# Fundamentals of Non-Local Total Variation Spectral Theory

• Jean-François Aujol
• Guy Gilboa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)

## Abstract

Eigenvalue analysis based on linear operators has been extensively used in signal and image processing to solve a variety of problems such as segmentation, dimensionality reduction and more. Recently, nonlinear spectral approaches, based on the total variation functional have been proposed. In this context, functions for which the nonlinear eigenvalue problem $$\lambda u \in \partial J(u)$$ admits solutions, are studied. When $$u$$ is the characteristic function of a set $$A$$, then it is called a calibrable set. If $$\lambda >0$$ is a solution of the above problem, then $$1/\lambda$$ can be interpreted as the scale of $$A$$. However, this notion of scale remains local, and it may not be adapted for non-local features. For this we introduce in this paper the definition of non-local scale related to the non-local total variation functional. In particular, we investigate sets that evolve with constant speed under the non-local total variation flow. We prove that non-local calibrable sets have this property. We propose an onion peel construction to build such sets. We eventually confirm our mathematical analysis with some simple numerical experiments.

### Keywords

Non-local Total variation Calibrable sets Scale Nonlinear eigenvalue problem

## Preview

### References

1. 1.
Andreu, F., Ballester, C., Caselles, V., Mazon, J.M.: Minimizing total variation flow. Differential and Integral Equations 14(3), 321–360 (2001)
2. 2.
Andreu-Vaillo, F., Caselles, V., Mazon, J.M.: Parabolic quasilinear equations minimizing linear growth functionals. Progress in Mathematics, vol. 223. Birkhauser (2002)Google Scholar
3. 3.
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Applied Mathematical Sciences, vol. 147. Springer (2002)Google Scholar
4. 4.
Aujol, J.-F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition - modeling, algorithms, and parameter selection. International Journal of Computer Vision 67(1), 111–136 (2006)
5. 5.
Aujol, J.F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. Journal of Mathematical Imaging and Vision 22(1), 71–88 (2005)
6. 6.
Boulanger, J., Elbau, P., Pontow, C., Scherzer, O.: Non-local functionals for imaging. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 131–154. Springer (2011)Google Scholar
7. 7.
Brox, T., Weickert, J.: A tv flow based local scale measure for texture discrimination. In: Pajdla, T., Matas, J.G. (eds.) ECCV 2004. LNCS, vol. 3022, pp. 578–590. Springer, Heidelberg (2004)
8. 8.
Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation (SIAM Interdisciplinary Journal) 4, 490–530 (2005)
9. 9.
Caselles, V., Chambolle, A., Moll, S., Novaga, M.: A characterization of convex calibrable sets in $$\mathbb{R}^n$$ with respect to anisotropic norms. Annales de l’Institut Henri Poincaré 25, 803–832 (2008)
10. 10.
Chan, T.F., Esedoglu, S.: Aspects of total variation regularized $${L}^1$$ function approximation. SIAM Journal on Applied Mathematics 65(5), 1817–1837 (2005)
11. 11.
Duval, V., Aujol, J.-F., Gousseau, Y.: The tvl1 model: a geometric point of view. SIAM Journal on Multiscale Modeling and Simulation 8(1), 154–189 (2009)
12. 12.
Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing. IEEE Transactions on Image Processing 17(7), 1047–1060 (2008)
13. 13.
Gilboa, G.: A total variation spectral framework for scale and texture analysis. SIAM Journal on Imaging Sciences 7(4), 1937–1961 (2014)
14. 14.
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. SIAM Multiscale Modeling and Simulation 7(3), 1005–1028 (2008)
15. 15.
Luo, B., Aujol, J.-F., Gousseau, Y.: Local scale measure from the topographic map and application to remote sensing images. SIAM Journal on Multiscale Modeling and Simulation (in press 2009)Google Scholar
16. 16.
Moll, J.S.: The anisotropic total variation flow. Mathematische Annalen 332, 177–218 (2005)
17. 17.
Osher, S.J., Sole, A., Vese, L.A.: Image decomposition and restoration using total variation minimization and the H$$^{-1}$$ norm. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal 1(3), 349–370 (2003)
18. 18.
Pontow, C., Scherzer, O.: Analytical Evaluations of Double Integral Expressions Related to Total Variation. Springer (2012)Google Scholar
19. 19.
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
20. 20.
Strong, D., Aujol, J.-F., Chan, T.F.: Scale recognition, regularization parameter selection, and Meyer’s $${G}$$ norm in total variation regularization. SIAM Journal on Multiscale Modeling and Simulation 5(1), 273–303 (2006)
21. 21.
Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Problems 19(6), 165–187 (2003)
22. 22.
van Gennip, Y., Guillen, N., Osting, B., Bertozzi, A.L.: Mean curvature, threshold dynamics, and phase field theory on finite graphs. Milan Journal of Mathematics 82(1), 3–65 (2014)

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Jean-François Aujol
• 1
• 2
• Guy Gilboa
• 3