Spherical Correlation of Visual Representations for 3D Model Retrieval

Transcription

1 Noname manuscript No. (wi be inserted by the editor) Spherica Correation of Visua Representations for 3D Mode Retrieva Ameesh Makadia Kostas Daniiidis the date of receipt and acceptance shoud be inserted ater Abstract In recent years we have seen a tremendous growth in the amount of freey avaiabe 3D content, in part due to breakthroughs for 3D mode design and acquisition. For exampe, advances in range sensor technoogy and design software have dramaticay reduced the manua abor required to construct 3D modes. As coections of 3D content continue to grow rapidy, the abiity to perform fast and accurate retrieva from a database of modes has become a necessity. At the core of this retrieva task is the fundamenta chaenge of defining and evauating simiarity between 3D shapes. Some effective methods deaing with this chaenge consider simiarity measures based on the visua appearance of modes. Whie coections of rendered images are discriminative for retrieva tasks, such representations come with a few inherent imitations such as restrictions in the image viewpoint samping and high computationa costs. In this paper we present a nove agorithm for mode simiarity that addresses these issues. Our proposed method expoits techniques from spherica signa processing to efficienty evauate a visua simiarity measure between modes. Extensive evauations on mutipe datasets are provided. Keywords 3D shape retrieva Visua simiarity Spherica Fourier transform Ameesh Makadia Googe Research New York, New York, NY 111 E-mai: [email protected] Kostas Daniiidis Department of Computer Science University of Pennsyvania, Phiadephia, PA 1914 E-mai: [email protected] 1 Introduction Laser-scanned objects, CAD modes, and even imagebased reconstructions are just a few of the sources contributing to rapidy growing, pubicy avaiabe 3D mode coections. Aong with these vast 3D coections comes the need for a fast, arge-scae mode retrieva and matching system. At the core of any content-based mode retrieva engine ies the chaenge of computing 3D shape simiarity. Many of the difficuties in this task can be identified as either goba or oca. Any shape representation or simiarity measure must compensate for goba variations such as change in scae, orientation, etc. The second big chaenge ies in oca variations caused by object articuations or perturbations to oca surface geometry. Such variations can be attributed to noise or even modeing technoogy. For exampe, a poygona mesh obtained from aser data wi be quite different from a poygona mode of the same object designed by a human. 3D shape matching for retrieva has been a topic of ongoing research eading to many interesting techniques [1 11], a of which address the chaenges mentioned above to various degrees. In this work we are inspired by the famiy of methods which compare 3D modes based on their visua simiarity [1, 9]. For exampe, [1] has shown state-of-the-art retrieva and cassification resuts on standard benchmarks. The premise behind this approach is a 3D mode representation consisting of a coection of images rendered from various viewpoints. Athough it may be a bit surprising that a coection of 2D images provides better discrimination than features based on 3D geometric information, a coser ook wi revea two possibe reasons for the strength of visua representations. First, rendering

2 views of a mode circumvents the probem of deaing with compex, noisy, and possiby corrupt oca 3D surface geometry. Second, if two categories of 3D modes can be differentiated even by one particuar discriminative view, a sufficient samping of renderings is ikey to capture this distinguishing information. However, despite their high performance reative to other retrieva methods, the image-based methods present their own chaenges and imitations. For exampe, the Light Fied Descriptor (LFD) of [1] is a representation that is not invariant to mode orientation. Thus, there is a high computationa cost that comes with evauating a distance measure for many possibe rotationa aignments between modes. The inspiring idea of our work has first been drafted in [12], where a very preiminary formuation was deveoped. In this paper we present methods for 3D shape comparison and retrieva that are buit upon a visua representation of modes. Specificay, simiar to [1], our representation is a coection of sihouette images rendered from various viewpoints on the sphere surrounding the mode. We define mode simiarity as the cross-correation of these rendered sihouette image coections. Our primary contributions are in the formuation and efficient evauation of this cross-correation simiarity measure. We wi show how mode simiarity can be evauated efficienty using techniques from spherica harmonic anaysis, taking advantage of the fact that spherica correation is equivaent to mutipication in the spherica Fourier domain. Furthermore, our mode comparison method can be extended in a simpe and intuitive way to deveop an iterative, coarseto-fine mode retrieva system for arge coections of modes. A thorough experimenta evauation of our proposed methods is presented for mutipe chaenging datasets, and the resuts show consistenty state-of-theart or near-state-of-the-art performance. 2 Prior work Content-based 3D mode retrieva continues to be an important practica as we as fundamenta probem. From the practica perspective, many arge web repositories (e.g. 3D Warehouse 1 ) ignore shape content for mode search, which often eads to search resuts of imited success and appicabiity. Many researchers have proposed to address the probem of 3D simiarity for the task of mode retrieva, and what foows is a brief overview of just a few of the existing methods in the iterature. Goba spherica representations are the most natura (and common) representations for 3D modes. The 1 Extended Gaussian Image (EGI, [7]) was perhaps the eariest such representation, but there exist many others such as the Compex EGI [6], spherica distributions of shape area [5], radia distance functions [3, 4], and the Light Fied Descriptor [1], just to name a few. Less common is the case where the underying representation is a 3D grid (see [2,13] for exampes). A arge subset of methods based on spherica representations utiize a Spherica Fourier representation to buid mode descriptors (see [2, 4] for exampes). On the opposite end of the spectrum exist those methods where 3D shapes are represented by oca features. Spin images [14,15] and 3D Shape Contexts [16 18] are exampes where surface points are described by shape distributions of a oca neighborhood. Whie oca descriptors make it easier to dea with object articuations or missing parts, there is the added chaenge of obtaining accurate correspondences. Recenty [9] incorporated oca SIFT features [19] from rendered depth images into a traditiona document retrieva bag-of-features approach to circumvent the direct correspondence probem. One of the chaenges for shape matching is the wide variety of transformations that must be accounted for when comparing 3D modes. In this regard, most of the approaches we have mentioned above can be divided into two categories. The first category contains those approaches where invariance to the possibe transformations are buit directy into the mode representation or the extracted descriptors. The second category contains those approaches that address the possibe transformations of a mode at the time when mode descriptors are being compared. For exampe, the most common transformations to which any 3D retrieva engine must be invariant are goba changes in size (scae), position (3D transation), and orientation (3D rotation). Most of the approaches we have discussed above propose methods to generate 3D mode descriptors which have buit-in invariances (i.e. any scaing, transation, or rotation of the 3D mode wi not ater the resuting mode feature descriptor). There are a number of ways this can be achieved. The most direct is to use descriptors that are inherenty invariant to such transformations. For exampe, histograms of distances between point pairs [8], or histograms of distances from surface points to the center of mass [5], are invariant to both rotation and transation. For those methods where the underying representation is not invariant to certain transformations, simpe measures can be taken: Scae can be normaized by isotropic scaing of a mode to fix the average distance from surface points to the center of mass, for exampe. Transation can be normaized by shifting the mode so that the center of mass 2

