Area and length minimizing flows for shape segmentation

IEEE Transactions on Image Processing

Volumn 7, Issue 3, 1998, Pages 433-443

  Siddiqi, Kaleem ^a   Lauzière, Yves Bérubé ^b   Tannenbaum, Allen ^c,d   Zucker, Steven W ^a,d  

^a YALE UNIVERSITY   (United States)

^b MCGILL UNIVERSITY   (Canada)

^c University of Minnesota ^*  (United States)

^d IEEE

Author keywords

Curve evolution; Edge capturing; Gradient flows; Snakes

Indexed keywords

COMPUTATIONAL METHODS; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS;

CURVE EVOLUTION THEORY; EUCLIDEAN CURVE SHORTENING EQUATION; GRADIENT EVOLUTION EQUATIONS;

IMAGE SEGMENTATION;

EID: 0032022522     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/83.661193     Document Type: Article

References (30)

1. B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," Lecture Notes Comput. Sci., vol. 427, pp. 402-407, 1990.
2. _, "Shape, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space," Int. J. Comput. Vis., vol. 15, pp. 189-224, 1995.
3. _, "On the evolution of curves via a function of curvature I: The classical case," J. Math. Anal. Applicat., vol. 163, pp. 438-458, 1992.
4. V. Caselles, F. Catte, T. Coll, and F. Dibos, "A geometric model for active contours in image processing," Numer. Math., vol. 66, pp. 1-31, 1993.
5. R. Malladi, J. A. Sethian, and B. C. Vemuri, "Shape modeling with front propagation: A level set approach," IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 158-175, Feb. 1995.
6. H. Tek and B. B. Kimia, "Deformable bubbles in the reaction-diffusion space," in Proc. Fifth Int. Conf. Computer Vision. Los Alamitos, CA: IEEE Comput. Soc. Press, 1995.
7. V. Caselles, R. Kimmel, and G. Sapiro, "Geodesic active contours," in Proc. Fifth Int. Conf. Computer Vision. Los Alamitos, CA: IEEE Comput. Soc. Press, 1995, pp. 694-699.
8. S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, "Gradient flows and geometric active contour models," in Fifth Int. Conf. Computer Vision. Los Alamitos, CA: IEEE Comput. Soc. Press, 1995, pp. 810-815.
9. _, "Conformal curvature flows: From phase transitions to active vision," Arch. Rational Mech. Anal., vol. 134, pp. 275-301, 1996.
10. J. Shah, "Recovery of shapes by evolution of zero-crossings," Dept. Mathematics, Tech. Rep., Northeastern Univ., Boston, MA, 1995.
11. M. Kass, A. Witkin, and D. Terzopoulos, "Snakes: Active contour models," Int. J. Comput. Vis., vol. 1, pp. 321-331, 1987.
12. D. Terzopoulos and A. Witkin, "Constraints on deformable models: Recovering shape and nonrigid motion," Artif. Intell., vol. 36, pp. 91-123, 1988.
13. A. Blake and A. Yuille, Active Vision, Cambodge, MA: MIT Press, 1992.
14. K. Siddiqi, A. Tannenbaum, and S. W. Zucker, "Hyperbolic 'smoothing' of shapes," ICCV, 1998.
15. S. Osher, "Riemann solvers, the entropy condition, and difference approximations," SIAM J. Numer. Anal., vol. 21, pp. 217-235, 1984.
16. J. A. Sethian, "An analysis of flame propagation," Ph.D. dissertation, Univ. Calif., Berkeley, 1982.
17. _, "Curvature and the evolution of fronts," Commun. Math. Phys., vol. 101, pp. 487-499, 1985.
18. _, "A review of recent numerical algorithms for hypersurfaces moving with curvature dependent speed," J. Differ. Geom., vol. 31, pp. 131-161, 1989.
19. M. Gage and R. S. Hamilton, "The heat equation shrinking convex plane curves," J. Differ. Geom., vol. 23, pp. 69-96, 1986.
20. M. Grayson, "The heat equation shrinks embedded plane curves to round points," J. Differ. Geom., vol. 26, pp. 285-314, 1987.
21. L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, "Axiomes et équations fondamentales du traitement d'images," C. R. Acad. Sci. Paris, vol. 315, pp. 135-138, 1992.
22. _, "Axiomatization et nouveaux opérateurs de la morphologie mathématique," C. R. Acad. Sci. Paris, vol. 315, pp. 265-268, 1992.
23. G. Sapiro and A. Tannenbaum, "On affine plane curve evolution," J. Funct. Anal., vol. 119, pp. 79-120, 1994.
24. P. Olver, G. Sapiro, and A. Tannenbaum, "Affine invariant edge maps and active contours," submitted for publication.
25. M. P. Do Carmo, Riemannian Geometry. Cambridge, MA: Birkhauser, 1992.
26. A. Yezzi, S. Kichenessamy, A. Kumar, P. Olver, and A. Tannenbaum, "A geometric snake model for segmentation of medical imagery," IEEE Trans. Med. Imag., to be published.
27. V. Caselles, R. Kimmel, G. Sapiro, and C. Sbert, "Minimal surfaces: A three dimensional segmentation approach," Tech. Rep. 973, Technion - Israel Inst. Technol. Haifa, June 1995.
28. L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, "Axioms and fundamental equations of image processing," Arch. Rational Mech. Anal., vol. 123, 1993.
29. L. Alvarez, P. L. Lions, and J. M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM J. Numer. Anal., vol. 29, pp. 845-866, 1992.
30. B. B. Kimia and K. Siddiqi, "Geometric heat equation and nonlinear diffusion of shapes and images," Comput. Vis. Image Understand., vol. 64, pp. 305-322, 1996.