This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Snake Pedals: Compact and Versatile Geometric Models with Physics-Based Control
May 2000 (vol. 22 no. 5)
pp. 445-459

Abstract—In this paper, we introduce a novel geometric shape modeling scheme which allows for representation of global and local shape characteristics of an object. Geometric models are traditionally well-suited for representing global shapes without local detail. However, we propose a powerful geometric shape modeling scheme which allows for the representation of global shapes with local detail and permits model shaping as well as topological changes via physics-based control. The proposed modeling scheme consists of representing shapes by pedal curves and surfaces—pedal curves/surfaces are the loci of the foot of perpendiculars to the tangents of a fixed curve/surface from a fixed point called the pedal point. By varying the location of the pedal point, one can synthesize a large class of shapes which exhibit both local and global deformations. We introduce physics-based control for shaping these geometric models by letting the pedal point vary and use a snake to represent the position of this varying pedal point. The model dubbed as a “snake pedal” allows for interactive manipulation via forces applied to the snake. We develop a fast numerical iterative algorithm for shape recovery from image data using this geometric shape modeling scheme. The algorithm involves the Levenberg-Marquardt (LM) method in the outer loop for solving the global parameters and the Alternating Direction Implicit (ADI) method in the inner loop for solving the local parameters of the model. The combination of the global and local scheme leads to an efficient numerical solution to the model fitting problem. We demonstrate the applicability of this modeling scheme via examples of shape synthesis and shape estimation from real image data.

