This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Scale Space Tracking and Deformable Sheet Models for Computational Vision
July 1993 (vol. 15 no. 7)
pp. 697-706

The deformable sheet, a physical model that provides a natural framework for addressing many vision problems that can be solved by smoothness-constrained optimization, is described. Deformable sheets are characterized by a global energy functional, and the smoothness constraint is represented by a linear internal energy term. Analogous to physical sheets, the model sheets are deformed by problem-specific external forces and, in turn, impose smoothness on the applied forces. The model unifies the properties of scale and smoothness into a single parameter that makes it possible to perform scale space tracking by properly controlling the smoothness constraint. Specifically, the desired scale space trajectory is found by solving a differential equation in scale. The simple analytic dependence on scale also provides a mechanism for adaptive step size control. Results from application of the deformable sheet model to various problems in computational vision are presented.

[1] D. Marr,Vision. San Francisco: Freeman, 1982.
[2] B. K. P. Horn,Robot Vision. Cambridge, MA: M.I.T. Press, 1986.
[3] D. Terzopoulos, A. Witkin, and K. Kass, "Stereo matching as constrained optimization using scale continuation methods," inProc. SPIE Conf. Optical Digital Patt. Recog.(Los Angeles, CA), Jan. 1987.
[4] A. Witkin, D. Terzopoulos, and M. Kass, "Signal matching through scale space,"Int. J. Comput. Vision, 1987, pp. 133-144, vol. 1.
[5] D. Terzopoulos, "Regularization of inverse visual problems involving discontinuities,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, no. 4, pp. 413-424, July 1986.
[6] S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images;"IEEE Trans. Patt. Anal. Machine Intell., vol. 6, pp. 721-741, 1985.
[7] M. Kass, A. Witkin, and D. Terzopoulos, "Snakes: Active contour models,"Int. J. Comput. Vision, vol. 1, pp. 321-331, 1987.
[8] R. Courant and D. Hilbert,Methods of Mathematical Physics, Volume 1. New York: Wiley, Interscience, 1953.
[9] G. Strang,Introduction to Applied Mathematics. Wellesley, MA: Wellesley Cambridge Press.
[10] R. Szeliski, "Regularization uses fractal priors," inProc. First Int. Conf. Comput. Vision(London, England), 1987.
[11] W.H. Press et al.,Numerical Recipes--The Art of Scientific Computing, Cambridge University Press, 1986, p. 254.
[12] C.W. Gear,Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971.
[13] D. Terzopoulos, "Image analysis using multigrid relaxation methods,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 129-139, Mar. 1986.
[14] W. E. L. Grimson,From Images to Surfaces.\quad Cambridge, MA:MIT Press, 1981.
[15] A. N. Tikhonov and V. Y. Arsenin,Solutions of Ill-posed Problems. Washington, DC: W. H. Winston, 1977.
[16] G. Wahba,Spline Models for Observational Data, Series in Applied Mathematics, Vol. 59, Philadelphia: SIAM, 1990.
[17] M. Bertero, T. A. Poggio, and V. Torre, "Ill-posed problems in early vision,"Proc. IEEE, vol. 76, pp. 869-889, Aug. 1988.
[18] J. L. Marroquinn, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computation vision,"J. Amer. Stat. Assoc., vol. 82, pp. 76-89, 1987.
[19] P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion,"IEEE Trans. Patt. Anal. Machine Intell., vol. 12, no. 7, pp. 629-639, July 1990.
[20] A. Papoulis,Signal Analysis. New York: McGraw-Hill, 1977.
[21] A. Blake and A. Zisserman,Visual Reconstruction. Cambridge, MA: MIT Press, 1987.

Index Terms:
deformable sheet models; computational vision; smoothness-constrained optimization; global energy functional; smoothness constraint; linear internal energy; scale space tracking; differential equation; computer vision; image processing; optimisation
Citation:
G. Whitten, "Scale Space Tracking and Deformable Sheet Models for Computational Vision," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 7, pp. 697-706, July 1993, doi:10.1109/34.221170
Usage of this product signifies your acceptance of the Terms of Use.