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Issue No.01 - Jan. (2014 vol.36)
pp: 171-180
R. Litman , Sch. of Electr. Eng., Tel Aviv Univ., Tel Aviv, Israel
A. M. Bronstein , Sch. of Electr. Eng., Tel Aviv Univ., Tel Aviv, Israel
Informative and discriminative feature descriptors play a fundamental role in deformable shape analysis. For example, they have been successfully employed in correspondence, registration, and retrieval tasks. In recent years, significant attention has been devoted to descriptors obtained from the spectral decomposition of the Laplace-Beltrami operator associated with the shape. Notable examples in this family are the heat kernel signature (HKS) and the recently introduced wave kernel signature (WKS). The Laplacian-based descriptors achieve state-of-the-art performance in numerous shape analysis tasks; they are computationally efficient, isometry-invariant by construction, and can gracefully cope with a variety of transformations. In this paper, we formulate a generic family of parametric spectral descriptors. We argue that to be optimized for a specific task, the descriptor should take into account the statistics of the corpus of shapes to which it is applied (the "signal") and those of the class of transformations to which it is made insensitive (the "noise"). While such statistics are hard to model axiomatically, they can be learned from examples. Following the spirit of the Wiener filter in signal processing, we show a learning scheme for the construction of optimized spectral descriptors and relate it to Mahalanobis metric learning. The superiority of the proposed approach in generating correspondences is demonstrated on synthetic and scanned human figures. We also show that the learned descriptors are robust enough to be learned on synthetic data and transferred successfully to scanned shapes.
Shape, Heating, Kernel, Manifolds, Measurement, Equations, Eigenvalues and eigenfunctions,Mahalanobis distance, Diffusion geometry, heat kernel signature (HKS), wave kernel signature (WKS), descriptor, deformable shapes, correspondence, retrieval, spectral methods, Laplace-Beltrami operator, metric learning, Wiener filter
R. Litman, A. M. Bronstein, "Learning Spectral Descriptors for Deformable Shape Correspondence", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.36, no. 1, pp. 171-180, Jan. 2014, doi:10.1109/TPAMI.2013.148
[1] N. Gelfand, N.J. Mitra, L.J. Guibas, and H. Pottmann, "Robust Global Registration," Proc. Third Eurographics Symp. Geometry Processing, pp. 197-206, 2005.
[2] C. Wang, A.M. Bronstein, M.M. Bronstein, and N. Paragios, "Discrete Minimum Distortion Correspondence Problems for Non-Rigid Shape Matching," Proc. Scale Space and Variational Method, vol. 6667, pp. 580-591, 2011.
[3] N.J. Mitra, L. Guibas, J. Giesen, and M. Pauly, "Probabilistic Fingerprints for Shapes," Proc. Fourth Eurographics Symp. Geometry Processing, vol. 256, pp. 121-130, 2006.
[4] A. Bronstein, M. Bronstein, L. Guibas, and M. Ovsjanikov, "Shape Google: Geometric Words and Expressions for Invariant Shape Retrieval," ACM Trans. Graphics, vol. 30, no. 1, p. 1, 2011.
[5] P. Skraba, M. Ovsjanikov, F. Chazal, and L. Guibas, "Persistence-Based Segmentation of Deformable Shapes," Proc. IEEE Conf. Computer Vision and Pattern Recognition Workshops, pp. 45-52, 2010.
[6] S. Belongie, J. Malik, and J. Puzicha, "Shape Context: A New Descriptor for Shape Matching and Object Recognition," Proc. Advances in Neural Information Processing Systems, pp. 831-837, 2001.
[7] A.E. Johnson and M. Hebert, "Using Spin Images for Efficient Object Recognition in Cluttered 3D Scenes," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 5, pp. 433-449, May 1999.
[8] S. Manay, B. Hong, A. Yezzi, and S. Soatto, "Integral Invariant Signatures," Proc. Eighth European Conf. Computer Vision, pp. 87-99, 2004.
[9] M. Pauly, R. Keiser, and M. Gross, "Multi-Scale Feature Extraction on Point-Sampled Surfaces," Computer Graphics Forum, vol. 22, no. 3, pp. 281-289, 2003.
[10] A. Hamza and H. Krim, "Geodesic Object Representation and Recognition," Proc. Discrete Geometry for Computer Imagery, pp. 378-387, 2003.
[11] A. Elad and R. Kimmel, "On Bending Invariant Signatures for Surfaces," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, no. 10, pp. 1285-1311, Oct. 2003.
[12] Y. Lipman and T. Funkhouser, "Möbius Voting for Surface Correspondence," ACM Trans. Graphics, vol. 28, no. 3, p. 72, 2009.
[13] P. Bérard, G. Besson, and S. Gallot, "Embedding Riemannian Manifolds by Their Heat Kernel," Geometric and Functional Analysis, vol. 4, no. 4, pp. 373-398, 1994.
[14] R. Coifman and S. Lafon, "Diffusion Maps," Applied and Computational Harmonic Analysis, vol. 21, no. 1, pp. 5-30, 2006.
[15] F. Mémoli, "Spectral Gromov-Wasserstein Distances for Shape Matching," Proc. 12th IEEE Int'l Conf. Computer Vision Workshops, pp. 256-263, 2009.
