Open Access
December 2010 Coarsening in high order, discrete, ill-posed diffusion equations
Catherine Kublik
Commun. Math. Sci. 8(4): 797-834 (December 2010).


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.


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Catherine Kublik. "Coarsening in high order, discrete, ill-posed diffusion equations." Commun. Math. Sci. 8 (4) 797 - 834, December 2010.


Published: December 2010
First available in Project Euclid: 2 November 2010

zbMATH: 1214.65050
MathSciNet: MR2744907

Primary: 35G20 , 35K55

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

Rights: Copyright © 2010 International Press of Boston

Vol.8 • No. 4 • December 2010
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