Relations Between Regularization and Diffusion Filtering

• Authors: Otmar Scherzer;Joachim Weickert
• Venue: Journal of Mathematical Imaging and Vision
• Year: 2000

Regularization may be regarded as diffusion filtering with animplicit time discretization where one single step is used. Thus, iteratedregularization with small regularization parameters approximatesa diffusion process. The goal of this paper is to analyse relationsbetween noniterated and iterated regularization and diffusionfiltering in image processing. In the linear regularization framework, we show that with iterated Tikhonovregularization noise can be better handled than with noniterated.In the nonlinear framework, two filtering strategies are considered:the total variation regularization technique and the diffusion filtertechnique of Perona and Malik. It is shown that the Perona-Malik equation decreases the total variation during its evolution.While noniterated and iterated total variation regularization iswell-posed, one cannot expect to find a minimizing sequence whichconverges to a minimizer of the corresponding energy functional forthe Perona–Malik filter. To overcome this shortcoming, a novel regularization technique ofthe Perona–Malik process is presented that allows to construct aweakly lower semi-continuous energy functional.In analogy to recently derived results for a well-posed class ofregularized Perona–Malik filters, we introduce Lyapunov functionalsand convergence results for regularization methods. Experiments on real-world images illustrate that iterated linearregularization performs better than noniterated, whileno significant differences between noniterated and iterated totalvariation regularization have been observed.