3D Model Retrieval Using Probability Density-Based Shape Descriptors Ceyhun Burak Akgu ¨l, Student Member, IEEE, Bu ¨lent Sankur, Senior Member, IEEE, Yu ¨cel Yemez, Member, IEEE, and Francis Schmitt Abstract—We address content-based retrieval of complete 3D object models by a probabilistic generative description of local shape properties. The proposed shape description framework characterizes a 3D object with sampled multivariate probability density functions of its local surface features. This density-based descriptor can be efficiently computed via kernel density estimation (KDE) coupled with fast Gauss transform. The nonparametric KDE technique allows reliable characterization of a diverse set of shapes and yields descriptors which remain relatively insensitive to small shape perturbations and mesh resolution. Density-based characterization also induces a permutation property which can be used to guarantee invariance at the shape matching stage. As proven by extensive retrieval experiments on several 3D databases, our framework provides state-of-the-art discrimination over a broad and heterogeneous set of shape categories. Index Terms—Shape matching, retrieval, surface representations, nonparametric statistics, geometric transformations, invariance, feature evaluation and selection, performance evaluation. Ç 1 INTRODUCTION F AST and accurate scanning technology equipped with shape modeling and rendering tools has enabled the means of acquiring, designing, and manipulating complete 3D models of real-world objects. Digital 3D models as a new modality of visual information find applications in several domains such as computer-aided design [1], cultural heritage archival [2], molecular modeling [3], and video games industry [4], [5]. With growing interest in 3D models, their effective retrieval from large databases is acquiring economic utility [4], [6], [7]. Text-based systems, much like in all other media applications, would remain severely limited in describing and retrieving 3D models [7]. Content-based systems, on the other hand, offer an effective and scalable complementary solution to the 3D retrieval problem. We address content-based retrieval of complete 3D object models by a probabilistic generative description of their local shape. We call the proposed method as the density-based framework (DBF) in that it describes 3D objects with multivariate probability density functions (pdfs) of chosen shape features. Our previous study [8] has shown that such an approach has a promising retrieval potential. In this paper, we analyze DBF in greater detail and provide extensive retrieval experiments to demon- strate that it can satisfactorily handle large collections of heterogeneous shape categories. In particular, we show that DBF is relatively insensitive to small shape perturba- tions and mesh resolution, that it is computationally efficient, and that it enjoys a permutation property which guarantees invariance to a certain class of 3D transforma- tions at the shape matching stage. As a consequence of these contributions, DBF qualifies as one of the best 3D shape descriptors, as established by retrieval experi- ments on several databases. Our starting point is that, as similar shapes induce similar feature distributions, two shapes can be compared by the distance between their feature pdfs. Histogram- based 3D shape descriptors [9], [10], [11], [12], [13], [14], [15] (see Section 2) have relied on this intuitively appealing idea but failed to provide fine grain discrimination required by the 3D retrieval task [7]. Compared to its histogram-based ancestors, DBF is original in two aspects: 1) It employs richer sets of multivariate shape features and 2) it adopts the kernel strategy to estimate the distribution [16]. As a further contribution, we experimentally show that these two aspects overcome the performance limitation of early histogram-based 3D shape descriptors. After transforming a given 3D model into a canonical coordinate frame and scale, our scheme first characterizes its surface locally using simple and direct features, without resorting to computationally intensive methods such as shape contexts [17] or spin images [18]. Our features are, in fact, as simple as distance-to-origin, radial, and normal directions, and principal curvatures (Sections 3.1 and 3.2). Without sacrificing computational simplicity, we construct more informative local characterizations by joining these simple features into multivariate ones. In a previous work IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 31, NO. 6, JUNE 2009 1117 . C.B. Akgu ¨l is with the Video Processing and Analysis Group, Philips Research Europe, High Tech Campus 36 (WOp122 O-1), 5656AE Eindhoven, The Netherlands. E-mail: ceyhun.akgul@philips.com. . B. Sankur is with the Department of Electrical and Electronic Engineering, Bo gazic ¸i University, Bebek 80815, Istanbul, Turkey. E-mail: bulent.sankur@boun.edu.tr. . Y. Yemez is with the Department of Computer Engineering, Koc ¸ University, Rumeli Feneri Yolu, 34450 Sariyer, Istanbul, Turkey. E-mail: yyemez@ku.edu.tr. . F. Schmitt was with the Image and Signal processing Department, Te´le´com ParisTech (Ecole Nationale Supe´rieure des Te´le´communications before his death. Manuscript received 2 June 2008; revised 12 Oct. 2008; accepted 8 Jan. 2009; published online 16 Jan. 2009. Recommended for acceptance by S. Belongie. For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number TPAMI-2008-06-0323. Digital Object Identifier no. 10.1109/TPAMI.2009.25. 0162-8828/09/$25.00 ß 2009 IEEE Published by the IEEE Computer Society Authorized licensed use limited to: ULAKBIM UASL - KOC UNIVERSITY. Downloaded on April 27, 2009 at 10:54 from IEEE Xplore. Restrictions apply.