Equivalence results for TV diffusion and TV regularisation

Abstract: It has been stressed that regularisation methods and diffusion processes approximate each other. In this paper we identify a situation where both processes are even identical: the space-discrete 1-D case of total variation (TV) denoising. This equivalence is proved by deriving identical analytical solutions for both processes. The temporal evolution confirms that space-discrete TV methods implement a region merging strategy with finite extinction time. Between two merging events, only extremal segments move. Their speed is inversely proportional to their size. Our results stress the distinguished nature of TV denoising. Furthermore, they enable a mutual transfer of all theoretical and algorithmic achievements between both techniques.