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Abstract and Applied Analysis
Volume 2007, Article ID 45153, 14 pages
Research Article

L1TV 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 L1TV functional can be used to explicitly compute the flat norm for codimension one boundaries. Furthermore, using L1TV, we also obtain the flat norm decomposition. Conversely, using the flat norm as the precise generalization of L1TV 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 L1TV functional. We illustrate the results and implications with examples and figures.