\(L^{1}\mathrm{TV}\) computes the flat norm for boundaries. (English) Zbl 1145.49028

Summary: We show that the recently introduced \(L^{1}\)TV functional can be used to explicitly compute the flat norm for codimension one boundaries. Furthermore, using \(L^{1}\)TV, we also obtain the flat norm decomposition. Conversely, using the flat norm as the precise generalization of \(L^{1}\)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 \(L^{1}\)TV functional. We illustrate the results and implications with examples and figures.


49Q20 Variational problems in a geometric measure-theoretic setting
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[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. · Zbl 1096.94004 · doi:10.1137/040604297
[2] K. R. Vixie and S. Esedoḡlu, “Some properties of minimizers for the L1TV functional,” preprint, 2007.
[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.
[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. · Zbl 1159.68610
[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. · doi:10.1109/78.564179
[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. · Zbl 1018.49025 · doi:10.1137/S0036142901389165
[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. · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[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.
[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.
[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.
[11] H. Federer, “Real flat chains, cochains and variational problems,” Indiana University Mathematics Journal, vol. 24, no. 4, pp. 351-407, 1974. · Zbl 0289.49044 · doi:10.1512/iumj.1974.24.24031
[12] F. Morgan, Geometric Measure Theory: A beginner/s Guide, Academic Press, San Diego, Calif, USA, 3rd edition, 2000. · Zbl 0974.49025
[13] H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer, New York, NY, USA, 1969. · Zbl 0176.00801
[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. · Zbl 1074.49011
[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. · Zbl 0546.49019
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