3 aigns with the origin. A simpe way to factor out orientation is to use PCA-aignment, where the principa axes of a mode are aigned with some common reference frame. This type of PCA-aignment is commony used for spherica or 3D grid representations where the mode orientation is difficut to factor out. For spherica representations, an aternative to PCA-aignment is to extract genera properties of a spherica function that are invariant to 3D rotations. For exampe, it is weknown that the magnitudes of Spherica Fourier coefficient vectors are invariant to rotation (see [2] for an appication to 3D mode comparison). The benefit of encoding transformation invariance into 3D mode descriptors is that such features can be directy compared using traditiona distance measures. Nearest neighbor retrieva over thousands of modes is sti a fairy efficient computation when the pairwise distance measure is the L 2 distance between sma feature vectors, for exampe. Furthermore, it is straightforward to utiize powerfu cassification machinery (e.g. an SVM cassifier) with such features. The probem with encoding invariance directy into the descriptors is that it often comes at a cost. As a genera rue, the more invariance captured by a feature the ess discriminative the descriptor. Another probem comes from possibe inaccuracies in the methods. For exampe, orientation normaization using PCA-aignment has been shown to be inaccurate [2]. The aternative to buiding invariant descriptors is to address possibe mode transformations at the time of simiarity (or distance) computation. This aows one the freedom to buid very robust and discriminative features from 3D modes. However, the penaty is that there is a computationa disadvantage when descriptors are compared since the possibe transformations must be accounted for. Typicay this is addressed by an optimization or search over transformation parameters. For exampe, the Light Fied Descriptor [1] represents a 3D mode with a coection of rendered sihouette images. When two modes are compared their respective sihouette coections must be compared for a possibe 3D rotationa aignments. As our work in this paper buids on a visua mode representation, it is cosey reated to the Light Fied Descriptor of [1]. Thus, in the foowing section we wi summarize the approach of [1] and highight some of the existing imitations which are addressed in this paper. The method of [1] can be described as having three steps: First, given a 3D mode, a coection of sihouette images are rendered from mutipe viewpoints surrounding the mode. Second, features are extracted for each image. These features are used for pairwise comparison of images. Third, for the comparison of two 3D modes, the pairwise distances between the modes respective image coections are aggregated to provide a composite distance. This computation is then repeated for mutipe rotations, and the minimum composite distance is seected as the fina distance between the modes. In the foowing subsections we wi attempt to fi in many of the detais of this approach. 3.1 Sihouette viewpoints There are a few constraints which hep determine the viewpoints from which sihouette images are rendered given a 3D mode. Ideay, the viewpoints shoud be distributed uniformy over the sphere, to imit redundancy. Furthermore, the sihouettes from two different modes wi be compared pairwise for a set of 3D rotations. This impies that there must exist some 3D rotations which map any viewpoint onto another (transitivity), whie aso mapping the coection onto itsef. For a coection of N viewpoints (N > 2), the set of rotations that satisfy this constraint make up a finite subgroup of the 3D rotation group SO(3). The finite subgroups of SO(3) are the cycic groups, the dihedra groups, and the symmetry groups of the Patonic soids [21]. Athough the cycic and dihedra groups do not imit the number of sihouette vertices, the corresponding rotations wi cover ony a sma subspace of SO(3). The Patonic soid with the most vertices is the reguar dodecahedron (2 vertices). The dodecahedra group (often referred to by it s dua, the icosahedra group), has order 6. In other words, for the configuration of 2 vertices aigned with the vertices of a dodecahedron, there are 6 unique 3D rotations which wi map this set of 2 vertices onto itsef. We shoud note that in practice ony 1 sihouettes are used since vertices p and p provide identica information. The coection of 1 sihouettes, aong with their individua sihouette descriptors, constitute the Light Fied Descriptor. A denser samping of the viewpoint space is obtained by repicating the consteation of 1 sihouettes at sma rotationa offsets from the initia position. 3.2 Sihouette descriptors 3 Light Fied Descriptors (LFD) A rendered sihouette image is a binary image with a singe connected component. Lacking any appearance information, purey shape-based descriptors are used for the comparison of sihouettes. 3

4 The Zernike moment descriptor is obtained by projecting the 2D sihouette onto a set of circuar, compex Zernike poynomias of increasing degree. A few exampes of Zernike poynomias are shown in figure Degree = Order = Degree = 1 Order = 1 Degree = 2 Order = 9 Contour Distance Function 8 Distance from centroid Degree = 2 Order = 2 Degree = 3 Order = 1 Degree = 3 Order = θ Degree = 4 Order = Degree = 4 Order = 2 Degree = 4 Order = 4 Fig. 2 On the top eft is a sampe sihouette obtained from rendering a 3D mode of a human hand. On the top right we show the detected contour (obtained by a tracing agorithm) in green. The centroid of the shape is given by the green circe in the midde of the hand. The green circe intersecting the shape contour specifies the first detected point aong the contour, which aso acts as the starting reference point for generating the contour distance function. The resuting contour distance function, measuring the distance of the contour points to the centroid, is shown in the bottom pot. Fig. 1 Nine Zernike poynomias (see [22] for detais). The poynomias are shown for various degrees and orders. The poynomias are compex, so we are ony showing the rea component here. Coors coser to red are higher vaues (positive), whie coors coser to bue are ower vaues (negative). The poynomias are defined on the circe, so the region outside the circe shoud not be considered. The (, ) poynomia is uniform since a projection onto this is equivaent to just an integration of the input function over the circe. In tota, the magnitude of 35 coefficients are kept for the descriptor. The contour distance function r(θ) measures the distance from the sihouette centroid to the contour. An exampe of a sihouette image aong with its extracted contour and corresponding distance function are shown in figure 2. Since r(θ) is a periodic function on the circe, it is natura to examine its Fourier transform ˆr(k). The magnitude of 1 coefficients are retained for the descriptor. Both the Zernike moment descriptor and the contour distance descriptor were evauated extensivey in [22], where it was shown an integrated approach utiizing both descriptors performed we for image retrieva tasks. 3.3 Mode comparison Given Light Fied Descriptors for two modes, there are 6 possibe rotationa aignments that must be considered. The fina 3D mode distance is the minimum distance over a Light Fied Descriptor pairs for a possibe rotationa aignments. To speed up comparisons and retrieva from arge databases, a muti-stage comparison approach is proposed in [1]. There are four controabe parameters: (versus accuracy): 1. Number of sihouettes. The number of images (up to ten) used in the comparison can be varied. 2. Number of LFDs can be varied. 3. Quantization of sihouette descriptors. Feature vectors can be quantized so that each coefficient is represented with 4 or 8 bits, for exampe. 4. Subset of feature vectors. Distances can be computed using a subset of the feature vector coefficients. At each stage of the iterative mode comparison, the above parameters are varied to provide additiona accuracy, the basic idea being many modes can be discarded after each iteration. For detais of the proposed six-stage retrieva process see [1]. 4

