databases, while using a compact signatue with few hundeds of dimensions. This makes ou appoach suitable fo etieval in web-scale databases. Figue 3 pesents the Pecision vs Recall plots of ou method and the state of the at methods fom [LGA 12]. It is inteesting to notice that ou method and the ZFDR one pesent quite compaable pefomances, howeve ou method s pecision is slightly highe fo low ecall values. Ou method clealy outpefoms LSDsum and 3DSP methods Aticulated Shapes In ode to evaluate the pefomance of ou 3D shape etieval appoach on non-igid 3D models, we use the McGill Aticulated Shape Benchmak database. We have compaed the pefomance of the PVLAT method with thee ecent algoithms on the McGill Database: Hybid BoW appoach fom [Lav12], the gaph-based appoach fom Agathos et al. [APP 09] and the hybid 2D/3D appoach fom Papadakis et al. [PPT 08]. Table 4 pesents the etieval pefomance of the thee algoithms compaed to ou PVLAT method. Fom this table, one can notice that the gaph-based algoithm [APP 09] pefoms the best. This is because of the obustness of the stuctual epesentation used in the algoithm against the aticulation defomations. Howeve, not only the computation complexity of gaph matching algoithms, gaph based epesentation have a limited disciminating powe when seaching fo geneic shapes, because only topology is taken into account. Moeove, fo gaph based methods, mino changes in topology may esult in significant diffeences in similaity. So that, they cannot be applied to abitay meshes, because topological poblems like holes distub the computation of the effective stuctue of the objects. Although the PVLAT method does not conc The Euogaphics Association 2013.

7 Methods Dictionay size (n) NN 1-Tie 2-Tie e-measue DCG VLAT VLAT PVLAT PVLAT CPVLAT CPVLAT Table 1: Poposed method pefomances with espect to the dictionay size. Methods Dictionay size (n) NN 1-Tie 2-Tie e-measue DCG BOF BOF BOF BOF VLAT PVLAT CPVLAT Table 2: Compaison with Bag of Featues. Pecision Bai(LSD_sum) Li(ZFDR) Redondo(3DSP_L2_1000_hik) Tatsuma(DVD_DB_GMR) Yanagimachi(DG1SIFT) VLAT PVLAT CVLAT We have pesented a novel appoach to geneic 3D object etieval using featue vectos constucted fom local desciptos of object depth maps. In ou appoach, the desciptos ae aggegated using Vectos of Locally Aggegated Tensos technique which showed to be a good appoximation of insightful similaity measues between desciptos. We futhe educe thei size using Pincipal Component Analysis (PCA) on the VLAT vectos, while impoving futhe thei discimination powe. We also popose to incease the compactness of the educed-size signatue by pojection in a well chosen sub-dimensional space. This subspace is obtained by a low ank appoximation of the Gam matix on a taining set. The appoach has been evaluated on two standad 3D shape etieval benchmaks, demonstating the method is suitable fo geneic shapes and poduces vey high etieval accuacy even with pesence of non-igid tansfomations. A lot of futue wok emains. As the poposed signatues ae vey poweful and highly compacted, we plan to test it on web-scale datasets Recall Figue 3: Pecision ecall gaph fo ou appoach on the SHREC 2012 dataset compaed to state-of-the-at gaphs. Acknowledgements This wok has been patly suppoted by the Cultue3DCloud poject. Hamid Laga is suppoted by the South Austalian State Govenment though its Pemies Science and Reseach Fund. side the stuctual infomation of the shape, it obtains vey compaable esults to the gaph-based algoithm and slightly bette than the two othe algoithms. The obustness of the PVLAT unde non-igid shapes can be pobably due to the local HOG featue s obustness against the small isometic tansfomations. 5. Conclusion Refeences [APP 09] AGATHOS A., PRATIKAKIS I., PAPADAKIS P., PERANTONIS S. J., AZARIADIS P. N., SAPIDIS N. S.: Retieval of 3d aticulated objects using a gaph-based epesentation. In 3DOR (2009), Euogaphics Association, pp , 8 [BBGO11] BRONSTEIN A. M., BRONSTEIN M. M., GUIBAS L. J., OVSJANIKOV M.: Shape google: Geometic wods and expessions fo invaiant shape etieval. ACM Tans. Gaph. 30 (2011). 1, 2 [CLVZ11] CHATFIELD K., LEMPITSKY V., VEDALDI A., ZIS- SERMAN A.: The devil is in the details: an evaluation of ecent featue encoding methods. In BMVC (2011), vol. 76, pp

