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Issue No.12 - Dec. (2012 vol.34)
pp: 2407-2419
Meizhu Liu , Dept. of CISE, Univ. of Florida, Gainesville, FL, USA
B. C. Vemuri , Dept. of CISE, Univ. of Florida, Gainesville, FL, USA
Shun-Ichi Amari , Math. Neurosci. Lab., RIKEN Brain Sci. Inst., Wako, Japan
F. Nielsen , Ecole Polytech., Palaiseau, France
ABSTRACT
In this paper, we consider the family of total Bregman divergences (tBDs) as an efficient and robust “distance” measure to quantify the dissimilarity between shapes. We use the tBD-based l1-norm center as the representative of a set of shapes, and call it the t-center. First, we briefly present and analyze the properties of the tBDs and t-centers following our previous work in [1]. Then, we prove that for any tBD, there exists a distribution which belongs to the lifted exponential family (lEF) of statistical distributions. Further, we show that finding the maximum a posteriori (MAP) estimate of the parameters of the lifted exponential family distribution is equivalent to minimizing the tBD to find the t-centers. This leads to a new clustering technique, namely, the total Bregman soft clustering algorithm. We evaluate the tBD, t-center, and the soft clustering algorithm on shape retrieval applications. Our shape retrieval framework is composed of three steps: 1) extraction of the shape boundary points, 2) affine alignment of the shapes and use of a Gaussian mixture model (GMM) [2], [3], [4] to represent the aligned boundaries, and 3) comparison of the GMMs using tBD to find the best matches given a query shape. To further speed up the shape retrieval algorithm, we perform hierarchical clustering of the shapes using our total Bregman soft clustering algorithm. This enables us to compare the query with a small subset of shapes which are chosen to be the cluster t-centers. We evaluate our method on various public domain 2D and 3D databases, and demonstrate comparable or better results than state-of-the-art retrieval techniques.
INDEX TERMS
statistical distributions, image retrieval, maximum likelihood estimation, pattern clustering, shape recognition, GMM, hierarchical total Bregman soft clustering, total Bregman divergences, tBD-based l1-norm center, lifted exponential family, lEF, statistical distributions, maximum a posteriori estimate, MAP, exponential family distribution, clustering technique, shape retrieval framework, Gaussian mixture model, Shape analysis, Databases, Clustering algorithms, Robustness, Convex functions, Harmonic analysis, Three dimensional displays, 3D shape retrieval, Total Bregman divergence, t-center, lifted exponential families, hard clustering, soft clustering, Gaussian mixture model
CITATION
Meizhu Liu, B. C. Vemuri, Shun-Ichi Amari, F. Nielsen, "Shape Retrieval Using Hierarchical Total Bregman Soft Clustering", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.34, no. 12, pp. 2407-2419, Dec. 2012, doi:10.1109/TPAMI.2012.44
REFERENCES
 [1] M. Liu, B.C. Vemuri, S.-I. Amari, and F. Nielsen, "Total Bregman Divergence and Its Applications to Shape Retrieval," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 3463-3468, 2010. [2] H. Chui and A. Rangarajan, "A Feature Registration Framework Using Mixture Models," Proc. IEEE Workshop Math. Methods in Biomedical Image Analysis, pp. 190-197, 2000. [3] B. Jian and B.C. Vemuri, "A Robust Algorithm for Point Set Registration Using Mixture of Gaussians," Proc. 10th IEEE Int'l Conf. Computer Vision, vol. 2, pp. 1246-1251, 2005. [4] B. Jian and B.C. Vemuri, "Robust