5 3.4 Limitations The viewpoint configuration described above is tighty couped with the set of possibe rotationa aignments. It is not simpe to vary the number or position of viewpoints, or the number or sampes of 3D rotations independenty. For exampe, increasing the number of rotationa aignments evauated whie keeping the sihouette viewpoints fixed impies sihouette (or feature) interpoation, however there is no cear approach for this. The fu comparison, even for just 1 sihouettes per mode, is too computationay expensive for a database search. However, the proposed six-iteration method is fairy compex and seems a bit ad-hoc and arbitrary. There does not exist a cear intuition behind the decisions made during each stage. Combined with an absence of a thorough evauation, the motivations behind this process, aong with the performance contributions from each parameter, are uncear. Inspired by the discriminative strength of visua mode representations, we present our mode comparison technique and address the key issues discussed above in the foowing sections. 4 Efficient 3D mode comparison In this section we wi detai our proposed 3D mode comparison technique. The outine is as foows: In section 4.1, we describe sihouette generation and feature extraction. In section 4.2 we wi define a simiarity measure for comparing two modes, and in 4.3 we show how this simiarity measure can be evauated efficienty borrowing techniques from spherica harmonic anaysis. Section 4.4 covers the samping requirements of our approach. In sections 4.5 and 4.6 we summarize the agorithm and provide some anaysis and observations. fexibiity in seecting this feature vector, for comparison we wi use the 45-dimensiona Zernike and contour descriptor as in [1]. 4.2 Simiarity measure For the moment, et us consider that our coection of sihouettes is not finite, but rather we have obtained images and feature vectors from a points on the sphere. In this continuous setting, we have a N-dimensiona feature vector at each point on the sphere (as described above, in our experiments N = 45). Formay, we wi write this sihouette feature representation as M(p) i, where p is a sphere point (p S 2 ) and i is the index into the N-dimensiona feature vector that describes the sihouette obtained from viewpoint p. To compare two 3D modes, we define their simiarity as the crosscorreation of their feature representations: G c = N [ ] M 1 (p) i M 2 (p) i dp p S 2 i=1 (1) In practice we evauate a normaized cross-correation, but for simpicity we eave out the normaization terms in our description here. Note, equation 1 evauates a simiarity measure over the two mode representations M 1 and M 2 in their native orientations. However, as we do not know the correct rotationa aignment, we must consider a possibiities: NX»Z G c(r) = M 1 (p) i M 2 (R T p) i dp, R SO(3) (2) i=1 p S 2 Here G c (R) measures the cross-correation for a possibe 3D rotationa aignments R SO(3). We define the simiarity between two modes as the maximum vaue of G c (R). Computationay, evauating G c (R) directy is cumbersome. For each 3D rotation we must rotate one mode representation (M 2 ) and perform a 3D integration. In the next subsections we see how to evauate G c (R) efficienty. 4.1 Sihouette rendering and feature extraction Our 3D mode representation is a coection of sihouette images rendered from viewpoints surrounding the mode. Consider a 3D mode centered at the origin. For any sphere point p S 2, we can render a sihouette via an orthographic projection of the mode onto the pane tangent to the sphere at p. In this way we generate sihouette images for any coection of spherica coordinates (we wi discuss the number of sihouettes and their ocations in subsequent sections). Furthermore, each sihouette we obtain wi be represented by a feature vector describing its shape. Athough we have a 4.3 Simiarity evauation To efficienty evauate the mode simiarity function G c (R) from equation 2, we recognize that the inner integra fits the definition of a correation between functions defined on the sphere. Isoating the inner integra gives G(R) = M 1 (p)m 2 (R T p)dp, R SO(3) (3) p S 2 To evauate G(R), we adopt an approach simiar to those described in [23 28], which show that the spherica correation integra is equivaent to a mutipication 5

