L1TV Computes the Flat Norm for Boundaries

We show that the recently introduced L1TV functional can be used toexplicitly compute the flat norm for co-dimension one boundaries. While thisobservation alone is very useful, other important implications for imageanalysis and shape statistics include a method for denoising sets which are notboundaries or which have higher co-dimension and the fact that using the flatnorm to compute distances not only gives a distance, but also an informativedecomposition of the distance. This decomposition is made to depend on scaleusing the "flat norm with scale" which we define in direct analogy to the L1TVfunctional. We illustrate the results and implications with examples andfigures.