Two New Nonlinear Nonlocal Diffusions for Noise Reduction
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Two new nonlocal nonlinear diffusion models for noise reduction are proposed, analyzed and implemented. They are both a close relative of the celebrated Perona-Malik equation. In a way, they can be viewed as a new regularization paradigm for Perona-Malik. They do preserve and enhance the most cherished features of Perona-Malik while delivering well-posed equations which admit a stable natural discretization. Unlike other regularizations, however, certain piecewise smooth functions are (meta)stable equilibria and, as a consequence, their dynamical behavior and that of their discrete implementations can be fully understood and do not lead to any “paradox”. The presence of nontrivial equilibria also explains why blurring is kept in check. One of the models has been proved to be well-posed. Numerical experiments are presented that illustrate the main features of the new models and that provide insight into their interesting dynamical behavior as well as demonstrate their effectiveness as a denoising tool.
KeywordsNonlinear diffusion Nonlocal diffusion De-noising
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- 4.Belahmidi, A.: Equations aux dérivées partielles appliquées à la restoration et à l’agrandissement des images. Ph.D. Thesis, Université Paris-Dauphine, Paris (2003) Google Scholar
- 6.Bellettini, G., Novaga, M., Paolini, M.: Convergence for long-times of a semidiscrete Perona-Malik equation in one dimension. Preprint (2008) Google Scholar
- 7.Bellettini, G., Novaga, M., Paolini, M., Tornese, C.: Classification of the equilibria for the semi-discrete Perona-Malik equation. Preprint (2008) Google Scholar
- 16.Guidotti, P.: A new well-posed nonlinear nonlocal diffusion. Submitted Google Scholar
- 20.Nitzberg, M., Shiota, T.: Nonlinear image smoothing with edge and corner enhancement. Tech. Report 90-2, Harvard University, Cambridge, MA (1990) Google Scholar
- 26.Weickert, J.: Anisotropic diffusion in image processing. Ph.D. Thesis, Universität Kaiserslautern, Kaiserslautern (1996) Google Scholar
- 27.Weickert, J.: Anisotropic Diffusion in Image Processing. ECMI Series. Teubner, Stuttgart (1998) Google Scholar