6 of Fourier transforms. We provide a brief summary of this resut here, but readers are referred to [24, 25] for reference. In traditiona Fourier anaysis, periodic functions on the ine (or equivaenty functions on the circe S 1 ), are expanded in a basis spanned by the Eigenfunctions of the Lapacian. Simiary, the Eigenfunctions of the spherica Lapacian provide a basis for M(p) L 2 (S 2 ) (here L 2 denotes square-integrabiity). These Eigenfunctions are the we known spherica harmonics (Ym : S 2 C), which form an Eigenspace of harmonic homogeneous poynomias of dimension Thus, the spherica harmonics for each form an orthonorma basis for any M(p) L 2 (S 2 ). The spherica harmonic for degree and order m (, m,, m Z), is given as Ym(θ, φ) = ( 1) m (2 + 1)( m)! P 4π( + m)! m(cosθ)e imφ Note we are using p and (θ, φ) interchangeaby to denote points on the sphere. In the above equation Pm are the associated Legendre functions and the normaization factor is chosen to satisfy the orthogonaity constraint. Readers are referred to [29, 3] for a in-depth treatment of spherica harmonics. Any function M(p) L 2 (S 2 ) can be expanded in a basis of spherica harmonics: M(p) = N m= ˆM m Y m (p) (4) ˆM m p S = M(p)Ym (p)dp (5) 2 The ˆM m are the coefficients of the Spherica Fourier Transform (SFT). Henceforth, we wi use ˆM to annotate vectors in C 2+1 containing a coefficients of degree, ordered from through +. As functions on the sphere can be expanded in spherica harmonics, functions defined on the rotation group can be expanded in the irreducibe unitary representations of the rotation group. For G(R) L 2 (SO(3)), we can write its Fourier expansion as G(R) = N Ĝ mk = m= k= R SO(3) Ĝ mk U mk (R) (6) G(R)Umk (R)dR (7) The Ĝ mk, with m, k =,..., are the (2+1) (2+ 1) coefficients of degree of the SO(3) Fourier transform. The Umk (R) are the eements of the irreducibe matrix representations of SO(3). We wi write U (R) for the (2+1) (2+1) matrix representation at degree, and Ĝ for the matrix of SO(3) Fourier coefficients at degree. There is a cose reationship between the Fourier representation of functions on the sphere and the matrix representations U (R). Specificay, as spherica functions are rotated by eements of SO(3), their Fourier coefficients are moduated by the irreducibe representations of SO(3): M(p) M(R T p) ˆM U (R) T ˆM (8) This reationship, aong with the orthogonaity of the spherica harmonics, aows us to expand equation 3 in terms of the corresponding Fourier transforms: Ĝ = ˆM 1 ( ˆM2 ) T (9) This shows that the SO(3) Fourier coefficients of G(R) can be obtained as a matrix product between the coefficient vectors of the two spherica functions M 1 and M Viewpoint samping A fast discrete agorithm for the spherica Fourier transform, based on a separation of variabes technique, can be attributed to [3]. The compexity of the transform is O(L 2 og 2 L), where L is the bandwidth of the spherica function. In practice, the seection of L simpy specifies that ony those coefficients of degree ess than L wi be retained from the Fourier transform. The samping requirement to achieve the noted compexity is that 2L sampes must be paced uniformy in each spherica coordinate (i.e. 2L sampes in coatitude, and 2L sampes in azimuth). Figure 3 shows the effect of this samping constraint on the distribution of spherica sihouette viewpoints. A B C Fig. 3 On the eft (A) is a representation of a uniformy samped spherica grid, with 16 sampes spaced uniformy in each dimension. This is the samping requirement for a fast spherica Fourier transform at bandwidth L = 8. (B) depicts the corresponding sampe support regions as they appear on the sphere. The highighted bins correspond to the highighted row in (A). The circes in (C) specify the actua sampe ocations on the sphere. 6

7 Simiar to the spherica transform, there exists a separation of variabes technique for a fast discrete SO(3) Fourier transform [24]. The compexity for such a technique is O(L 3 og 2 L), where as before L is the function bandwidth. This fast discrete transform is given for a standard Euer ange parameterization of SO(3). In particuar, the three anges α, γ [, 2π) and β [, π], can generate any 3D rotation throughr = R z (γ)r y (β)r z (α). Here R z and R y represent rotations about the Z and Y axis, respectivey. For a fast discrete SO(3) transform of a function with bandwidth L, the samping theorem requires 2L sampes uniformy spaced in each of the three Euer anges α, β, and γ. As with the spherica transform, this uniform samping in Euer anges eads to a nonuniform samping in rotation space. φ = φ = 6 φ = 12 φ = 18 φ = 24 φ = 3 The fast spherica and rotationa Fourier transforms detaied in [24, 3] provide us the machinery necessary θ = 15 for evauating the spherica correation integra (equation 3) in the Fourier domain (equation 9). θ = Pairwise mode comparison summary We now summarize our pairwise 3D shape comparison formuation as detaied above, and provide some practica considerations and compexity anaysis. The ony parameter we need to set is the frequency bandwidth L, which wi initiay dictate the samping frequency in the sihouette viewpoint space, as we as the 3D rotation space. Given two modes which we wish to compare, the first step is generating sihouette images and features from each mode. As per the Fourier samping theorem, given a bandwidth L, we wi need to generate 2L 2L = 4L 2 orthographicay rendered sihouette images from each mode. For any sihouette viewpoint p S 2, the antipoda point p provides redundant information. Thus in practice, we ony need to render haf the sihouettes (a tota of 2L 2 images). Figure 4 provides an exampe of a 3D mode and its sihouette images. Subsequenty, each sihouette image is repaced with its N-dimensiona feature vector representation. After feature extraction, we have our initia mode representations M 1 (p) i and M 2 (p) i. To obtain the Fourier domain representation of M(p) i, we must take a separate spherica Fourier transform for each sihouette feature index i = 1,...,N, eaving us with the coefficients ( ˆM m) i. For a fixed L, we wi have a tota of NL 2 Fourier coefficients. As we observed earier, the spherica mode representation exhibits the even property M(p) i = M( p) i. This redundancy transates to the Fourier space: ( ˆM m) i =, odd. Additionay, the spherica Fourier transform θ = 75 Fig. 4 An exampe of a 3D mode and its corresponding sihouette images. On the top is a 3D mode downoaded from Googe s 3D Warehouse. For a bandwidth L = 3, 2L 2 = 18 sihouettes wi be rendered. In this figure, the top row corresponds to sihouettes from viewpoints with fixed coatitude (15 ) but varying azimuth. is defined generay for compex-vaued functions. For rea-vaued functions the coefficient vectors ( ˆM m ) i exhibit an hermitian property: ( ˆM m) i = ( 1) m ( ˆM m) i. These two facts show that ony the coefficients ( ˆM m) i for even and m are necessary, greaty reducing the storage space of our Fourier representation. The Fourier representations of the two modes, ( ˆM 1 m ) i and ( ˆM 2 m) i, are the necessary input for evauating the correation simiarity measure. From equation 9, we showed that the SO(3) Fourier coefficients of G c (R) can be obtained in the spectra domain as: Ĝ c = N i=1 ˆM 1 i ( ˆM 2 i )T (1) To obtain the sampes of our desired correation function G c (R), we must take an inverse SO(3) Fourier transform as a fina step. Note, we have the option of taking the inverse SO(3) transforms before or after the summation i. In other words, if we et ISOFT(.) represent the inverse SO(3) Fourier transform operator, 7

