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Issue No.02 - February (2009 vol.31)
pp: 337-350
Adrian M. Peter , University of Florida, Gainesville
Anand Rangarajan , University of Florida Dept. of CISE, Gainesville Gainesville
Shape matching plays a prominent role in the comparison of similar structures. We present a unifying framework for shape matching that uses mixture models to couple both the shape representation and deformation. The theoretical foundation is drawn from information geometry wherein information matrices are used to establish intrinsic distances between parametric densities. When a parameterized probability density function is used to represent a landmark-based shape, the modes of deformation are automatically established through the information matrix of the density. We first show that given two shapes parameterized by Gaussian mixture models (GMMs), the well-known Fisher information matrix of the mixture model is also a Riemannian metric (actually, the Fisher-Rao Riemannian metric) and can therefore be used for computing shape geodesics. The Fisher-Rao metric has the advantage of being an intrinsic metric and invariant to reparameterization. The geodesic?computed using this metric?establishes an intrinsic deformation between the shapes, thus unifying both shape representation and deformation. A fundamental drawback of the Fisher-Rao metric is that it is not available in closed form for the GMM. Consequently, shape comparisons are computationally very expensive. To address this, we develop a new Riemannian metric based on generalized \phi-entropy measures. In sharp contrast to the Fisher-Rao metric, the new metric is available in closed form. Geodesic computations using the new metric are considerably more efficient. We validate the performance and discriminative capabilities of these new information geometry-based metrics by pairwise matching of corpus callosum shapes. We also study the deformations of fish shapes that have various topological properties. A comprehensive comparative analysis is also provided using other landmark-based distances, including the Hausdorff distance, the Procrustes metric, landmark-based diffeomorphisms, and the bending energies of the thin-plate (TPS) and Wendland splines.
Information geometry, Fisher information, Fisher-Rao metric, Havrda-Charvá t entropy, Gaussian mixture models, shape analysis, shape matching, landmark shapes.
Adrian M. Peter, Anand Rangarajan, "Information Geometry for Landmark Shape Analysis: Unifying Shape Representation and Deformation", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.31, no. 2, pp. 337-350, February 2009, doi:10.1109/TPAMI.2008.69
[1] F.L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge Univ. Press, 1991.
[2] A. Peter and A. Rangarajan, “Shape Matching Using the Fisher-Rao Riemannian Metric: Unifying Shape Representation and Deformation,” Proc. IEEE Int'l Symp. Biomedical Imaging (ISBI '06), pp. 1164-1167, 2006.
[3] A. Peter and A. Rangarajan, “A New Closed-Form Information Metric for Shape Analysis,” Proc. Int'l Conf. Medical Image Computing and Computer Assisted Intervention (MICCAI '06), pp.249-256, 2006.
[4] C. Small, The Statistical Theory of Shape. Springer, 1996.
[5] F.L. Bookstein, “Principal Warps: Thin-Plate Splines and the Decomposition of Deformations,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 6, pp. 567-585, June 1989.
[6] K. Rohr, H.S. Stiehl, R. Sprengel, T.M. Buzug, J. Weese, and M.H. Kuhn, “Landmark-Based Elastic Registration Using Approximating Thin-Plate Splines,” IEEE Trans. Medical Imaging, vol. 20, no. 6, pp. 526-534, June 2001.
[7] R.H. Davies, C. Twining, T.F. Cootes, and C.J. Taylor, “An Information Theoretic Approach to Statistical Shape Modelling,” Proc. British Machine Vision Conf. (BMVC '01), vol. 1, pp. 3-12, 2001.
[8] V. Camion and L. Younes, “Geodesic Interpolating Splines,” Proc. Int'l Conf. Energy Minimization Methods for Computer Vision and Pattern Recognition (EMMCVPR '01), pp. 513-527, 2001.
[9] S. Joshi and M. Miller, “Landmark Matching via Large Deformation Diffeomorphisms,” IEEE Trans. Image Processing, vol. 9, no. 8, pp. 1357-1370, Aug. 2000.
[10] H. Chui and A. Rangarajan, “A New Point Matching Algorithm for Non-Rigid Registration,” Computer Vision and Image Understanding, vol. 89, nos. 2-3, pp. 114-141, Mar. 2003.
[11] H. Guo, A. Rangarajan, and S. Joshi, “3-D Diffeomorphic Shape Registration on Hippocampal Data Sets,” Proc. Int'l Conf. Medical Image Computing and Computer Assisted Intervention (MICCAI '05), pp. 984-991, 2005.
[12] K. Siddiqi, A. Shokoufandeh, S.J. Dickinson, and S.W. Zucker, “Shock Graphs and Shape Matching,” Proc. IEEE Int'l Conf. Computer Vision (ICCV '98), pp. 222-229, 1998.
[13] A. Srivastava, S. Joshi, W. Mio, and X. Liu, “Statistical Shape Analysis: Clustering, Learning and Testing,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 27, no. 4, pp. 590-602, Apr. 2005.
[14] P. Thompson and A.W. Toga, “A Surface-Based Technique for Warping Three-Dimensional Images of the Brain,” IEEE Trans. Medical Imaging, vol. 5, no. 4, pp. 402-417, Aug. 1996.
[15] J. Burbea and R. Rao, “Entropy Differential Metric, Distance and Divergence Measures in Probability Spaces: A Unified Approach,” J. Multivariate Analysis, vol. 12, pp. 575-596, 1982.
[16] M.E. Havrda and F. Charvát, “Quantification Method of Classification Processes: Concept of Structural $\alpha\hbox{-}{\rm entropy}$ ,” Kybernetica, vol. 3, pp. 30-35, 1967.
