The L1TV functional, which is the standard Rudin-Osher-Fatemi Total Variation image functional with an L1 data fidelity term, has recently been very carefully studied and shown to have very nice properties. Also very recently, J. Glaunes and S. Joshi have used the flat norm from geometric measure theory to compute distances in shape spaces. In this talk I will explain our recent discovery that in fact the L1TV functional computes the flat norm for co-dimension 1 sets which are themselves boundaries of a full dimensional set. Furthermore, using L1TV, we also obtain the flat norm decomposition. Conversely, using the flat norm as the precise generalization of the L1TV functional, we obtain a method for denoising non-boundary or higher co-dimension sets. Finally the flat norm decomposition of shapes can made to depend on scale using a new "flat norm with scale" which we define in direct analogy to the L1TV functional. I will illustrate the results and implications with examples and figures.
This is joint work with Simon Morgan.