Point Set Registration Using Gaussian Mixture Models," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 33, no. 8, pp. 1633-1645, Aug. 2011. [5] A. Peter, A. Rangarajan, and J. Ho, "Shape l'Ane Rouge: Sliding Wavelets for Indexing and Retrieval," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2008. [6] H. Ling and D. Jacobs, "Using the Inner-Distance for Classification of Articulated Shapes," Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 719-726, 2005. [7] H. Ling and K. Okada, "An Efficient Earth Movers Distance Algorithm for Robust Histogram Comparison," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 3, pp. 840-853, May 2007. [8] X. Bai, X. Yang, L.J. Latecki, W. Liu, and Z. Tu, "Learning Context Sensitive Shape Similarity by Graph Transduction," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 32, no. 5, pp. 861-874, May 2010. [9] G. McNeill and S. Vijayakumar, "Hierarchical Procrustes Matching for Shape Retrieval," Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 885-894, 2006. [10] I. Biederman and G. Ju, "Surface vs. Edge-Based Determinants of Visual Recognition," Cognitive Psychology, vol. 20, pp. 38-64, 1988. [11] D.M. Gavrila, "A Bayesian, Exemplar-Based Approach to Hierarchical Shape Matching," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 8, pp. 1408-1421, Aug. 2007. [12] D. Macrini, K. Siddiqi, and S. Dickinson, "From Skeletons to Bone Graphs: Medial Abstraction for Object Recognition," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2008. [13] D. Weinland and E. Boyer, "Action Recognition Using Exemplar-Based Embedding," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-7, 2008. [14] T.B. Sebastian, P.N. Klein, and B.B. Kimia, "Recognition of Shapes by Editing Their Shock Graphs," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 5, pp. 550-571, May 2004. [15] A. Temlyakov, B.C. Munsell, J.W. Waggoner, and S. Wang, "Two Perceptually Motivated Strategies for Shape Classification," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 2289-2296, 2010. [16] Z. Tu and A. Yuille, "Shape Matching and Recognition: Using Generative Models and Informative Features," Proc. European Conf. Computer Vision, vol. 3, pp. 195-209, 2004. [17] C. Xu, J. Liu, and X. Tang, "2D Shape Matching by Contour Flexibility," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 1, pp. 180-186, Jan. 2009. [18] R. Gopalan, P. Turaga, and R. Chellappa, "Articulation-Invariant Representation of Non-Planar Shapes," Proc. 11th European Conf. Computer Vision, vol. 6313, pp. 286-299, 2010. [19] X. Yang, S.K. Tezel, and L. Latecki, "Locally Constrained Diffusion Process on Locally Densified Distance Spaces with Applications to Shape Retrieval," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 357-364, 2009. [20] H. Ling, X. Yang, and L.J. Latecki, "Balancing Deformability and Discriminability for Shape Matching," Proc. 11th European Conf. Computer Vision, pp. 411-424, 2010. [21] X. Yang and L.J. Latecki, "Affinity Learning on a Tensor Product Graph with Applications to Shape and Image Retrieval," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 2873-2880, 2011. [22] B.C. Munsell, A. Temlyakov, M. Styner, and S. Wang, "Pre-Organizing Shape Instances for Landmark-Based Shape Correspondence," Int'l J. Computer Vision, vol. 97, pp. 210-228, 2011. [23] K. Siddiqi and S.M. Pizer, Medial Representations: Mathematics, Algorithms and Applications. Springer, 2008. [24] M. Leyton, "A Process-Grammar for Shape," Artificial Intelligence, vol. 34, pp. 213-247, 1988. [25] P.J. Flynn