Abstract and Applied Analysis

Abstract and Applied Analysis / 2007 / Article

Research Article | Open Access

Volume 2007 |Article ID 045153 | https://doi.org/10.1155/2007/45153

Simon P. Morgan, Kevin R. Vixie, "L1TV Computes the Flat Norm for Boundaries", Abstract and Applied Analysis, vol. 2007, Article ID 045153, 14 pages, 2007. https://doi.org/10.1155/2007/45153

L1TV Computes the Flat Norm for Boundaries

Academic Editor: Samuel Shen
Received14 Jan 2007
Accepted22 May 2007
Published08 Jul 2007


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.


  1. T. F. Chan and S. Esedoḡlu, “Aspects of total variation regularized L1 function approximation,” SIAM Journal on Applied Mathematics, vol. 65, no. 5, pp. 1817–1837, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. K. R. Vixie and S. Esedoḡlu, “Some properties of minimizers for the L1TV functional,” preprint, 2007. View at: Google Scholar
  3. W. K. Allard, “On the regularity and curvature properties of level sets of minimizers for denoising models using total variation regularization—I: theory,” preprint, 2006. View at: Google Scholar
  4. W. Yin, D. Goldfarb, and S. Osher, “Image cartoon-texture decomposition and feature selection using the total variation regularized L1 functional,” submitted to SIAM Multiscale Modeling & Simulation. View at: Google Scholar
  5. S. Alliney, “A property of the minimum vectors of a regularizing functional defined by means of the absolute norm,” IEEE Transactions on Signal Processing, vol. 45, no. 4, pp. 913–917, 1997. View at: Publisher Site | Google Scholar
  6. M. Nikolova, “Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers,” SIAM Journal on Numerical Analysis, vol. 40, no. 3, pp. 965–994, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1–4, pp. 259–268, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. J. Glaunès, Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l' anatomie numérique, Ph.D. thesis, l' Université Paris 13 en Mathématiques, Paris, France, 2005. View at: Google Scholar
  9. M. Vaillant and J. Glaunès, “Surface matching via currents,” in Proceedings of the 19th International Conference on Information Processing in Medical Imaging (IPMI '05), vol. 3565 of Lecture Notes in Computer Science, pp. 381–392, Springer, Glenwood Springs, Colo, USA, July 2005. View at: Google Scholar
  10. J. Glaunès and S. Joshi, “Template estimation form unlabeled point set data and surfaces for computational anatomy,” to appear in Journal of Mathematical Imaging and Vision. View at: Google Scholar
  11. H. Federer, “Real flat chains, cochains and variational problems,” Indiana University Mathematics Journal, vol. 24, no. 4, pp. 351–407, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. F. Morgan, Geometric Measure Theory: A beginner's Guide, Academic Press, San Diego, Calif, USA, 3rd edition, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  13. H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer, New York, NY, USA, 1969. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  14. F. Lin and X. Yang, Geometric Measure Theory—An Introduction, vol. 1 of Advanced Mathematics (Beijing/Boston), Science Press, Beijing, China; International Press, Boston, Mass, USA, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  15. L. Simon, Lectures on Geometric Measure Theory, vol. 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University, Australian National University, Centre for Mathematical Analysis, Canberra, Australia, 1983. View at: Google Scholar | Zentralblatt MATH | MathSciNet

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.

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