New methods of controlled total variation reduction for digital functions

Abstract

We consider the functional of total variation for functions of either one or two discrete variables. We introduce a new principle for construction of evolution processes that gradually reduce the total variation of a function. The method is derived from analysis of relaxation of the functional along the directions given by the Haar wavelets - an approach that does not require regularization of either the functional or its singular gradient. We further present a rigorous analysis of the analytical properties of the resulting evolution schemes. The purely discrete theory grows out of universal observations about the relation between the total variation and the Haar functions. An additional advantage of the method is that it can be directly implemented as an effective numerical procedure. Moreover, it can be intertwined in a natural way with a nonlinear phase-space filtering which further increases control of the deformation of a signal. Finally, we report that the algorithms can be used as remarkable tools for signal processing and image enhancement.