and A. Jain, "CAD-Based Computer Vision: From CAD Models to Relational Graphs," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 2, pp. 114-132, Feb. 1991. [26] V. Ferrucci and A. Paoluzzi, "Extrusion and Boundary Evaluation for Multidimensional Polyhedra," Computer Aided Design, vol. 1, pp. 40-50, 1991. [27] J.C. Damski and J.S. Gero, "A Logic-Based Framework for Shape Representation," Computer-Aided Design, vol. 28, no. 3, pp. 169-181, 1996. [28] A.P.D. Pobil and M.A. Serna, Spatial Representation and Motion Planning, pp. 169-181, Springer-Verlag, 1995. [29] S. Maji, A.C. Berg, and J. Malik, "Classification Using Intersection Kernel Support Vector Machines is Efficient," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2008. [30] S.H. Joshi, E. Klassen, A. Srivastava, and I. Jermyn, "A Novel Representation for Riemannian Analysis of Elastic Curves in ${\hbox{\rlap{I}\kern 2.0pt{\hbox{R}}}}^n$ ," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-7, 2007. [31] L. Younes, "Computable Elastic Distance between Shapes," SIAM J. Applied Math., vol. 58, no. 2, pp. 565-586, 1998. [32] L. Chen, R. Feris, and M. Turk, "Efficient Partial Shape Matching Using Smith-Waterman Algorithm," Proc. IEEE Conf. Computer Vision and Pattern Recognition Worshops, pp. 1-6, 2008. [33] J. Shotton, A. Blake, and R. Cipolla, "Multiscale Categorical Object Recognition Using Contour Fragments," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 30, no. 7, pp. 1270-1281, July 2008. [34] A. Opelt, A. Pinz, and A. Zisserman, "A Boundary-Fragment-Model for Object Detection," Proc. Ninth European Conf. Computer Vision, vol. 2, pp. 575-588, 2006. [35] V. Ferrari, T. Tuytelaars, and L. Van Gool, "Object Detection by Contour Segment Networks," Proc. Ninth European Conf. Computer Vision, vol. 3, pp. 14-28, 2006. [36] G. Mori, S. Belongie, and J. Malik, "Efficient Shape Matching Using Shape Contexts," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 27, no. 11, pp. 1832-1837, Nov. 2005. [37] L. Ding and M. Belkin, "Component Based Shape Retrieval Using Differential Profiles," Proc. First ACM Int'l Conf. Multimedia Information Retrieval, pp. 216-222, 2008. [38] B.C. Vemuri, M. Liu, S.-I. Amari, and F. Nielsen, "Total Bregman Divergence and Its Applications to DTI Analysis," IEEE Trans. Medical Imaging, vol. 30, no. 2, pp. 475-483, Feb. 2011. [39] P.D. Groen, "An Introduction to Total Least Squares," Nieuw Archief voor Wiskunde, vol. 14, pp. 237-253, 1996. [40] J. Lancaster et al., "k-Tree Method for High-Speed Spatial Normalization," Human Brain Mapping, pp. 358-363, 1998. [41] F. Nielsen, P. Piro, and M. Barlaud, "Bregman Vantage Point Trees for Efficient Nearest Neighbor Queries," Proc. IEEE Int'l Conf. Multimedia and Expo, pp. 878-881, 2009. [42] L.J. Latecki, R. Lakamper, and U. Eckhardt, "Shape Descriptors for Non-Rigid Shapes with a Single Closed Contour," Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 424-429, 2000. [43] O.J.O. Soderkvist, "Computer Vision Classification of Leaves from Swedish Trees," Linkoping Univ., 2001. [44] P. Shilane, P. Min, M. Kazhdan, and T. Funkhouser, "The Princeton Shape Benchmark," Proc. Shape Modeling Int'l, pp. 167-178, 2004. [45] A. Banerjee, S. Merugu, I.S. Dhillon, and J. Ghosh, "Clustering with Bregman Divergences," J. Machine Learning Research, vol. 6, pp. 1705-1749, 2005. [46] J. Zhang, "Divergence Function, Duality, and Convex Analysis," Neural Computation, vol. 16, pp. 159-195, 2004. [47] B.A. Frigyik, S. Srivastava, and M.R. Gupta, "Functional Bregman