[16] A.M. Bronstein, M.M. Bronstein, R. Kimmel, M. Mahmoudi, and G. Sapiro, "A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-Rigid Shape Matching," Int'l J. Computer Vision, vol. 89, nos. 2-3, pp. 266-286, 2010.
[17] B. Lévy, "Laplace-Beltrami Eigenfunctions Toward an Algorithm that 'Understands' Geometry," Proc. IEEE Int'l Conf. Shape Modeling and Applications, pp. 13-13, 2006.
[18] R. Rustamov, "Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation," Proc. Symp. Geometry Processing, pp. 225-233, 2007.
[19] J. Sun, M. Ovsjanikov, and L. Guibas, "A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion," Computer Graphics Forum, vol. 28, no. 5, pp. 1383-1392, 2009.
[20] M. Aubry, U. Schlickewei, and D. Cremers, "The Wave Kernel Signature: A Quantum Mechanical Approach to Shape Analysis," Proc. IEEE Int'l Conf. Computer Vision Workshops, pp. 1626-1633, 2011.
[21] A. Bronstein et al., "SHREC 2010: Robust Correspondence Benchmark," Proc. Eurographics Workshop 3D Object Retrieval, pp. 87-91, 2010.
[22] A. Bronstein et al., "SHREC 2010: Robust Large-Scale Shape Retrieval Benchmark," Proc. Eurographics Workshop 3D Object Retrieval, pp. 71-78, 2010.
[23] T. Dey, K. Li, C. Luo, P. Ranjan, I. Safa, and Y. Wang, "Persistent Heat Signature for Pose-Oblivious Matching of Incomplete Models," Computer Graphics Forum, vol. 29, no. 5, pp. 1545-1554, 2010.
[24] M.M. Bronstein and I. Kokkinos, "Scale-Invariant Heat Kernel Signatures for Non-Rigid Shape Recognition," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1704-1711, 2010.
[25] J. Aflalo, A.M. Bronstein, M.M. Bronstein, and R. Kimmel, "Deformable Shape Retrieval by Learning Diffusion Kernels," Proc. Scale Space and Variational Methods, 2011.
[26] M. Kac, "Can One Hear the Shape of a Drum?" The Am. Math. Monthly, vol. 73, no. 4, pp. 1-23, 1966.
[27] M. Reuter, F. Wolter, and N. Peinecke, "Laplace-Beltrami Spectra as 'Shape-DNA' of Surfaces and Solids," Computer-Aided Design, vol. 38, no. 4, pp. 342-366, 2006.
[28] A. Sharma and R. Horaud, "Shape Matching Based on Diffusion Embedding and on Mutual Isometric Consistency," Proc. IEEE Conf. Computer Vision and Pattern Recognition Workshops, pp. 29-36, 2010.
[29] D. Raviv, M.M. Bronstein, A.M. Bronstein, and R. Kimmel, "Volumetric Heat Kernel Signatures," Proc. ACM Workshop 3D Object Retrieval, pp. 39-44, 2010.
[30] M. Leordeanu and M. Hebert, "A Spectral Technique for Correspondence Problems Using Pairwise Constraints," Proc. IEEE Int'l Conf. Computer Vision, vol. 2, pp. 1482-1489, 2010.
[31] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, with Engineering Applications. MIT Press, 1949.
[32] L. Yang and R. Jin, Distance Metric Learning: A Comprehensive Survey, pp. 1-51, Michigan State Univ., 2006.
[33] K. Weinberger, J. Blitzer, and L. Saul, "Distance Metric Learning for Large Margin Nearest Neighbor Classification," Proc. Advances in Neural Information Processing Systems, vol. 18, pp. 1473-1480, 2005.
[34] J.V. Davis, B. Kulis, P. Jain, S. Sra, and I.S. Dhillon, "Information-Theoretic Metric Learning," Proc. 24th Int'l Conf. Machine learning, pp. 209-216, 2007.
[35] C. Strecha, A. Bronstein, M. Bronstein, and P. Fua, "LDAHash: Improved Matching with Smaller Descriptors," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 34, no. 1, pp. 66-78, Jan. 2012.
[36] X. He, D. Cai, and J. Han, "Learning a Maximum Margin Subspace for Image Retrieval," IEEE Trans. Knowledge and Data Eng., vol. 20, no. 2, pp. 189-201, Feb. 2008.
[37] A. Bronstein, M. Bronstein, and R. Kimmel, Numerical Geometry of Non-Rigid Shapes. Springer, 2008.
[38] D. Anguelov, P. Srinivasan, D. Koller, S. Thrun, J. Rodgers, and J. Davis, "Scape: Shape Completion and Animation of People," ACM Trans. Graphics, vol. 24, no. 3, pp. 408-416, 2005.
[39] Y. Eldar, M. Lindenbaum, M. Porat, and Y.Y. Zeevi, "The Farthest Point Strategy for Progressive Image Sampling," IEEE Trans. Image Processing, vol. 6, no. 9, pp. 1305-1315, Sept. 1997.
[40] P. Shilane, P. Min, M. Kazhdan, and T. Funkhouser, "The Princeton Shape Benchmark," Proc. IEEE Shape Modeling Applications, pp. 167-178, 2004.
[41] Y. Weiss, A. Torralba, and R. Fergus, "Spectral Hashing," Proc. Advances in Neural Information Processing Systems, 2008.
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