The Community for Technology Leaders
RSS Icon
Issue No.07 - July (2011 vol.33)
pp: 1415-1428
Anuj Srivastava , Florida State University, Tallahassee
Eric Klassen , Florida State University, Tallahassee
Shantanu H. Joshi , UCLA School of Medicine, Los Angeles
Ian H. Jermyn , Durham University, Durham
This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the {\hbox{\rlap{I}\kern 2.0pt{\hbox{L}}}}^2 metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses.
Elastic curves, Riemannian shape analysis, elastic metric, Fisher-Rao metric, square-root representations, path straightening method, elastic geodesics, parallel transport, shape models.
Anuj Srivastava, Eric Klassen, Shantanu H. Joshi, Ian H. Jermyn, "Shape Analysis of Elastic Curves in Euclidean Spaces", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.33, no. 7, pp. 1415-1428, July 2011, doi:10.1109/TPAMI.2010.184
[1] S. Amari, Differential Geometric Methods in Statistics, vol. 28. Springer, 1985.
[2] A. Bhattacharya, "On a Measure of Divergence between Two Statistical Populations Defined by Their Probability Distributions," Bull. of Calcutta Math. Soc., vol. 35, pp. 99-109, 1943.
[3] A.M. Bronstein, M.M. Bronstein, and R. Kimmel, "Three-Dimensional Face Recognition," Int'l J. Computer Vision, vol. 64, no. 1, pp. 5-30, 2005.
[4] N.N. Čencov, Statistical Decision Rules and Optimal Inferences. Am. Math. Soc., 1982.
[5] H. Drira, B. Ben Amor, A. Srivastava, and M. Daoudi, "A Riemannian Analysis of 3d Nose Shapes for Partial Human Biometrics," Proc. IEEE Int'l Conf. Computer Vision, 2009.
[6] I.L. Dryden and K.V. Mardia, Statistical Shape Analysis. John Wiley & Sons, 1998.
[7] M. Frenkel and R. Basri, "Curve Matching Using Fast Marching Method," Proc. Fourth Int'l Workshop Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 35-51, 2003.
[8] S.H. Joshi, E. Klassen, A. Srivastava, and I. Jermyn, "Removing Shape-Preserving Transformations in Square-Root Elastic (SRE) Framework for Shape Analysis of Curves," Proc. Sixth Int'l Conf. Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 387-398, 2007.
[9] S.H. Joshi, E. Klassen, A. Srivastava, and I.H. Jermyn, "A Novel Representation for Riemannian Analysis of Elastic Curves," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-7, 2007.
[10] H. Karcher, "Riemannian Center of Mass and Mollifier Smoothing," Comm. Pure and Applied Math., vol. 30, no. 5, pp. 509-541, 1977.
[11] D. Kaziska and A. Srivastava, "Joint Gait-Cadence Analysis for Human Identification Using an Elastic Shape Framework," Comm. Statistics—Theory and Methods, vol. 39, no. 10, pp. 1817-1831, 2010.
[12] D.G. Kendall, "Shape Manifolds, Procrustean Metrics and Complex Projective Spaces," Bull. of the London Math. Soc., vol. 16, no. 2, pp. 81-121, 1984.
[13] M. Kilian, N.J. Mitra, and H. Pottmann, "Geometric Modeling in Shape Space," Proc. ACM SIGGRAPH, 2007.
[14] E. Klassen, A. Srivastava, W. Mio, and S.H. Joshi, "Analysis of Planar Shapes Using Geodesic Paths on Shape Spaces," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 3, pp. 372-383, Feb. 2004.
[15] S. Lang, Fundamentals of Differential Geometry. Springer, 1999.
[16] W. Liu, A. Srivastava, and J. Zhang, "Protein Structure Alignment Using Elastic Shape Analysis," Proc. ACM Conf. Bioinformatics and Computational Biology, Aug. 2010.
[17] A.C.G. Mennuci, Metrics of Curves in Shape Optimization and Analysis. 2009.
[18] P.W. Michor and D. Mumford, "Riemannian Geometries on Spaces of Plane Curves," J. European Math. Soc., vol. 8, pp. 1-48, 2006.
[19] J.W. Milnor, Topology from the Differentiable Viewpoint. Princeton Univ. Press, 1997.
[20] W. Mio, A. Srivastava, and S.H. Joshi, "On Shape of Plane Elastic Curves," Int'l J. Computer Vision, vol. 73, no. 3, pp. 307-324, 2007.
[21] I. Mpiperis, S. Malassiotis, and M.G. Strintzis, "3-D Face Recognition with the Geodesic Polar Representation," IEEE Trans. Information Forensics and Security, vol. 2, no. 3, pp. 537-547, Sept. 2007.
[22] R.S. Palais, "Morse Theory on Hilbert Manifolds," Topology, vol. 2, pp. 299-349, 1963.
[23] C. Samir, A. Srivastava, and M. Daoudi, "Three-Dimensional Face Recognition Using Shapes of Facial Curves," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 11, pp. 1858-1863, Nov. 2006.
[24] C. Samir, A. Srivastava, M. Daoudi, and S. Kurtek, "On Analyzing Symmetry of Objects Using Elastic Deformations," Proc. Int'l Conf. Computer Vision Theory and Applications , Feb. 2009.
[25] S. Savarese and F.-F. Li, "View Synthesis for Recognizing Unseen Poses of Object Classes," Proc. 10th European Conf. Computer Vision, 2008.
[26] 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.
[27] J. Shah, "An ${\rm H}^2$ Type Riemannian Metric on the Space of Planar Curves," Proc. Workshop Math. Foundations of Computational Anatomy, 2006.
[28] A. Srivastava, I. Jermyn, and S.H. Joshi, "Riemannian Analysis of Probability Density Functions with Applications in Vision," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1-8, June 2007.
[29] A. Srivastava and I.H. Jermyn, "Looking for Shapes in Two-Dimensional, Cluttered Point Clouds," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 9, pp. 1616-1629, Sept. 2009.
[30] A. Srivastava, S.H. Joshi, W. Mio, and X. Liu, "Statistical Shape Anlaysis: Clustering, Learning and Testing," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 27, no. 4, pp. 590-602, Apr. 2005.
[31] A. Srivastava, C. Samir, S.H. Joshi, and M. Daoudi, "Elastic Shape Models for Face Analysis Using Curvilinear Coordinates," J. Math. Imaging and Vision, vol. 33, no. 2, pp. 253-265, Feb., 2009.
[32] G. Sundaramoorthi, A.C.G. Mennucci, S. Soatto, and A. Yezzi, "A New Geometric Metric in the Space of Curves, and Applications to Tracking Deforming Objects by Prediction and Filtering," 2010.
[33] L. Younes, "Computable Elastic Distance between Shapes," SIAM J. Applied Math., vol. 58, no. 2, pp. 565-586, 1998.
[34] L. Younes, P.W. Michor, J. Shah, D. Mumford, and R. Lincei, "A Metric on Shape Space with Explicit Geodesics," Matematica e Applicazioni, vol. 19, no. 1, pp. 25-57, 2008.
[35] L. Younes, A. Qiu, R.L. Winslow, and M.I. Miller, "Transport of Relational Structures in Groups of Diffeomorphisms," J. Math. Imaging and Vision, vol. 32, no. 1, pp. 41-56, 2008.
293 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool