We show that the recently introduced TV functional can be used to explicitly compute the flat norm for codimension one boundaries. Furthermore, using TV, we also obtain the flat norm decomposition. Conversely, using the flat norm as the precise generalization of 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 TV functional. We illustrate the results and implications with examples and figures.
"TV Computes the Flat Norm for Boundaries." Abstr. Appl. Anal. 2007 1 - 14, 2007. https://doi.org/10.1155/2007/45153