8 the foowing are equivaent: N ( ) G c (R) = ISOFT ˆM 1 i ( ˆM 2 i )T i=1 = ISOFT ( N i=1 ˆM 1 i ( ˆM 2 i )T ) (11) (12) Whie these computations have numericay identica resuts, there is a cear advantage to equation 12. Evauating equation 11 has O(NL 3 og 2 L) compexity. This comes from having N separate SO(3) Fourier transforms, each of which has compexityo(l 3 og 2 L). On the other hand, evauating equation 12 requires ony one Fourier transform. Athough the inner summation N i=1 over coefficient vectors has compexityo(nl3 ), the constant factor is minima. The tota compexity of equation 12 is O(NL 3 )+O(L 3 og 2 L) = O(L 3 (og 2 L+ N)). In practice, the computationa burden ies in the Fourier transform, and thus we see a speedup by a factor of approximatey 1 when evauating equation 12 in pace of equation 11. We can aso compare this compexity against the origina definition of G c (R) as given in equation 2. To evauate equation 2 in the spatia domain woud have a compexity of O(NL 5 ) (this woud aso be the compexity of evauating [1] for simiar numbers of sihouette and rotation sampes). In the fina step, the maximum vaue from the sampes of G c (R) (as obtained via equation 12) is seected as the simiarity score between the two input modes. 4.6 Samping fexibiity We wi now discuss how our approach addresses one of the key issues brought up earier, namey the dependence between the number of sihouette viewpoint sampes and the number of sampes in the 3D rotation space. First, we note that our deveopment aows for an arbitrariy dense samping of the viewing sphere and 3D rotation space. For a fixed bandwidth L we wi have 2L sihouette viewpoint sampes uniformy spaced in the each of the two spherica coordinates, as we as 2L sampes uniformy spaced in each of the three Euer anges. This straightforward formuation aows us to achieve a samping where the maximum distance between any sihouette viewpoint and its nearest neighbor is arbitrariy sma (simiary with 3D rotations), simpy by varying the bandwidth L. Furthermore, we can achieve an independence between the number of sihouette sampes and the number of 3D rotations sampes of G c (R). In other words, we are not forced to have the same bandwidth parameter L for both the sihouette mode representationm(p) i and the 3D correation function G c (R). For exampe, assume L is the chosen bandwidth of the mode representations M 1,2 (p) i, which impies a tota of 2L 2 sihouette images. Let L > L. We can easiy generate Ĝ c for =,..., L 1 as in equation 1 by setting Ĝ c =, L < L. In this approach, the extra sampes obtained in the 3D rotation space by having a higher bandwidth L are interpoated using the Fourier coefficients of M(p) i up to bandwidth L. Thus, our approach provides a simpe mechanism for independenty varying the number of sihouette viewpoint sampes and the number of 3D rotation sampes. If desired, many more sampes of G c (R) can be interpoated from few sihouette images. Contrast this to a direct spatia approach (e.g. [1]), where there are strict dependencies between sihouette view sampes and possibe 3D rotations, and no simpe mechanism for interpoation exists. 5 A natura coarse-to-fine estimation of simiarity The deveopment in the previous section presents a nove approach for determining the simiarity between a pair of 3D modes. Such a technique can be very important for an appication such as 3D mode retrieva, where the chaenge is to identify the most simiar modes to a query from a very arge database. In such a setting, it can be computationay infeasibe to perform a fu simiarity evauation between the query and every database mode just to identify the few most simiar modes. Instead, when searching for nearest neighbors we woud ike to discard arge numbers of candidate modes with few computations. To this end, our mode comparison approach can easiy be extended to form an iterative coarse-to-fine evauation of mode simiarity. The basic idea is very intuitive, and comes from the observation that the degree of a Fourier coefficient indicates the frequency component that is represented. In other words, a coarse estimation of simiarity using ony ow-frequency signa information can be obtained by using ony the ow-degree Fourier coefficients. Subsequenty, a higher precision can be achieved by introducing high-frequency signa information in the way of the high-degree Fourier coefficients. To buid a 3D mode retrieva system for a arge database of modes, we can proceed as foows: In a pre-processing step, each mode in the database is represented with the Fourier coefficients ( ˆM m ) i at some bandwidth L. Given a query mode, the first iteration for retrieva invoves evauating a coarse simiarity between the query and every database mode. This coarse simiarity is obtained by computing G c (R) at some sma bandwidth L < L. Those modes furthest from the query can be discarded. In each subsequent 8

9 iteration, a finer simiarity score is computed between the query and remaining database candidates by evauating G c (R) at an increased bandwidth. In the fina iteration a ranked ist of nearest neighbors is created by evauating G c (R) at the fu bandwidth L for the few remaining candidates. In this way we can discard the arge majority of database modes with imited computation. 6 Rotationa invariants The mode comparison and retrieva approaches we have presented above utiize representations that are not invariant to 3D rotations. As we discussed earier, a genera aternative is to buid rotationa invariance directy into the mode feature representation. Feature descriptors which are invariant to mode transformations aways ead to faster comparison (since no search over the transformation space needs to be done onine), and are aso suitabe for use with standard off-the-shef cassification techniques (e.g. SVM cassifiers). For spherebased 3D mode representations there are two ways to normaize for orientation. The simpest and most common approach is to aign the mode s principa axes with a fixed reference frame. An aternative to this PCAaignment is to identify the rotation-invariant terms in the spherica Fourier domain (see [2,31] for detais and other appications for such invariants). We saw earier (equation 8) how the Fourier coefficients of a function transform under a 3D rotation of the origina function. The Fourier anaogue to 3D rotations are given by the matrix transformationsu (R). We know that these unitary transformations wi not ater the distribution of spectra energy among coefficient degrees: U (R) ˆM 2 = ˆM 2, R SO(3) where. 2 indicates L 2 vector norm. We can buid a rotation-invariant mode feature vector by retaining ony the magnitudes of the Fourier coefficient vectors ( ˆM i ). The tota size of such a mode descriptor is L 2 N. For exampe, consider a mode for which we render a very arge number of sihouettes (e.g. L = 17 means we must render 2L 2 = 578 images). Assuming N = 45 as we have used throughout this paper, our mode representation is just one feature vector of 45 8 = 36 dimensions. The distance between two modes is defined as the Eucidean (L 2 ) distance between their respective feature vectors. 7 Experiments In this section we wi study the effectiveness our proposed 3D mode comparison technique with three chaenging evauations (the Princeton Shape Benchmark, the Shape Retrieva Contest in 26, and a coection of modes downoaded from Googe s 3D Warehouse). We begin with a study on the de-facto evauation benchmark for 3D shape retrieva, the Princeton Shape Benchmark [32]. 7.1 Princeton Shape Benchmark The Princeton Shape Benchmark [32] provides a coection of 3D modes designed for the standardized evauation of retrieva, matching, custering, and recognition agorithms. The database consists of 1814 manuay categorized 3D modes coected from the Web. The database is segregated into a training set consisting of 97 modes and spanning 9 mode casses, and a test set consisting of the remaining 97 modes and spanning 92 mode casses. In the test set, the argest category contains 26 modes ( potted pant ), and the smaest category contains 4 modes (there are 17 categories with just 4 modes). Figure 5 shows a few exampes of modes in the benchmark. Fig. 5 Thumbnai images of six different modes in the Princeton Shape Benchmark [32]. The top row consists of thumbnais from the potted pant cass, which constitutes the argest cass in the PSB test set. The bottom row consists of thumbnais from the Newtonian toy cass, which is one of 17 casses in the test set tied for having the fewest modes (four). To stay consistent with pubished evauations on the benchmark, we restricted ourseves to evauations over the 97 mode test set. In principe, our approach is training-free, and thus coud be evauated over the entire 1814 mode benchmark. To evauate the robustness of our method, we initiaize every 3D mode with a randomy generated 3D 9

