Abstract
We consider the Cauchy problem for the Perona–Malik equation
in a bounded open set \({\Omega \subseteq \mathbb{R}^{n}}\) , with Neumann boundary conditions.
If n = 1, we prove some a priori estimates on u and u x . Then we consider the semi-discrete scheme obtained by replacing the space derivatives by finite differences. Extending the previous estimates to the discrete setting we prove a compactness result for this scheme and we characterize the possible limits in some cases. Finally, for n > 1 we give examples to show that the corresponding estimates on \({\nabla u}\) are in general false.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alberti G., Müller S. (2001) A new approach to variational problems with multiple scales. Commun. Pure Appl. Math. 54, 761–825
Alvarez L., Guichard F., Lions P.L., Morel J.M. (1993) Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123, 199–257
Bellettini, G., Fusco, G., Guglielmi, N.: Numerical experiments and conjectures on the dynamics defined by some singularly perturbed non-convex functionals of the gradient (in preparation)
Bellettini, G., Novaga, M., Paolini, E.: The gradient flow of a non-convex functional in one dimension (preprint)
De Giorgi, E.: Su alcuni problemi instabili legati alla teoria della visione. In: Bruno, T., Buonocore, P., Carbone, L., Esposito, V. (eds). Atti del convegno in onore di Carlo Ciliberto (Napoli, 1995), La Città del Sole, Napoli, (1997), pp. 91–98
Esedoglu S. (2001) An analysis of the Perona–Malik scheme. Commun. Pure Appl. Math. 54, 1442–1487
Esedoglu, S.: Stability properties of Perona–Malik scheme (preprint)
Gobbino M. (1998) Gradient flow for the one-dimensional Mumford–Shah functional. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 27, 145–193
Gobbino, M.: Entire solutions of the one-dimensional Perona–Malik equation (preprint)
Höllig K. (1983) Existence of infinitely many solutions for a forward–backward heat equation. Trans. Am. Math. Soc. 278, 299–316
Kawohl B. (2004) From Mumford–Shah to Perona–Malik in image processing. Math. Methods Appl. Sci. 27, 1803–1814
Kawohl B., Kutev N. (1998) Maximum and comparison principle for one-dimensional anisotropic diffusion. Math. Ann. 311, 107–123
Kichenassamy S. (1997) The Perona–Malik paradox. SIAM J. Appl. Math. 57, 1328–1342
Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, R.I. 1967
Perona P., Malik J. (1990) Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghisi, M., Gobbino, M. Gradient estimates for the Perona–Malik equation. Math. Ann. 337, 557–590 (2007). https://doi.org/10.1007/s00208-006-0047-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-006-0047-1