L 1 TV Computes the Flat Norm for Boundaries

Abstract

We show that the recently introduced $L^1$TV functional can be used to explicitly compute the flat norm for codimension one boundaries. Furthermore, using $L^1$TV, we also obtain the flat norm decomposition. Conversely, using the flat norm as the precise generalization of $L^1$TV functional, we obtain a method for denoising nonboundary or higher codimension sets. The flat norm decomposition of differences can made to depend on scale using the flat norm with scale which we define in direct analogy to the $L^1$TV functional. We illustrate the results and implications with examples and figures.