Area-preserving curve-shortening flows: From phase separation to image processing

Interfaces and Free Boundaries

Volumn 4, Issue 4, 2002, Pages 325-343

  Dolcetta, Italo Capuzzo ^a   Vita, Stefano Finzi ^a   March, Riccardo ^b  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b CNR   (Italy)

Author keywords

Image processing; Phase separation theory; Planar curve evolution

Indexed keywords

EID: 17244367314     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (40)

1. ALIKAKOS, N., BATES, P., & CHEN, X. Convergence of the Cahn-Hilliard equation to the Hele- Shaw model. Arch. Rational Mech. Anal. 128, (1994) 165-205.
2. ALLEN, S. M. & CAHN, J. W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Met. 27, (1979) 1085-1095.
3. ALVAREZ, L., GUICHARD, F., LIONS, P. L., & MOREL, J. M. Axioms and fundamental equations of image processing. Arch. Rational Mech. Anal. 123, (1993) 199-257.
4. BLOWEY, J. F. & ELLIOTT, C. M. The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. Part II: Numerical analysis. Euro. J. Appl. Math. 3, (1992) 147-179.
5. BRONSARD, L. & KOHN, R. V. Motion by mean curvature as a singular limit of Ginzburg- Landau dynamics. J. Diff. Eqs. 90, (1991) 211-237.
6. BRONSARD, L. & STOTH, B. Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg- Landau equation. SIAM J. Math. Anal. 28, (1997) 769-807.
7. CAPUZZO DOLCETTA, I. & FERRETTI, R. Optimal stopping time formulation of adaptive image filtering. Appl. Math. and Optimiz 43, (2001) 245-258.
8. CAPUZZO DOLCETTA, I., FINZI VITA, S., & MARCH, R. A phase transition approach to planar curve evolution without shrinkage. Proc. of the MTNS98 Conference. Padova (1998) pp. 451-454.
9. CASELLES, V., CATTÉ , F., COLL, T., & DIBOS, F. A geometric model for active contours in image processing. Numer. Math. 66, (1993) 1-31.
10. CHEN, X. The Hele-Shaw problem and area-preserving curve-shortening motions. Arch. Rational Mech. Anal. 123, (1993) 117-151.
11. CHEN, X. Global asymptotic limit of solutions of the Cahn-Hilliard equations. J. Diff. Geom. 47, (1996) 262-311.
12. CHEN, Y.-G., GIGA, Y., & GOTO, S. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Diff. Geom. 33, (1991) 749-786.
13. DE MOTTONI, P. & SCHATZMAN, M. Evolution géométrique d'interfaces. C.R. Acad. Sci. Paris Sér. I Math. 309, (1989) 453-458.
14. ELLIOTT, C. M., FRENCH, D. A., & MILNER, F. A. A second order splitting method for the Cahn- Hilliard equation. Num. Math. 54, (1989) 575-590.
15. EVANS, L. C., SONER, H. M., & SOUGANIDIS, P. E. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45, (1992) 1097-1123.
16. EVANS, L. C. & SPRUCK, J. Motion of level sets by mean curvature. J. Diff. Geom. 33, (1991) 635-681.
17. FINZI VITA, S. A Cahn-Hilliard equation with non homogeneous mobility and its application to image processing. Proc. of the ENUMATH 2001 Conference, submitted. Ischia (2001).
18. GAGE, M. Curve shortening makes convex curves circular. Invent. Math. 76, (1984) 357-364.
19. GAGE, M. On an area-preserving evolution equation for plane curves. Contemp. Math. 51, (1986) 51-62.
20. GAGE, M. & HAMILTON, R. S. The heat equation shrinking convex plane curves. J. Diff. Geom. 23, (1986) 69-96.
21. GRAYSON, M. The heat equation shrinks embedded plane curves to round points. J. Diff. Geom. 26, (1987) 285-314.
22. GRAYSON, M. Shortening embedded curves. Annals of Mathematics 129, (1989) 71-111.
23. KICHENASSAMY, S., KUMAR, A., OLVER, P., TANNENBAUM, A., & YEZZI, JR. A. Conformal curvature flows: from phase transitions to active vision. Arch. Rational Mech. Anal. 134, (1996) 275-301.
24. KIMIA, B. B. & SIDDIQI, K. Geometric heat equation and nonlinear diffusion of shapes and images. Computer Vision and Image Understanding 64, (1996) 305-322.
25. KIMIA, B. B., TANNENBAUM, A., & ZUCKER, S. W. Shapes, shocks, and deformations, I: the components of shape and the reaction-diffusion space. Int. J. Comput. Vision 15, (1995) 189-224.
26. MALLADI, R., SETHIAN, J. A., & VEMURI, B. C. Shape modeling with front propagation: a level set approach. IEEE Trans. PAMI 17, (1995) 158-175.
27. MARCH, R. & DOZIO, M. A variational method for the recovery of smooth boundaries. Image and Vision Computing Journal 15, (1997) 705-712.
28. MARR, D. & HILDRETH, E. Theory of edge detection. Proc. Roy. Soc. Lond. Ser. B 207, (1980) 187-217.
29. MOKHATARIAN, F. & MACKWORTH, A. A theory of multi-scale, curvature based shape representation for planar curves. IEEE Trans. PAMI 14, (1992) 789-805.
30. NOCHETTO, R. H., PAOLINI, M., & VERDI, C. A dynamic mesh algorithm for curvature dependent evolving interfaces. J. Comput. Phys. 123, (1996) 296-310.
31. NOCHETTO, R. H. & VERDI, C. Approximating curvature driven interfaces with applications to shape recovery. Curvature flows and related topics (Levico, 1994), Gakuto Internat. Ser. Math. Sci. Appl. 5, (1995) 159-177.
32. NOVICK-COHEN, A. On the viscous Cahn-Hilliard equation. In: BALL, J. (ed), Material Instabilities in Continuum Mechanics and Related Mathematical Problems. pp. 329-342. Clarendon, Oxford (1988).
33. OSHER, S. J. & SETHIAN, J. A. Fronts propagation with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comp. Phys. 79, (1988) 12-49.
34. PEGO, R. L. Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. Lond. Ser. A 422, (1989) 261-278.
35. RUBINSTEIN, J. & STERNBERG, P. Nonlocal reaction-diffusion equations and nucleation. IMA J. of Appl. Math. 48, (1992) 249-264.
36. RUBINSTEIN, J., STERNBERG, P., & KELLER, J. B. Fast reaction, slow diffusion, and curve shortening. SIAM J. Appl. Math. 49, (1989) 116-133.
37. SAPIRO, G. & TANNENBAUM, A. Affine invariant scale-space. Int. J. Comput. Vision 11, (1993) 25-44.
38. SAPIRO, G. & TANNENBAUM, A. Area and length preserving geometric invariant scale-spaces. IEEE Trans. PAMI 17, (1995) 67-72.
39. SHAH, J. Uses of elliptic approximations in computer vision. In: SERAPIONI, R. & TOMARELLI, F. (eds), Variational Methods for Discontinuous Structures. pp. 19-34. Birkhauser, Boston (1996).
40. TORRE, V. & POGGIO, T. On edge detection. IEEE Trans. PAMI 8, (1986) 147-163.