8 Methods NN 1-Tie 2-Tie e-measue DCG DG1SIFT [LGA 12] PVLAT DVD+DB [LGA 12] ZFDR [LGA 12] CPVLAT VLAT DSP_L3_200_hik [LGA 12] LSD-sum [LGA 12] Table 3: Compaison with state-of-the-at methods as epoted in the shape etieval context SHREC 2012 [LGA 12]. Methods NN 1-Tie 2-Tie DCG Gaph-based algoithm [APP 09] PVLAT Hybid BoW algoithm [Lav12] Hybid 2D/3D algoithm [PPT 08] Table 4: Results on McGill dataset. [DT05] DALAL N., TRIGGS B.: Histogams of oiented gadients fo human detection. In Poceedings of the 2005 IEEE Compute Society Confeence on Compute Vision and Patten Recognition (CVPR 05) - Volume 1 - Volume 01 (2005), CVPR 05, pp [JPD 12] JEGOU H., PERRONNIN F., DOUZE M., SANCHEZ J., PEREZ P., SCHMID C.: Aggegating local image desciptos into compact codes. IEEE Tans. Patten Anal. Mach. Intell. 34, 9 (2012), , 2 [Lav12] LAVOUÉ G.: Combination of bag-of-wods desciptos fo obust patial shape etieval. The Visual Compute 28, 9 (2012), , 2, 6, 8 [LG09] LI X., GODIL A.: Exploing the bag-of-wods method fo 3d shape etieval. In Poceedings of the 16th IEEE intenational confeence on Image pocessing (2009), ICIP 09, IEEE Pess, pp , 2 [LGA 12] LI B., GODIL A., AONO M., BAI X., FURUYA T., LI L., LÓPEZ-SASTRE R. J., JOHAN H., OHBUCHI R., REDONDO-CABRERA C., TATSUMA A., YANAGIMACHI T., ZHANG S.: Shec 12 tack: Geneic 3d shape etieval. In 3DOR (2012), Euogaphics Association, pp , 6, 8 [LGS10] LIAN Z., GODIL A., SUN X.: Visual similaity based 3d shape etieval using bag-of-featues. In Poceedings of the 2010 Shape Modeling Intenational Confeence (Washington, DC, USA, 2010). 1, 2, 3 [Low04] LOWE D. G.: Distinctive image featues fom scaleinvaiant keypoints. Int. J. Comput. Vision 60, 2 (Nov. 2004), [LZQ06] LIU Y., ZHA H., QIN H.: Shape topics: A compact epesentation and new algoithms fo 3d patial shape etieval. In Poceedings of the 2006 IEEE Compute Society Confeence on Compute Vision and Patten Recognition - Volume 2 (2006), CVPR 06, IEEE Compute Society, pp [MS05] MIKOLAJCZYK K., SCHMID C.: A pefomance evaluation of local desciptos. IEEE Tans. Patten Anal. Mach. Intell. 27, 10 (Oct. 2005), [NPG12a] NEGREL R., PICARD D., GOSSELIN P.: Compact tenso based image epesentation fo similaity seach. In ICIP (Olando, Floida, USA, Septembe 2012). 2, 3 [NPG12b] NEGREL R., PICARD D., GOSSELIN P.: Using spatial pyamids with compacted vlat fo image categoization. In ICPR (Tsukuba Science City, Japan, Novembe 2012). 2 [OOFB08] OHBUCHI R., OSADA K., FURUYA T., BANNO T.: Salient local visual featues fo shape-based 3d model etieval. In Shape Modeling Intenational (2008), pp , 2 [PPT 08] PAPADAKIS P., PRATIKAKIS I., THEOHARIS T., PAS- SALIS G., PERANTONIS S.: 3d object etieval using an efficient and compact hybid shape descipto. In 3DOR (2008), Euogaphics Association. 6, 8 [PSM10] PERRONNIN F., SÁNCHEZ J., MENSINK T.: Impoving the fishe kenel fo lage-scale image classification. In ECCV (2010), pp [SZ03] SIVIC J., ZISSERMAN A.: Video google: A text etieval appoach to object matching in videos. In Poceedings of ICCV 03 (Washington, DC, USA, 2003), pp , 2, 3 [SZM 08] SIDDIQI K., ZHANG J., MACRINI D., SHOKOUFAN- DEH A., BOUIX S., DICKINSON S.: Retieving aticulated 3-d models using medial sufaces. Mach. Vision Appl. 19, 4 (2008), , 6 [TCF10] TOLDO R., CASTELLLANI U., FUSIELLO A.: The bag of wods appoach fo etieval and categoization of 3D objects. The Visual Compute 26, 10 (2010), , 2 [TDCV12] TABIA H., DAOUDI M., COLOT O., VANDEBORRE J.-P.: Thee-dimensional object etieval based on vecto quantization of invaiant desciptos. SPIE Jounal of Electonic Imaging 21, 2 (Apil-June 2012), [TDVC10] TABIA H., DAOUDI M., VANDEBORRE J.-P., COL- LOT O.: Local visual patch fo 3D shape etieval. In ACM Intenational Wokshop on 3D Object Retieval (in conjunction with ACM Multimedia 2010) (Fienze, Italy, Octobe ). 1 [TV08] TANGELDER J. W. H., VELTKAMP R. C.: A suvey of content based 3d shape etieval methods. In Multimedia Tools and Applications (2008), pp [WYY 10] WANG J., YANG J., YU K., LV F., HUANG T., GONG Y.: Locality-constained linea coding fo image classification. In CVPR (2010), pp