Coarsening in high order, discrete, ill-posed diffusion equations

Communications in Mathematical Sciences

Volumn 8, Issue 4, 2010, Pages 797-834

  Kublik, Catherine ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Backward diffusion equations; Coarsening; Image processing; Perona Malik model; You Kaveh model

Indexed keywords

EID: 77956941432     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2010.v8.n4.a1     Document Type: Article

References (24)

1. H. Amann, Time-delayed Perona-Malik type problems, Acta. Math. Univ. Comeniance, LXXVI, 15-38, 2007.
2. A. Belahmidi and A. Chambolle, Time-delay regularization of anisotropic diffusion, ESAIM Mathematical Modelling and Numerical Analysis, 39(2), 231-251, 2005.
3. G. Bellettini and G. Fusco, A regularized Perona-Malik functional: some aspects of the gradient dynamics, H. Broer, J. Mahwin, A. Vanderbauwhede and S.V. Lunel (eds), EQUADIFF 2003 Proceedings of the International Conference on Differential Equations, Hasselt, Belgium, July, 2003.
4. G. Bellettini, M. Novaga and E. Paolini, Global solutions to the gradient flow equation of a nonconvex functional, SIAM J. Math. Anal., 37(5), 1657-1687, 2006.
5. F. Catt, P.L. Lions, J.M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29, 182-193, 1992.
6. S. S. Esedoglu, An analysis Esedoḡlu, An analysis of the Perona-Malik scheme, Commun. Pure Appl. Math, 54, 1442-1487, 2001.
7. S. Esedoḡlu and J. Greer, Upper bounds of the coarsening rate of discrete ill-posed nonlinear diffusion equations, Commun. Pure Appl. Math., 62(1), 57-81, 2009.
8. S. Esedoglu and D. Slepčev, Refined upper bounds of the coarsening rate of discrete ill-posed nonlinear diffusion equations, Nonlinearity, 21(1), 2759-2776, 2008.
9. M. Ghisi and M. Gobbino, A class of local classical solutions for the one-dimensional Perona-Malik equation, preprint, 2006.
10. M. Ghisi and M. Gobbino, Gradient estimates for the Perona-Malik equation, Math. Ann., 337(3), 557-590, 2007.
11. M. Gobbino, Entire solutions of the one-dimensional Perona-Malik equation, Commun. Part. Differ. Equ., 32(4-6), 719-743, 2007.
12. J. Goodman, A. Kurganov and P. Rosenau, Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity, 12, 247-268, 1999.
13. J. Greer and A. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36(1), 38-68, 2004.
14. B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann., 311, 107-123, 1998.
15. S. Kichenassamy, The Perona-Malik paradox, SIAM J. Appl. Math., 57(5), 1328-1342, 1997.
16. R.V. Kohn and F. Otto, Upper bounds on coarsening rates, Commun. Math. Phys., 229(3), 375-395, 2002.
17. R.V. Kohn and X. Yan, Upper bound on the coarsening rate for an epitaxial growth model, Commun. Pure Appl. Math., 56(11), 1549-1564, 2003.
18. M. Nitzberg and T. Shiota, Nonlinear image filtering with edge and corner enhancement, IEEE Trans. Pattern Anal. Machine Intell., 14, 629-639, 1992.
19. F. Otto, T. Rump and D. Slepčev, Coarsening rates for a droplet model: rigorous upper bounds, SIAM J. Math. Anal., 32(12), 503-529, 2006.
20. P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, Technical report, Dept. of EECS, U.C. Berkeley, 1987.
21. P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern. Anal. Machine Intell., 12, 629-639, 1990.
22. P. Perona, J. Malik and T. Shiota, Geometry-Driven diffusion in computer vision, chapter Anisotropic diffusion, Kluwer, 73-92, 1995.
23. T.P. Witelski, D.G. Schaeffer and M. Shearer, A discrete model for an ill-posed nonlinear parabolic PDE, Phys. D., 160(3-4), 189-221, 2001.
24. Y.L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Transactions on Image Processing, 9(10), 1723-1730, 2000.