Communications in Mathematical Sciences

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

Catherine Kublik

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We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order $2n$. The fourth order $(n=2)$ version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation $(n=1)$ corresponds to another famous model from image processing, namely Perona and Malik’s anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the second order model, their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the discrete in space version of these high order equations in any space dimension, for a large class of diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are typically observed.

Article information

Commun. Math. Sci., Volume 8, Number 4 (2010), 797-834.

First available in Project Euclid: 2 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35G20: Nonlinear higher-order equations 35K55: Nonlinear parabolic equations

Backward diffusion equations coarsening image processing Perona-Malik model You-Kaveh model


Kublik, Catherine. Coarsening in high order, discrete, ill-posed diffusion equations. Commun. Math. Sci. 8 (2010), no. 4, 797--834.

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