[1] E. Bardinet, N. Ayache, and L.D. Cohen, “Fitting of Iso-Surface Using Superquadrics and Free-Form Deformations,” Proc. IEEE Workshop Biomedical Image Analysis WBIA '94, pp. 184-193, 1994.
[2] I. Cohen and L.D. Cohen, “Hyperquadric Model for 2D and 3D Data Fitting,” Proc. Int'l Conf. Pattern Recognition, pp. 403-405, 1994.
[3] D. DeCarlo and D. Metaxas, “Blended Deformable Models,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, pp. 443-448, 1996.
[4] A. Pentland and J. Williams, “Good Vibrations: Modal Dynamics for Graphics and Animation,” Proc. ACM SIGGRAPH, pp. 215-222, 1989.
[5] D. Terzopoulos and D. Metaxas, “Dynamic 3D Models with Local and Global Deformations: Deformable Superquadrics,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 7, pp. 703-714, July 1991.
[6] B.C. Vemuri and A. Radisavljevic, “Multiresolution Stochastic Hybrid Shape Models with Fractal Priors,” ACM Trans. Graphics, vol. 13, no. 2, pp. 177-207, Apr. 1994.
[7] D. Metaxas, J. Park, and L. Axel, “Analysis of Left Ventricular Wall Motion Based on Volumetric Deformable Models and MRI-Spamm,” Medical Image Analysis, vol. 1, no. 1, pp. 53-72, 1996.
[8] T.W. Sederberg and S.R. Parry, “Free-Form Deformations of Solid Geometric Models,” Computer Graphics, vol. 20, no. 4, pp. 151-160, Aug. 1986.
[9] S. Han, D.B. Goldgof, and K.W. Bowyer, “Using Hyperquadrics for Shape Recovery from Range Data,” IEEE Proc. Third Int'l Conf. Computer Vision, pp. 492-496, 1993.
[10] T. O'Donnel, T. Boult, and A. Gupta, ”Global Models with Parametric Offsets as Applied to Cardiac,” IEEE Proc. Conf. Computer Vision and Pattern Recognition, pp. 293-299, 1996.
[11] P. Radeva, A. Amini, and J. Huang, “Deformable B-Solids and Implicit Snakes for 3D Localization and Tracking of Spamm MRI Data,” Computer Vision and Image Understanding, vol. 66, no. 2, pp. 163-178, 1997.
[12] A.A. Amini, Y. Chen, R.W. Curwen, V. Mani, and J. Sun, “Coupled B-Snake Grids and Constrained Thin-Plate Splines for Analysis of 2D Tissue Deformations from Tagged MRI,” IEEE Trans. Medical Imaging, vol. 17, no. 3, pp. 344-356, 1998.
[13] L.H. Staib and J.S. Duncan, “Parametrically Deformable Contour Model,“ Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR), pp. 98-103, 1989.
[14] G. Szekely, A. Kelemen, C. Brechbuhler, and G. Gerig, “Segmentation of 2D and 3D Objects from MRI Volume Data Using Constrained Elastic Deformations of Flexible Fourier Contour and Surface Models,” Medical Image Analysis, vol. 1, no. 1, pp. 19-34, 1996.
[15] A. Kelemen, G. Szekely, and G. Gerig, “3D Model-Based Segmentation of Brain MRI,” Proc. IEEE Workshop Biomedical Image Analysis, B.Vemuri, ed., pp. 4–13, 1998.
[16] V. Caselles, F. Catte, T. Coll, and F. Dibos, “A Geometric Model for Active Contours in Image Processing,” Numerische Mathematik, vol. 66, pp. 1-31, 1993.
[17] R. Malladi, J. Sethian, and B.C. Vemuri, "Shape Modeling with Front Propagation: A Level Set Approach," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, pp. 158-175, 1995.
[18] T. McInerney and D. Terzopoulos, "Topologically Adaptable Snakes," Proc. IEEE ICCV-95, pp. 840-845, 1995.
[19] L.D. Cohen, “Auxiliary Variables and Two-Step Iterative Algorithms in Computer Vision Problems,” J. Math. Imaging and Vision, vol. 6, pp. 59-83, 1996.
[20] W.H. Press, S.S. Teukolsky, W.T. Vetterling, and B.P Flannery, Numerical Recipes in C. Cambridge Univ. Press, 1992.
[21] R.E. Lynch, J.R. Rich, and D.H. Thomas, “Tensor Product Analysis of Alternating Direction Implicit Methods,” J. Soc. Industrial and Applied Math., vol. 13, no. 4, pp. 995-1,007, 1965.
[22] A. Lu and E.L. Wachspress, “Solution of Lyapunov Equations by Alternating Direction Implicit Iteration,” Computers Math. Applications, vol. 21, no. 9, pp. 43-58, 1991.
[23] A. Gray, Modern Differential Geometry of Curves and Surfaces. Boca Raton: Fla., CRC Press, 1993.
[24] M. Kass, A. Witkin, D. Terzopoulos, “Snakes: An Active Contour Model,” Int'l J. Computer Vision, vol. 1, pp. 321-331, 1987.
[25] P.S. Marc, S. Menet, and G. Medioni, “B-Snakes: Implementation and Application to Stereo,” Proc. Image Understanding Workshop, pp. 720-726, 1990.
[26] Y. Guo and B.C. Vemuri, “Hybrid Geometric Active Models for Shape Recovery in Medical Images,” Proc. 16th Int'l Conf. Information Processing in Medical Imaging (IPMI), pp. 112-125, 1999.
[27] T. Jones and D. Metaxas, “Automated 3D Segmentation Using Deformable Models and Fuzzy Affinity,” Proc. 15th Int'l Conf. Information Processing in Medical Imaging (IPMI), pp. 113-126, 1997.
[28] W.R. Dyksen, “Tensor Product Generalized ADI Methods for Separable Elliptic Problems,” Siam J. Numerical Analysis, vol. 24, no. 1, pp. 59-76, 1987.
[29] S.H. Lai and B.C. Vemuri, “An$o(n)$Iterative Solution to the Poisson Equation in Low-Level Vision Problems,” Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 9-14, June 1994 (also Technical Report 93-035, Dept. of Computer Information Science and Eng., Univ. of Florida).
[30] D.V. Ouellette, “Schur Complements and Statistics,” Linear Algebra and Its Applications, vol. 36, pp. 187-295, 1981.

Index Terms:
Geometric models, snakes, pedal curves/surfaces, alternating direction implicit method, Levenberg-Marquardt method.
Citation:
Baba C. Vemuri, Yanlin Guo, "Snake Pedals: Compact and Versatile Geometric Models with Physics-Based Control," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 5, pp. 445-459, May 2000, doi:10.1109/34.857002
Usage of this product signifies your acceptance of the Terms of Use.