Relations between regularization and diffusion filtering

Regularization may be regarded as diffusion filtering with an implicit time discretization where one single step is used. Thus, iterated regularizatio n with small regularization parameters approximates a diffusion process. Th e goal of this paper is to analyse relations between noniterated and iterat ed regularization and diffusion filtering in image processing. In the linea r regularization framework, we show that with iterated Tikhonov regularizat ion noise can be better handled than with noniterated. In the nonlinear fra mework, two filtering strategies are considered: the total variation regula rization technique and the diffusion filter technique of Perona and Malik. It is shown that the Perona-Malik equation decreases the total variation du ring its evolution. While noniterated and iterated total variation regulari zation is well-posed, one cannot expect to find a minimizing sequence which converges to a minimizer of the corresponding energy functional for the Pe rona-Malik filter. To overcome this shortcoming, a novel regularization tec hnique of the Perona-Malik process is presented that allows to construct a weakly lower semi-continuous energy functional. In analogy to recently deri ved results for a well-posed class of regularized Perona-Malik filters, we introduce Lyapunov functionals and convergence results for regularization m ethods. Experiments on real-world images illustrate that iterated linear re gularization performs better than noniterated, while no significant differe nces between noniterated and iterated total variation regularization have b een observed.