[17] Y.W.K. Woods and M. McClain, “Information-Theoretic Matching of Two Point Sets,” IEEE Trans. Image Processing, vol. 11, no. 8, pp.868-872, Aug. 2002.
[18] F. Wang, B.C. Vemuri, A. Rangarajan, I.M. Schmalfuss, and S.J. Eisenschenk, “Simultaneous Nonrigid Registration of Multiple Point Sets and Atlas Construction,” Proc. European Conf. Computer Vision (ECCV '06), pp. 551-563, 2006.
[19] B. Jian and B.C. Vemuri, “A Robust Algorithm for Point Set Registration Using Mixture of Gaussians,” Proc. IEEE Int'l Conf. Computer Vision (ICCV '05), vol. 2, pp. 1246-1251, 2005.
[20] J. Lin, “Divergence Measures Based on the Shannon Entropy,” IEEE Trans. Information Theory, vol. 37, no. 1, pp. 145-151, Jan. 1991.
[21] N. Paragios, M. Rousson, and V. Ramesh, “Non-Rigid Registration Using Distance Functions,” Computer Vision and Image Understanding, vol. 89, nos. 2-3, pp. 142-165, Mar. 2003.
[22] J. Glaunes, A. Trouvé, and L. Younes, “Diffeomorphic Matching of Distributions: A New Approach for Unlabeled Point-Sets and Sub-Manifolds Matching,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR), vol. 2, pp. 712-718, 2004.
[23] G. Wahba, Spline Models for Observational Data, Soc. for Industrial and Applied Math., 1990.
[24] T. Cootes and C. Taylor, “A Mixture Model for Representing Shape Variation,” Proc. British Machine Vision Conf. (BMVC '97), pp. 110-119, 1997.
[25] G.J. McLachlan and K.E. Basford, Mixture Models: Inference and Applications to Clustering. Marcel Dekker, 1988.
[26] M.A.T. Figueiredo and A.K. Jain, “Unsupervised Learning of Finite Mixture Models,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 3, pp. 381-396, Mar. 2002.
[27] C.R. Rao, “Information and Accuracy Attainable in Estimation of Statistical Parameters,” Bull. of the Calcutta Math. Soc., vol. 37, pp.81-91, 1945.
[28] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, 2002.
[29] N.N. Čencov, Statistical Decision Rules and Optimal Inference, Am. Math. Soc., 1982.
[30] S.-I. Amari and H. Nagaoka, Methods of Information Geometry, Am. Math. Soc., 2001.
[31] I.J. Myung, V. Balasubramanian, and M.A. Pitt, “Counting Probability Distributions: Differential Geometry and Model Selection,” Proc. Nat'l Academy of Sciences, vol. 97, pp. 11 170-11 175, 2000.
[32] S.J. Maybank, “The Fisher-Rao Metric for Projective Transformations of the Line,” Int'l J. Computer Vision, vol. 63, no. 3, pp. 191-206, 2005.
[33] W. Mio, D. Badlyans, and X. Liu, “A Computational Approach to Fisher Information Geometry with Applications to Image Analysis,” Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR '05), pp. 18-33, 2005.
[34] C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras, “Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing,” J. Math. Imaging and Vision, vol. 25, no. 3, pp. 423-444, 2006.
[35] A. Srivastava, I. Jermyn, and S. Joshi, “Riemannian Analysis of Probability Density Functions with Applications in Vision,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR), pp. 1-8, 2007.
[36] R. Courant and D. Hilbert, Methods of Math. Physics, vol. 2. Wiley Interscience, 1989.
[37] W. Mio and X. Liu, “Landmark Representation of Shapes and Fisher-Rao Geometry,” Proc. IEEE Int'l Conf. Image Processing (ICIP '06), pp. 2113-2116, 2006.
[38] B.K.P. Horn, Robot Vision. MIT Press, 1986.
[39] S.I.R. Costa, S. Santos, and J.E. Strapasson, “Fisher Information Matrix and Hyperbolic Geometry,” Proc. IEEE Information Theory Workshop, pp. 28-30, 2005.
[40] Catalog of Fishes, Dept. of Ichthyology, California Academy of Sciences, /, 2007.
[41] D.G. Kendall, “Shape-Manifolds, Procrustean Metrics and Complex Projective Spaces,” Bull. of the London Math. Soc., vol. 16, pp.81-121, 1984.
[42] D.P. Huttenlocher, G.A. Klanderman, and W.J. Rucklidge, “Comparing Images Using the Hausdorff Distance,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 9, pp. 850-863, Sept. 1993.
[43] M. Fornefett, K. Rohr, and H.S. Stiehl, “Radial Basis Functions with Compact Support for Elastic Registration of Medical Images,” Image and Vision Computing, vol. 19, no. 1, pp. 87-96, Jan. 2001.
[44] A. Banerjee, I. Dhillon, J. Ghosh, and S. Sra, “Clustering on the Unit Hypersphere Using Von Mises-Fisher Distributions,” J.Machine Learning Research, vol. 6, pp. 1345-1382, 2005.
[45] G. Lebanon, “Riemannian Geometry and Statistical Machine Learning,” PhD dissertation, Carnegie Mellon Univ., 2005.
[46] A. Peter, A. Rangarajan, and J. Ho, “Shape L'Âne Rouge: Sliding Wavelets for Indexing and Retrieval,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR), pp. 1-8, 2008.
[47] GatorBait 100, Univ. of Florida, , 2008.
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