10 rotation before rendering mode sihouettes. This is important because athough mode orientation is unknown, it is quite common in the benchmark to see many modes that are aigned with the ground pane in their native orientations. In order to provide a proper evauation of our correation-based method, a random rotation of each mode wi cance out any orientation bias in the benchmark. In genera, evauation over the test set is performed by removing one mode to act as the query, and ranking the remaining modes from most simiar to east simiar. This ranked ist can be evauated in a number of ways (a few of which we wi detai beow). Performance for a particuar method or set of parameters is given by averaging the performance over a query modes. The evauation measures for the benchmark are as foows (see [32] for detais): 1. Nearest Neighbor measures the accuracy of the first retrieved neighbor. 2. First Tier and Second Tier. The ratio of modes in the query cass that appear in the first K resuts. If C is the number of modes in the query s cass, K = C 1 for first tier and K = 2 ( C 1) for second tier. 3. E-measure is a combined measure of the precision and reca for a fixed number of resuts (here the evauation neighborhood size is 32). The E-measure is defined as 2/( 1 P + 1 R ), where precision (P) and reca (R) are defined in the usua document retrieva way. 4. Discounted Cumuative Gain (DCG) is an evauation measure of the entire ranked ist that weights positive resuts at the top of the ist higher than positive resuts ower on the ist. The evauation measures described above emphasize positive resuts earier in the ranked retrieva. This is in ine with the idea that for many search appications, users wi often ony be interested in the quaity of the first few returned resuts. Perturbations towards the end of the ist wi have itte effect on the perceived quaity of the retrieva. The performance criteria isted above were used to evauate a number of existing 3D shape comparison and retrieva agorithms in [32]. We have reprinted these pubished resuts aong with the performance of our proposed mode simiarity measure in tabe 1. We ran the retrieva agorithm for 8 different bandwidth parameters, ranging from L = 3 up to L = 17. In this setting, the seection of bandwidth was kept fixed for the entire agorithm (i.e. the same bandwidth L was used for feature generation and evauating the correation function G c (R)). Our evauation indicates that we do see an improvement over the cosest method, L = 17 L = 15 L = 13 L = 11 L = 9 L = 7 L = 5 L = 3 Fig. 6 - curves for our proposed shape comparison agorithm on the Princeton Shape Benchmark test set. The evauation code used to generate these pots was taken from the evauation utiities provided with the benchmark [32]. As we can see, there is itte variation in resuts between the owest and highest bandwidths. LFD [1]. Furthermore, we see cose-to state of the art resuts at even ower bandwidth settings. Interestingy, the reative performance difference between retrievas run at the higher bandwidths (L 9, for exampe) are very sma. This fact can be ceary observed in figure 6, where we show precision-reca curves for the various bandwidth settings. The gap in performance at various bandwidth settings indicates that our proposed coarse-to-fine scheme may prove just as effective as performing retrieva at the highest resoution of L = 17. What remains to be determined are the specifics of the incrementa coarseto-fine retrieva. Based on the resuts in tabe 1 and figure 6, we can do most of the ranking and simiarity computation at the ow bandwidths and use the high resoution bandwidth for a fina fine-tuning (i.e. reranking) of a few modes. What we woud expect to see is that the DCG scores may not reach those of the highest bandwidth since DCG is a measure of the entire ranked isting. However, we woud expect to see nearest-neighbor scores mosty unaffected. We experimented with the foowing incrementa scheme: first, the simiarity scores are computed for a mode pairs at the owest bandwidth (L = 3). In the second pass, we identify the most simiar 2%, and rerank these modes by computing simiarity at L = 5. In the third and fina pass, we re-rank the 1% most simiar modes at L = 17. The resuts are shown in tabe 2. As expected, the goba measurements ike DCG fe somewhere between the ow and high-bandwidth resuts. Surprisingy, however, the nearest-neighbor resuts outperformed a agorithms incuding the fu L = 1

11 Shape Storage Discrimination (%) Descriptor Size Nearest First Second (bytes) Neighbor Tier Tier E-MeasureDCG L=17 27, L=15 21, L=13 16, L=11 11, L=9 7, L=7 4, LFD 4, L=5 2, REXT 17, SHD 2, GEDT 32, L= EXT SECSHEL 32, VOXEL 32, SECTORS CEGI 2, EGI 1, D SHELLS Tabe 1 Retrieva resuts of our proposed simiarity measure, as we as a number of comparison methods, on the Princeton Shape Benchmark [32]. The rows denoted by L= correspond to evauations of G c(r) using different vaues for the bandwidth parameter L. In this evauation, seection of L specifies the samping in the sihouette viewpoint space as we as the 3D rotation space. The resuts for the competing agorithms are taken from [32]. The agorithms are sorted by the Discounted Cumuative Gain Score. The test data consists of 97 modes cassified into 92 categories. Surprisingy, even the ow bandwidth correations are outperforming most agorithms. See [32] for an overview of a the comparison agorithms. Nearest First Second E-Measure DCG Neighbor Tier Tier 67.5% 39.4% 48.% 28.% 64.7% Tabe 2 Discrimination resuts for coarse-to-fine simiarity computation. In this experiment three stages were used. First, a modes were ranked according to simiarity computed at L = 3. Subsequenty, the best 2% of the ranked modes were re-ranked with simiarity computed at L = 5. In the fina fine-tuning step, the cosest 1% of the modes were again re-ranked with simiarity computed at L = 17. As expected, the goba measurements ike DCG fe somewhere between the ow-res and high-res resuts. Surprisingy, however, the nearest-neighbor resuts outperformed a agorithms incuding the high-res computation. 17 evauation. These resuts indicate that much of the work for retrieva is being done at the ow frequencies. Whie the high-frequency mode coefficients may not contribute as much to the overa scheme, they are very vauabe as a fine-tuning mechanism for re-ranking the top resuts. In addition to these quantitative resuts, we show some exampes of the top retrieva resuts for various query modes in tabe SHREC 26 In addition to the extensive evauation on the PSB, our correation based shape retrieva agorithm was entered into SHREC, the 3D Shape Retrieva Contest [33, 34] in 26. The purpose of the contest was to study different retrieva methods under a wide variety of evauation criteria. A set of 3 3D modes served as the query set. The corpus against which ranked retrieva was to be performed was a permutation of the Princeton Shape Benchmark. The ranked retrieva ists for each of the 3 queries was evauated in a number of ways. Individua per-query resuts were tabuated, as we as aggregate resuts over a 3 queries for each contest entry. In tota, 17 aggregate evauation criteria were used, and our proposed correation method was the top performer in 11 of these 17 categories [33]. A subset of the aggregate resuts are shown in tabe 4. Quaitative resuts of our retrieva agorithm as it performed in this contest are shown in tabe 5. The tabe shows the 1 cosest modes in the database for 5 of the 3 queries used in the contest. An aternative to the correation-based comparison approach is to encode rotation-invariance directy into the feature descriptor. This can be done by computing rotation-invariant mode descriptors as described in section 6. Whie the onine comparison cost is much ess (Eucidean distance of a sma feature vector versus spherica correation), the discrimination performance is expected to be much worse. This is in fact the observed effect in figure 7. Given a ranked ist generated from rotation-invariant feature vectors, we observe how 11

