Total variation filtering

These notes describe the derivation of a simple algorithm for signal denoising (filtering) based on total variation (TV). Total variation based filtering was introduced by Rudin, Osher, and Fatemi [8]. TV denoising is an effective filtering method for recovering piecewise-constant signals. Many algorithms have been proposed to implement total variation filtering. The one described in these notes is by Chambolle [3]. (Note: Chambolle described another algorithm in [2]). Although the algorithm can be derived in several different ways, the derivation presented here is based on descriptions given in [1, 10]. The derivation is based on the min-max property and the majorization-minimization procedure. Total variation is often used for image filtering and restoration, however, to simplify the presentation of the TV filtering algorithm these notes concentrate on one-dimensional signal filtering only. In addition, the algorithm described here may converge slowly for some problems. Faster algorithms for TV filtering have recently been developed, for example [1,10]. The development of fast, robust algorithms for TV and related non-linear filtering is an active topic of research.