Evolution Equations and Control Theory (EECT)

On a parabolic-hyperbolic filter for multicolor image noise reduction

Pages: 251 - 272, Volume 5, Issue 2, June 2016      doi:10.3934/eect.2016004

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Valerii Maltsev - Taras Shevchenko National University of Kyiv, Faculty of Cybernetics, 4D Glushkov Ave, 03680 Kyiv, Ukraine (email)
Michael Pokojovy - Karlsruhe Institute of Technology, Department of Mathematics, Englerstrasse 2, 76131 Karlsruhe, Germany (email)

Abstract: We propose a novel PDE-based anisotropic filter for noise reduction in multicolor images. It is a generalization of Nitzberg & Shiota's (1992) model being a hyperbolic relaxation of the well-known parabolic Perona & Malik's filter (1990). First, we consider a `spatial' mollifier-type regularization of our PDE system and exploit the maximal $L^{2}$-regularity theory for non-autonomous forms to prove a well-posedness result both in weak and strong settings. Again, using the maximal $L^{2}$-regularity theory and Schauder's fixed point theorem, respective solutions for the original quasilinear problem are obtained and the uniqueness of solutions with a bounded gradient is proved. Finally, the long-time behavior of our model is studied.

Keywords:  Image processing, nonlinear partial differential equations, weak solutions, strong solutions, maximal regularity.
Mathematics Subject Classification:  Primary: 35G61, 35M33, 65J15; Secondary: 35B30, 35D30, 35D35.

Received: March 2016;      Revised: May 2016;      Available Online: June 2016.