Stochastic differential equations and geometric flows

IEEE Transactions on Image Processing

Volumn 11, Issue 12, 2002, Pages 1405-1416

  Unal, Gozde ^a,b   Krim, Hamid ^a   Yezzi, Anthony ^b  

^a North Carolina State University   (United States)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Geometric image and shape flows; Nonlinear filtering; Shape analysis; Stochastic differential equations

Indexed keywords

COMPUTER VISION; NONLINEAR FILTERING; PARTIAL DIFFERENTIAL EQUATIONS; RANDOM PROCESSES;

GEOMETRIC IMAGES;

IMAGE PROCESSING;

EID: 0036991088     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIP.2002.804568     Document Type: Article

References (32)

1. S. Osher and J. A. Sethian, "Fronts propagating with curvature dependent speed: Algorithms based on the Hamilton-Jacobi formulation," J. Comput. Phys., vol. 49, pp. 12-49, 1988.
2. J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences. Cambridge, U.K.: Cambridge University Press, 1966.
3. S. Osher, "Riemann solvers, the entropy condition and difference approximations," SIAM J. Num. Anal., vol. 21, pp. 217-235, 1984.
4. M. A. Grayson, "The heat equation shrinks embedded plane curves to round points," J. Diff. Geom., vol. 26, pp. 285-314, 1987.
5. B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "On the evolution of curves via a function of curvature, I the classical case," J. Math. Anal. Applicat., vol. 163, pp. 438-458, 1992.
6. B. B. Kimia and K. Siddiqi, "Geometric heat equation and nonlinear diffusion of shapes and images," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 1994.
7. B. B. Kimia, A. Tannenbaum, and S. Zucker, "Shapes, shocks, and deformations I," Int. J. Comput. Vis., no. 15, pp. 189-224, 1995.
8. L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel, "Axioms and fundamental equations of image processing," Arch. Rational Mech. Anal., vol. 123, pp. 199-257, 1993.
9. L. Alvarez, P. L. Lions, and J. M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM J. Num. Anal., vol. 29, no. 3, pp. 845-866, 1992.
10. A.P. Witkin, "Scale space filtering," in Proc. Int. Conf. on AI, Karlsruhe, Germany, 1983, pp. 1019-1023.
11. G. Sapiro and A. Tannenbaum, "Affine invariant scale space," Int. J. Comput. Vis., vol. 11, pp. 25-44, 1993.
12. _, "On invariant curve evolution and iamge analysis," Indiana Univ. J. Math., vol. 93, 1993.
13. H. Krim and Y. Bao, "A stochastic approach to signal denoising," presented at the ICASSP, Phoenix, AZ, 1999.
14. _, "Smart nonlinear diffusion; a probabilistic approach,", 2001, submitted for publication.
15. A. Bruckstein, G. Sapiro, and D. Shaked, "Evolutions of planar polygons," Int. J. Pattern Recognit. Artif. Intell., vol. 9, no. 6, pp. 991-1014, 1995.
16. G. Unal, A. Yezzi, and H. Krim, "Unsupervised texture segmentation by information-theoretic active polygons,", 2002, submitted for publication.
17. H. Blum, "Biological shape and visual science," J. Theor. Biol., pp. 205-287, 1973.
18. K. Siddiqi and B. B. Kimia, "Parts of visual form: Computational aspects," in Proc. IEEE Conf. on Computer Vision and Pattern Recognition, 1993.
19. R. L. Ogniewicz and M. Ilg, "Voronoi skeletons: Theory and applications," Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 63-69, 1992.
20. L. Da, F. Costa, and L. F. Estrozi, "Multiresolution shape representation without border shifting," Electron. Lett., vol. 35, no. 21, pp. 1829-1830, 1999.
21. S. M. Pizer, A. L. Thall, and D. T. Chen, "M-Reps: A New Object Representation for Graphics," Univ. North Carolina, Chapel Hill, vol. 17, Tech. Rep. TR99-030, 1999.
22. J. Gomes, "Warping and morphing of graphical objects, course notes," in Proc. SIGGRAPH'95, 1995.
23. P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Machine Intell., vol. 12, no. 7, pp. 629-639, 1990.
24. B. Oksendal, Stochastic Differential Equations, an Introduction with Applications. Berlin, Germany: Springer, 1998.
25. D. Mumford, "Elastica and computer vision," in Algebraic Geometry and Its Applications, C. Bajaj, Ed. New York: Springer Verlag, 1994, pp. 491-506.
26. L. R. Williams and D. W. Jacobs, "Stochastic completion fields: A neural model of illusory contour shape and salience," Neural Comput., vol. 9, pp. 837-858, 1997.
27. I.J. Cox, J. M. Rehg, and S. Hingorani, "A Bayesian multiple hypothesis approach to edge grouping and contour segmentation," Int. J. Comput. Vis., vol. 11, pp. 5-24, 1993.
28. M. G. Crandall, H. Ishii, and P. L. Lions, "User's guide to viscosity solutions of second order partial differential equations," Bull. Amer. Math. Soc., vol. 27, no. 1, pp. 1-67, 1992.
29. Y.-G. Chen, Y. Giga, and S. Goto, "Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations," J. Diff. Geom., vol. 33, pp. 749-786, 1991.
30. J. Ranch, Partial Differential Equations. New York: Springer-Verlag, 1991.
31. D. Peng, S. Osher, B. Merriman, and H.-K. Zhao, "The geometry of Wulff crystal shapes and its relations with Riemann problems," Contemp. Math., vol. 238, pp. 251-303, 1999.
32. D. Adalsteinsson and J. A. Sethian, "A fast level set method for propagating interfaces," J. Comput. Phys., vol. 118, pp. 269-277, 1995.