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Issue No.06 - June (2008 vol.30)
pp: 1093-1108
ABSTRACT
This paper proposes a deterministic observer framework for visual tracking based on non-parametric implicit (level-set) curve descriptions. The observer is continuous-discrete, with continuous-time system dynamics and discrete-time measurements. Its state-space consists of an estimated curve position augmented by additional states (e.g., velocities) associated with every point on the estimated curve. Multiple simulation models are proposed for state prediction. Measurements are performed through standard static segmentation algorithms and optical-flow computations. Special emphasis is given to the geometric formulation of the overall dynamical system. The discrete-time measurements lead to the problem of geometric curve interpolation and the discrete-time filtering of quantities propagated along with the estimated curve. Interpolation and filtering are intimately linked to the correspondence problem between curves. Correspondences are established by a Laplace-equation approach. The proposed scheme is implemented completely implicitly (by Eulerian numerical solutions of transport equations) and thus naturally allows for topological changes and subpixel accuracy on the computational grid.
INDEX TERMS
computer vision, observers, active contours
CITATION
Patricio A. Vela, Marc Niethammer, "Geometric Observers for Dynamically Evolving Curves", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.30, no. 6, pp. 1093-1108, June 2008, doi:10.1109/TPAMI.2008.28
REFERENCES
[1] Applied Optimal Estimation, A. Gelb, ed., 15th ed. MIT Press, 1999.
[2] D.G. Luenberger, “An Introduction to Observers,” IEEE Trans. Automatic Control, vol. 16, no. 6, pp. 596-602, 1971.
[3] R.E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” J. Basic Eng., vol. 82, no. 1, pp. 35-45, 1960.
[4] M. Arulampalam, S. Maskell, N. Gordon, T. Clapp, D. Sci, T. Organ, and S. Adelaide, “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking,” IEEE Trans. Signal Processing, vol. 50, no. 2, pp. 174-188, 2002.
[5] S.K. Mitter, “Filtering and Stochastic Control: A Historical Perspective,” IEEE Control Systems Magazine, vol. 16, no. 3, pp.67-76, 1996.
[6] S.V. Lototsky, “Problems in Statistic of Stochastic Differential Equations,” PhD dissertation, Univ. of Southern California, 1996.
[7] R.N. Miller, E.F. Carter, and S.T. Blue, “Data Assimilation into Nonlinear Stochastic Models,” Tellus A, vol. 51, pp. 167-194, 1999.
[8] A.V. Wouwer and M. Zeitz, “State Estimation in Distributed Parameter Systems,” Control Systems, Robotics and Automation, Theme in Encyclopedia of Life Support Systems, EOLSS, 2001.
[9] A. Yilmaz, X. Li, and M. Shah, Object Contour Tracking Using Level Sets. 2004.
[10] A. Blake and M. Isard, Active Contours. Springer, 1998.
[11] Y. Rathi, N. Vaswani, A. Tannenbaum, and A. Yezzi, “Tracking Deforming Objects Using Particle Filtering for Geometric Active Contours,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 8, pp. 1470-1475, Aug. 2007.
[12] Y. Rathi, N. Vaswani, and A. Tannenbaum, “A Generic Framework for Tracking Using Particle Filter with Dynamic Shape Prior,” IEEE Trans. Image Processing, vol. 16, no. 5, pp. 1370-2007, 2007.
[13] N. Vaswani, A. Yezzi, Y. Rathi, and A. Tannenbaum, “Time-Varying Finite Dimensional Basis for Tracking Contour Deformations,” Proc. Conf. Decision and Control, pp. 1665-1672, 2006.
[14] S.K. Zhou, R. Chellappa, and B. Moghaddam, “Visual Tracking and Recognition Using Appearance-Adaptive Models in Particle Filters,” IEEE Trans. Image Processing, vol. 13, no. 11, pp. 1491-1506, 2004.
[15] F. Dornaika and F. Davoine, “On Appearance Based Face and Facial Action Tracking,” IEEE Trans. Circuits and Systems for Video Technology, vol. 16, no. 9, pp. 1107-1124, 2006.
[16] D. Cremers, “Dynamical Statistical Shape Priors for Level Set-Based Tracking,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 8, pp. 1262-1273, Aug. 2006.
[17] A.-R. Mansouri, “Region Tracking via Level Set PDEs without Motion Computation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 7, pp. 947-961, July 2002.
[18] N. Papadakis and E. Mémin, “Variational Optimal Control Technique for the Tracking of Deformable Objects,” Proc. Int'l Conf. Computer Vision, 2007.
[19] N. Peterfreund, “Robust Tracking of Position and Velocity with Kalman Snakes,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 6, pp. 564-569, June 1999.
[20] J.D. Jackson, A.J. Yezzi, and S. Soatto, “Tracking Deformable Moving Objects under Severe Occlusions,” Proc. Conf. Decision and Control, 2004.
[21] P.W. Michor and D. Mumford, Riemannian Geometries on Spaces of Plane Curves, http://arxiv.org/abs/math0312384, 2008.
[22] A. Yezzi and A. Mennucci, “Conformal Metrics and True “Gradient Flows” for Curves,” Proc. Int'l Conf. Computer Vision, pp. 913-919, 2005.
[23] L. Younes, “Computable Elastic Distances between Shapes,” SIAM J. Applied Math., vol. 58, no. 2, pp. 565-586, 1998.
[24] L. Younes, “Optimal Matching between Shapes via Elastic Deformations,” Image and Vision Computing, vol. 5-6, pp. 381-389, 1999.
[25] 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, Mar. 2004.
[26] I. Cohen, N. Ayache, and P. Sulger, “Tracking Points on Deformable Objects Using Curvature Information,” Technical Report 1595, INRIA, 1991.
