L1TV computes the flat norm for boundaries

Abstract

We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which have higher co-dimension and the fact that using the flat norm to compute distances not only gives a distance, but also an informative decomposition of the distance. This decomposition is made to depend on scale using the "flat norm with scale" which we define in direct analogy to the L1TV functional. We illustrate the results and implications with examples and figures.