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Article
Coarsening in High Order, Discrete, Ill-Posed Diffusion Equations
Communications in Mathematical Sciences
  • Catherine Kublik, University of Dayton
Document Type
Article
Publication Date
1-1-2010
Abstract
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.
Inclusive pages
797-834
ISBN/ISSN
1539-6746
Document Version
Published Version
Comments

Article is included for download from the repository with the permission of the publisher.

Permission documentation is on file.

Publisher
International Press of Boston Inc.
Peer Reviewed
Yes
Keywords
  • Backward diffusion equations,
  • coarsening; image processing,
  • Perona-Malik model,
  • You-Kaveh model
Citation Information
Catherine Kublik. "Coarsening in High Order, Discrete, Ill-Posed Diffusion Equations" Communications in Mathematical Sciences Vol. 8 Iss. 4 (2010)
Available at: http://works.bepress.com/catherine_kublik/4/