[27] R. Basri, L. Costa, D. Geiger, and D. Jacobs, “Determining the Similarity of Deformable Shapes,” Vision Research, vol. 38, pp.2365-2385, 1998.
[28] G. Charpiat, O. Faugeras, and R. Keriven, “Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics,” Foundations of Computational Math., pp. OF1-OF58, 2004.
[29] H.D. Tagare, D. O'Shea, and D. Groissier, “Non-Rigid Shape Comparison of Plane Curves in Images,” J. Math. Imaging and Vision, vol. 16, no. 1, pp. 57-68, 2002.
[30] 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. 116-124, Jan. 2003.
[31] M.I. Miller and L. Younes, “Group Actions, Homeomorphisms, and Matching: A General Framework,” Int'l J. Computer Vision, vol. 41, pp. 61-84, 2001.
[32] M.F. Beg, M.I. Miller, A. Trouvé, and L. Younes, “Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms,” Int'l J. Computer Vision, vol. 61, no. 2, pp.139-157, 2005.
[33] S. Angenent, S. Haker, and A. Tannenbaum, “Minimizing Flows for the Monge-Kantorovich Problem,” SIAM J. Math. Analysis, vol. 35, pp. 61-97, 2003.
[34] R. Abraham, J.E. Marsden, and R. Ratiu, Manifolds, Tensor Analysis, and Applications, second ed. Springer, 1988.
[35] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Math. Sciences, vol. 153, Springer, 2003.
[36] J. Sethian, Level Sets Methods and Fast Marching Methods. Cambridge Univ. Press, 1999.
[37] M. Niethammer, A. Tannenbaum, and S. Angenent, “Dynamic Active Contours for Visual Tracking,” IEEE Trans. Automatic Control, vol. 51, no. 4, pp. 562-579, 2006.
[38] A. Yezzi and S. Soatto, “Deformotion: Deforming Motion, Shape Average and the Joint Registration and Approximation of Structures in Images,” Int'l J. Computer Vision, vol. 53, no. 2, pp.153-167, 2003.
[39] D. Cremers and S. Soatto, “A Pseudo-Distance for Shape Priors in Level Set Segmentation,” Proc. Int'l Workshop Variational, Geometric and Level Set Methods in Computer Vision, pp. 169-176, 2003.
[40] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, “Conformal Curvature Flows: From Phase Transitions to Active Vision,” Archive for Rational Mechanics and Analysis, vol. 134, pp. 275-301, 1996.
[41] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic Active Contours,” Int'l J. Computer Vision, vol. 13, pp. 5-22, 1997.
[42] T.F. Chan and L.A. Vese, “Active Contours without Edges,” IEEE Trans. Image Processing, vol. 10, no. 2, pp. 266-277, 2001.
[43] B.K.P. Horn and B.G. Schunck, “Determining Optical Flow,” Atificial Intelligence, vol. 23, pp. 185-203, 1981.
[44] E. Pichon, D. Nain, and M. Niethammer, A Laplace Equation Approach for the Validation of Image Segmentation, in preparation.
[45] A. Duci, A.J. Yezzi, S.K. Mitter, and S. Soatto, “Shape Representation via Harmonic Embedding,” Proc. Int'l Conf. Computer Vision, pp. 656-662, 2003.
[46] A. Yezzi and J.L. Prince, “A PDE Approach for Measuring Tissue Thickness,” Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 87-92, 2001.
[47] L. Evans, Partial Differential Equations. Am. Math. Soc., 1998.
[48] C.Y. Kao, S. Osher, and Y.-H. Tsai, “Fast Sweeping Methods for Static Hamilton-Jacobi Equations,” Technical Report 03-75, University of California, Los Angeles, 2003.
[49] M. Sussman, P. Smereka, and S. Osher, “A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow,” J.Computational Physics, vol. 114, pp. 146-159, 1994.
[50] P.A. Vela, M. Niethammer, G.D. Pryor, A.R. Tannenbaum, R. Butts, and D. Washburn, “Knowledge-Based Segmentation for Tracking through Deep Turbulence,” to be published in, IEEE Trans. Control Systems Technology, 2007.
[51] M. Rousson and R. Deriche, “A Variational Framework for Active and Adaptive Segmentation of Vector Valued Images,” Proc. IEEE Workshop Motion and Video Computing, 2002.
[52] J. Li, M. Dao, C. Lim, and S. Suresh, “Spectrin-Level Modeling of the Cytoskeleton and Optical Tweezers Stretching of the Erythrocyte,” Biophysical J., vol. 88, no. 5, pp. 3707-3719, 2005.
[53] Y. Tseng, J.-H. Lee, I. Jiang, T. Kole, and D. Wirtz, “Micro-Organization and Visco-Elasticity of the Interphase Nucleus Revealed by Particle Nanotracking,” J. Cell Science, vol. 117, no. 10, pp. 2159-2167, 2004.
[54] G. Dong, N. Ray, and S. Acton, “Intravital Leukocyte Detection Using the Gradient Inverse Coefficient of Variation,” IEEE Trans. Medical Imaging, vol. 24, no. 7, pp. 910-924, 2005.
[55] S. Suresh, J. Spatz, J. Mills, A. Micoulet, M. Dao, C. Lim, M. Beil, and T. Seufferlein, “Connections between Single-Cell Biomechanics and Human Disease States: Gastrointestinal Cancer and Malaria,” Acta Biomaterialia, vol. 1, p. 1630, 2005.
[56] W. Enkelmann, “Investigations of Multigrid Algorithms for the Estimation of Optical Flow Fields in Image Sequences,” Computer Vision, Graphics, and Image Processing, vol. 43, no. 2, pp. 150-177, 1988.
[57] Sequential Monte Carlo Methods in Practice, A. Doucet, ed. Springer, 2001.