On a parabolic-hyperbolic filter for multicolor image noise reduction

Valerii  Maltsev; Michael  Pokojovy

doi:10.3934/eect.2016004

Article Contents

2016, Volume 5, Issue 2: 251-272. Doi: 10.3934/eect.2016004

This issue Previous Article Exponential stability of a coupled system with Wentzell conditions Next Article New methods for local solvability of quasilinear symmetric hyperbolic systems

On a parabolic-hyperbolic filter for multicolor image noise reduction

Valerii Maltsev^1, and
Michael Pokojovy^2,

1.
Taras Shevchenko National University of Kyiv, Faculty of Cybernetics, 4D Glushkov Ave, 03680 Kyiv
2.
Karlsruhe Institute of Technology, Department of Mathematics, Englerstrasse 2, 76131 Karlsruhe

Received: February 29, 2016

Revised: April 30, 2016

Published: May 2016

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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.

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