Abstract and Applied Analysis
Volume 2007 (2007), Article ID 45153, 14 pages
TV Computes the Flat Norm for Boundaries
1Department of Mathematics, University of Minnesota, Minneapolis 55455, MN, USA
2Los Alamos National Laboratory, Los Alamos 87545, NM, USA
Received 14 January 2007; Accepted 22 May 2007
Academic Editor: Samuel Shen
Copyright © 2007 Simon P. Morgan and Kevin R. Vixie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.