Divergence," Proc. Int'l Symp. Information Theory, vol. 54, pp. 5130-5139, 2008. [48] F. Nielsen and S. Boltz, "The Burbea-Rao and Bhattacharyya Centroids," IEEE Trans. Information Theory, vol. 57, no. 8, pp. 5455-5466, Aug. 2011. [49] S. Amari, "Integration of Stochastic Models by Minimizing $\alpha$ -Divergence," Neural Computation, vol. 19, no. 10, pp. 2780-2796, 2007. [50] F. Nielsen and R. Nock, "On the Smallest Enclosing Information Disk," Information Processing Letters, vol. 105, pp. 93-97, 2008. [51] M.J. Wainwright and M.I. Jordan, "Graphical Models, Exponential Families and Variational Inference," Foundations and Trends in Machine Learning, vol. 1, pp. 1-305, 2008. [52] M. Collins, R. Schapire, and Y. Singer, "Logistic Regression, Adaboost, and Bregman Distances," Machine Learning, vol. 48, nos. 1-3, pp. 253-285, 2001. [53] S.D. Pietra, V.D. Pietra, and J. Lafferty, "Duality and Auxiliary Functions for Bregman Distances," Technical Report CMU-CS-01-109, School of Computer Science, Carnegie Mellon Univ., 2001. [54] T.M. Cover and J. Thomas, Elements of Information Theory. Wiley Interscience, 1991. [55] F. Nielsen and V. Garcia, "Statistical Exponential Families: A Digest with Flash Cards," arXiv.org:0911.4863, 2009. [56] S. Amari and H. Nagaoka, Methods of Information Geometry. Am. Math. Soc., 2001. [57] O. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory. Wiley, 1978. [58] A. Strehl and J. Ghosh, "Cluster Ensembles—A Knowledge Reuse Framework for Combining Partitionings," J. Machine Learning Research, vol. 3, no. 3, pp. 583-617, 2002. [59] C.D. Manning, P. Raghavan, and H. Schütze, Introduction to Information Retrieval. Cambridge Univ. Press, 2008. [60] J. Ho, M. Yang, A. Rangarajan, and B. Vemuri, "A New Affine Registration Algorithm for Matching 2D Point Sets," Proc. Eighth IEEE Workshop Applications of Computer Vision, vol. 89, pp. 25-30, 2007. [61] P. Felzenszwalb and J. Schwartz, "Hierarchical Matching of Deformable Shapes," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, 2007. [62] X. Bai, B. Wang, X. Wang, W. Liu, and Z. Tu, "Co-Transduction for Shape Retrieval," Proc. 11th European Conf. Computer Vision, pp. 861-874, 2010. [63] T.B. Sebastian, P.N. Klein, and B.B. Kimia, "On Aligning Curves," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, no. 1, pp. 116-125, Jan. 2003. [64] S. Belongie, J. Malik, and J. Puzicha, "Shape Matching and Object Recognition Using Shape Contexts," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 4, pp. 509-522, Apr. 2002. [65] F. Mokhtarian, S. Abbasi, and J. Kittler, "Efficient and Robust Retrieval by Shape Content through Curvature Scale Space," Proc. Int'l Workshop Image Databases and Multi-Media Search, pp. 51-58, 1997. [66] P. Kontschieder, M. Donoser, and H. Bischof, "Beyond Pairwise Shape Similarity Analysis," Proc. Ninth Asian Conf. Computer Vision, pp. 655-666, 2009. [67] S. Papadakis, I. Pratikakis, and T. Theoharis, "Efficient 3D Shape Matching and Retrieval Using a Concrete Radialized Spherical Projection Representation," Pattern Recognition, vol. 40, no. 9, pp. 2437-2452, 2007. [68] C.B. Akgul, B. Sankur, Y. Yemez, and F. Schmitt, "3D Model Retrieval Using Probability Density-Based Shape Descriptors," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 6, pp. 1117-1133, June 2009. [69] D. Vranic, "Tools for 3D Model Retrieval," http:// merkur01.inf. uni-konstanz.de3Dtools /, 2005. [70] R. Osada, T. Funkhouser, B. Chazelle, and D. Dobkin, "Shape Distributions," ACM Trans. Graphics, vol. 21, no. 4, pp. 807-832, 2002.