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Applied Math Seminar
Nonlinear Diffusions and Denoising: From Perona-Malik to some recent
developments.
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The celebrated Perona-Malik equation was one of the first nonlinear PDE-based models proposed for use in image processing. It has attracted considerable and steady interest in the image processing as well as in the mathematical communities. This is partly due to the apparent discrepancy between its mathematical properties (it is an ill-posed forward-backward diffusion equation) and its practical effectiveness. A variety of fixes in the form of regularizations/relaxations will be discussed which were suggested by various researchers over the years. While these are successful at curing the mathematical ill-posedness, they sacrifice some salient and welcome features of Perona-Malik. A new model will be presented which shares its well-posedness with more traditional types of regularization but also improves on some short-comings of Perona-Malik beyond preserving its most cherished qualities. While the new equations are well-posed and deliver an effective denoising and deblurring tool, they still raise a number of interesting mathematical questions. |