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Frequency Domain Formulation of Active Parametric Deformable Models
December 2004 (vol. 26 no. 12)
pp. 1568-1578
Active deformable models are simple tools, very popular in computer vision and computer graphics, for solving ill-posed problems or mimic real physical systems. The classical formulation is given in the spatial domain, the motor of the procedure is a second-order linear system, and rigidity and elasticity are the basic parameters for its characterization. This paper proposes a novel formulation based on a frequency-domain analysis: The internal energy functional and the Lagrange minimization are performed entirely in the frequency domain, which leads to a simple formulation and design. The frequency-based implementation offers important computational savings in comparison to the original one, a feature that is improved by the efficient hardware and software computation of the FFT algorithm. This new formulation focuses on the stiffness spectrum, allowing the possibility of constructing deformable models apart from the elasticity and rigidity-based original formulation. Simulation examples validate the theoretical results.

[1] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active Contour Models,” Int'l J. Computer Vision, pp. 321-331, 1988.
[2] D. Terzopoulos, A. Witkin, and M. Kass, “Constraints on Deformable Models: Recovering 3D Shape and Nonrigid Motion,” Artificial Intelligence, vol. 36, pp. 91-123, 1988.
[3] D. Terzopoulos, “The Computation of Visible-Surface Representations,” IEEE Trans. Pattern Analysis Machine Intelligence, vol. 10, no. 4, pp. 417-438, July/Aug. 1988.
[4] D. Metaxas, Physics-Based Deformable Models, Applications to Computer Vision, Graphics and Medical Imaging. Boston: Kluwer Academic, 1996.
[5] L.D. Cohen and I. Cohen, “Finite-Element Methods for Active Contour Models and Balloons for 2D and 3D Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 11, pp. 1131-1147, Nov. 1993.
[6] Deformable Models in Medical Images Analysis, Singh, Goldgof, and D. Terzopoulos, eds. Los Alamitos, Calif.: IEEE CS, 1995.
[7] P. Faloutsos, M. van de Panne, and D. Terzopoulos, “Dynamic Free-Form Deformations for Animation Synthesis,” IEEE Trans. Visualization and Computer Graphics, vol. 3, no. 3, pp. 201-214, July/Sept. 1997.
[8] B. Eberhardt, A. Weber, and W. Straßer, “A Fast, Flexible Particle-System Model for Cloth Draping,” IEEE Computer Graphics and Applications, vol. 16, pp. 52-59, Sept. 1996.
[9] Q. Yu and D. Terzopoulos, “Synthetic Motion Capture for Interactive Virtual Worlds,” Proc. Computer Animation, Los Alamitos, Calif.: IEEE CS Press, pp. 2-10, 1998.
[10] J. Liang, T. McInerney, and D. Terzopoulos, “United Snakes,” Proc. Seventh IEEE Int'l Conf. Computer Vision, pp. 933-940, 1999.
[11] D. Terzopoulos, “Deformable Models: Classic, Topology-Adaptive and Generalized Formulations,” Geometric Level Set Methods, Osher and Paragios, eds., pp. 21-40, New York: Springer-Verlag, 2003.
[12] W. Neuenschwander, P. Fua, G. Székely, and O. Kübler, “Velcro Surfaces: Fast Initialization of Deformable Models,” Computer Vision, Graphics, and Image Understanding, pp. 237-245, Feb. 1997.
[13] T. McInerney and D. Terzopoulos, “T-Snakes: Topology Adaptive Snakes,” Medical Image Analysis, vol. 4, pp. 73-91, 2000.
[14] W. Mao, J. Evans, L. Hassebrook, and C. Knapp, “A Multistage, Optimal Active Contour Model,” IEEE Trans. Image Processing, vol. 5, pp. 1586-1591, Nov. 1996.
[15] B. Olstad and A.H. Torp, “Encoding of A Priori Information in Active Contour Models,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 9, pp. 863-872, Sept. 1996.
[16] P. Brigger, J. Hoeg, and M. Unser, “B-Spline Snakes: A Flexible Tool for Parametric Contour Detection,” IEEE Trans. Image Processing, vol. 9, pp. 1484-1496, Sept. 2000.
[17] D.J. Williams and M. Shah, “A Fast Algorithm for Active Contours and Curvature Estimation,” Image Understanding, vol. 55, pp. 14-26, 1992.
[18] K. Lam and H. Yan, “Fast Greedy Algorithm for Active Contours,” Electronic Letters, vol. 30, pp. 21-23, 1994.
[19] A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems. New York: Wiley, 1977.
[20] A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing, second ed. Upper Saddle River, N.J.: Prentice-Hall, 1999.
[21] P. Duhamel and M. Vetterli, “Fast Fourier Transforms: A Tutorial Review and a State of the Art,” Signal Processing, vol. 19, pp. 259-299, 1990.
[22] S. Haykin, Adaptive Filter Theory, third ed. Upper Saddle River, N.J.: Prentice-Hall, 1996.
[23] B.M. Baas, “ An Approach to Low-Power, High-Performance, FFT Processor Design,” PhD dissertation, Stanford Univ., Stanford, Cailf., Feb. 1999.
[24] R. Matusiak, “Extended Precision Radix-4 FFT Implemented on TMS320C62xx,” Report SPRA297, Texas Instruments, Nov. 1998.
[25] L. Weruaga, J. Morales, L. Nunez, and R. Verdú, “Estimating Volumetric Motion in Human Thorax with Parametric Matching Constraints,” IEEE Trans. Medical Imaging, vol. 22, pp. 766-772, June 2003.

Index Terms:
Active deformable models, snakes, frequency domain, Fast Fourier transform.
Luis Weruaga, Rafael Verd?, Juan Morales, "Frequency Domain Formulation of Active Parametric Deformable Models," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 12, pp. 1568-1578, Dec. 2004, doi:10.1109/TPAMI.2004.124
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