12 Query Tabe 3 Retrieva resuts for 1 test queries in the Princeton Shape Benchmark (PSB). The first coumn shows each query s thumbnai. The ten modes to the right of each query are the cosest modes (in order) in the 97-mode PSB test set. Shape Descriptor Mean (highy reevant ony) Mean First Tier (highy reevant ony) Dynamic Average Normaized DCG at 5 Normaized DCG at 5 L= Shiane et a [35] Zaharia et a [35] Daras et a [35] Papadakis et a [35] Chaouch et a [35] Laga et a [35] Jayanti et a [35] Tabe 4 A subset of the pubished resuts from SHREC26 (3D Shape Retrieva Contest 26) [33,35]. The resuts are averaged over the 3 test queries. For each query, the database modes were manuay abeed as highy reevant, marginay reevant, or irreevant. The evauation criteria foow the same definitions as with the evauations for the Princeton Shape Benchmark, with a few exceptions. For a given query, if the reca ratio within the top i neighbors is given as r i, then the Dynamic Average is defined as the mean over a r i. Normaized DCG represents the Discounted Cumuative Gain divided by the idea or optima possibe Discounted Cumuative Gain score. For more contest resuts see [33, 35], and for a detaied expanation of the evauation criteria see [34]. Each entrant into the competition was aowed mutipe entries, either to be used for different agorithms, or just different parameter settings for the same genera approach. To be fair to the comparison methods, we have ony shown here the best performing method in each evauation coumn for a the entrants. For our proposed method, we are showing the resuts at L = 17. many of the top-ranked modes must be observed before seeing 5% of the top K modes retrieved using our correation simiarity measure (the pot shows the median over a 3 queries). For exampe, in order to see 5 of the top 1 retrieved modes using a correation simiarity, we have to examine the first 213 retrieved modes using rotation-invariant descriptors. As expected, the computationa benefit of buiding invariance directy into the descriptor is offset by the oss in discrimination power. 7.3 Googe 3D Warehouse Whie the Princeton Shape Benchmark has become one of the standard evauation datasets for 3D mode retrieva agorithms, the benchmark can be criticized for it s ack of variation within casses in addition to other factors such as a ack of articuated object casses. We therefore evauated our proposed mode comparison method on a more chaenging rea-word dataset. This set consists of 772 3D modes downoaded from Googe s 3D 12

13 Query Tabe 5 Top 1 retrieva resuts for 5 of the 3 queries from the SHREC 26 contest. On the eft coumn we show the query modes, and to the right of each query mode are the 1 nearest neighbors in the database best K modes in correation method Fig. 7 This pot shows how many modes in the ranked ist (obtained by comparing rotation-invariant vectors) you need to traverse before finding 5% of the best K matches in the ranked ist obtained with the proposed correation scheme. The pot shows the median over a test queries. For exampe, 5% of the best 1 matches from the correation method wi appear in the first 213 matches from the ranked ist obtained by comparing rotation-invariant mode descriptors. Fig. 8 Sampes of the 3D mode dataset created from 3D Warehouse. Each coumn shows sampes from a singe category. The three categories shown here are stadium, pane, and garage. These sampe images indicate a few reasons why this coection is very chaenging. For exampe, we see that there is a arge variation in scae and compexity within and between categories. Aso, modes on 3D Warehouse are not just individua objects as in the Princeton Shape Benchmark. Here individua mode fies such as the garage can consist of mutipe pieces ike the garage, ground pane, etc. This makes matching based soey on shape content very chaenging. Warehouse, which is a repository for 3D modes on the Web. A the modes are grouped into a tota of 25 casses. The categorization of modes corresponds simpy to the search term used to find and the mode. The argest category is airpane with 58 modes, and the smaest is fish with ony 6 modes. Figure 8 shows a few exampes from the dataset. One of the biggest chaenges posed by this repository is that individua mode fies can consist of mutipe objects. For exampe, the figure shows that the garage mode consists of mutipe structures in addition to the garage object (such as the ground pane). Our primary evauation criteria for this set was precision versus reca. We averaged resuts over individua casses, as we as over a 772 modes. For comparison, we evauated the method proposed by [13], where 3D Zernike descriptors are used to represent the modes. Note, for this dataset a of our simiarity measure evauations are performed at L = 11. Resuts averaged per cass are shown in figure 9, whie resuts averaged over a the modes are shown in figure 1. The resuts indicate consistenty high performance for most casses of modes in the set. 7.4 Timings The agorithms proposed in this paper can pay a big roe in a system for 3D mode search and retrieva. Hence, it is important to be aware of the execution times. There are two stages of computation. The preprocessing stage invoves computing the mode features. This is an offine step regarding database cre- 13

14 airpane (58).8 bench (48).5 bike (23).5 bridge (35) caste (44) chair (3).8 1 church (32).8 1 computer (16).8 1 door (11).8 1 fish (6) garage (12) guitar (37) gun (48) amp (31).8 1 mug (34) office (2) payground (25).8 1 rubix (39).8 1 screw (44) skyscraper (37).8 1 stadium (54).8 1 tank (15) tree (13) truck (29) window (31) Fig. 9 versus reca curves for each of the 25 casses of modes in the 3D Warehouse dataset. The number of modes in each cass appears in parentheses next to the cass name. The soid green ine is the curve from our proposed method with bandwidth setting L = 11. The red ine is the curve of the method proposed in [13]..7.5 A Modes (772) 3D Zernike L = Fig. 1 versus reca curves averaged over a the modes in the 3D Warehouse dataset. The tota number of modes in the dataset is 772. The soid green ine is the curve from our proposed method with bandwidth setting L = 11. The red ine is the curve of the method proposed in [13]. ation, but the computation time is sti important since the query modes may not have been seen before. In our agorithm feature generation invoves generating mode sihouettes, extracting features from the sihouettes, and then performing a spherica Fourier transform. For the highest resoution we tested in this paper (bandwidth L = 17, which corresponds to 578 sihouettes), the tota sihouette feature extraction time was on average 1.58 seconds. Since our feature vectors are 45 dimensiona, we must take 45 Fourier transforms, for which the tota time is.42 seconds. The remaining preprocessing piece is generating the sihouettes. In figure 11 we show the time it takes to render a the sihouettes for one mode. As expected, the times are dependent on the number of poygons. For modes with reative few poygons (e.g. 2,), the rendering time is about one second. For modes with over 3, poygons the rendering time is amost three seconds (for 578 sihouettes). However, the OpenGL code used for this process is far from optimized and does not take advantage of the many options avaiabe with modern graphics cards, so there remains room for improvement. The onine cost of the mode comparison agorithm is the computation time required to estimate the correation function G c (R) and identify the maximum vaue. Timings for evauating equation 12 at various bandwidth settings, as we as a comparison with [1], are 14

15 Sihouette rendering time (seconds) K poygons 2K poygons 185K poygons 316K poygons can aso ead to a very intuitive and effective coarse-tofine 3D mode retrieva system. A thorough evauation on mutipe benchmarks shows our proposed methods combine the discriminative power of a visua mode representation with efficient computation. 8.1 Acknowedgments Bandwidth (B) Fig. 11 Sihouette rendering times for various bandwidths and mode sizes. For a bandwidth L, the number of sihouettes rendered is 2L 2. For exampe, it takes ess than 3 seconds to generate 578 sihouettes of a mode with 316K poygons. Method Time (s) L=3.5 L=5.8 L=7.58 L=9.113 L= L= L= L= LFD [1].28 Tabe 6 Time in seconds to obtain a simiarity score between two modes given their precomputed mode feature representations. The top rows give times for different bandwidths, which represent different eves of precision and resoution. The bottom row gives the timing for an impementation of direct sihouette feature comparison as given in [1]. This computation invoves computing the L 1 distance between coections of 1 sihouette feature vectors over a tota of 546 possibe rotationa aignments. An optimized impementation timing (e.g., using ookup tabes) was reported as.13 seconds in [32]. The cosest setting of ours is L = 7, which uses 98 sihouette images, whereas 1 sihouettes are needed in [1]. Our approach (.58s, L = 7), is significanty faster than either LFD impementation (.28s, our impementation, and.13s optimized impementation). given in tabe 6. The machine used to generate these timings is a Appe Powerbook aptop computer with 2GB of RAM. 8 Concusion In this paper we presented a new simiarity measure for comparing 3D shapes based on a visua representation, as we as a nove estimation technique for the efficient evauation of the simiarity measure. We showed how an anaysis in the spherica Fourier domain provides a fexibiity to a components of our formuation, and We thank Corey Godfeder for the impementation of the 3D Zernike descriptors described in [13]. References 1. D.-Y. Chen X.-P. Tian, Y.T.S., Ouhyoung, M.: On visua simiarity based 3D mode retrieva. In: Eurographics (23) 2. Kazhdan, M., Funkhouser, T., Rusinkiewicz, S.: Rotation invariant spherica harmonic representation of 3D shape descriptors. In: Symposium on Geometry Processing (23) 3. Vranic, D.V.: An improvement of rotation invariant 3D shape descriptor based on functions on concentric spheres. In: Proceedings of Internationa Conference on Image Processing, pp (23) 4. Saupe, D., Vranic, D.V.: 3D mode retrieva with spherica harmonics and moments. In: Proceedings of the 23rd DAGM-Symposium on Pattern Recognition, pp Springer-Verag, London, UK (21) 5. Ankerst, M., Kastenmüer, G., Kriege, H.P., Seid, T.: Nearest neighbor cassification in 3D protein databases. In: Proceedings of the Seventh Internationa Conference on Inteigent Systems for Moecuar Bioogy, pp AAAI Press (1999) 6. Kang, S.B., Ikeuchi, K.: Determining 3-D object pose using the compex extended gaussian image. In: Proceedings of the 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 91) (1991) 7. Horn, B.K.P.: Extended gaussian images. IEEE 72, (1984) 8. Osada, R., Funkhouser, T., Chazee, B., Dobkin, D.: Matching 3D modes with shape distributions. In: SMI 1: Proceedings of the Internationa Conference on Shape Modeing & Appications, pp IEEE Computer Society, Washington, DC, USA (21) 9. Ohbuchi, R., Osada, K., Furuya, T., Banno, T.: Saient oca visua features for shape-based 3D mode retrieva. In: IEEE Internationa Conference on Shape Modeing & Appications (28) 1. Ohbuchi, R., Minamitani, T., Takei, T.: Shape-simiarity search of 3D modes by using enhanced shape functions. In: TPCG 3: Proceedings of the Theory and Practice of Computer Graphics, p. 97. IEEE Computer Society, Washington, DC, USA (23) 11. Tangeder, J.W., Vetkamp, R.C.: A survey of content based 3d shape retrieva methods. Mutimedia Toos App. 39(3), (28) 12. Makadia, A., Visontai, M., Daniiidis, K.: Harmonic sihouette matching for 3D modes. In: 3DTV. Kos (27) 13. Novotni, M., Kein, R.: 3D zernike descriptors for content based shape retrieva. In: SM 3: Proceedings of the eighth ACM symposium on Soid modeing and appications, pp ACM, New York, NY, USA (23). DOI http: //doi.acm.org/1145/

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