Abstract
We propose a nonlinear multiscale decomposition of signals defined on the vertex set of a general weighted graph. This decomposition is inspired by the hierarchical multiscale (BV,L 2) decomposition of Tadmor, Nezzar, and Vese (Multiscale Model. Simul. 2(4):554–579, 2004). We find the decomposition by iterative regularization using a graph variant of the classical total variation regularization (Rudin et al, Physica D 60(1–4):259–268, 1992). Using tools from convex analysis, and in particular Moreau’s identity, we carry out the mathematical study of the proposed method, proving the convergence of the representation and providing an energy decomposition result. The choice of the sequence of scales is also addressed. Our study shows that the initial scale can be related to a discrete version of Meyer’s norm (Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, 2001) which we introduce in the present paper. We propose to use the recent primal-dual algorithm of Chambolle and Pock (J. Math. Imaging Vis. 40:120–145, 2011) in order to compute both the minimizer of the graph total variation and the corresponding dual norm. By applying the graph model to digital images, we investigate the use of nonlocal methods to the multiscale decomposition task. Since the only assumption needed to apply our method is that the input data is living on a graph, we are also able to tackle the task of adaptive multiscale decomposition of irregularly sampled data sets within the same framework. We provide in particular examples of 3-D irregular meshes and point clouds decompositions.
Similar content being viewed by others
Notes
A very preliminary version of this work was published in [24].
Star as a superscript denotes the convex conjugate function while star as a subscript denotes the dual norm.
By a trivial decomposition we mean the decomposition f=u+v where \(u=\overline{f}\) is the vector whose all components are equal to the mean of f.
The model is taken from the sample dataset of the Cyberware Head & Face Color 3D Scanner available at: http://www.cyberware.com/.
References
Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123, 199–257 (1993)
Arias, P., Caselles, V., Sapiro, G.: A variational framework for non-local image inpainting. In: Energy Minimization Methods in Computer Vision and Pattern Recognition. LNCS, vol. 5681, pp. 345–358 (2009)
Aujol, J.F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22, 71–88 (2005)
Aujol, J.F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)
Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer, Berlin (2006)
Bougleux, S., Elmoataz, A., Melkemi, M.: Discrete regularization on weighted graphs for image and mesh filtering. In: Scale Space and Variational Methods in Computer Vision. LNCS, vol. 4485, pp. 128–139 (2007)
Buades, A., Coll, B., Morel, J.M.: Image denoising methods. A new non-local principle. SIAM Rev. 52(1), 113–147 (2010)
Candes, E.J., Romberg, J.K.: Signal Recovery from Random Projections, pp. 76–86. SPIE, Bellingham (2005)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)
Chambolle, A.: Total variation minimization and a class of binary mrf models. In: Energy Minimization Methods in Computer Vision and Pattern Recognition. LNCS, vol. 3757, pp. 136–152 (2005)
Chambolle, A., De Vore, R., Lee, N.Y., Lucier, B.: Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7(3), 319–335 (1998)
Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)
Chan, R., Chan, T., Yip, A.: Numerical methods and applications in total variation image restoration. In: Handbook of Mathematical Methods in Imaging, pp. 1059–1094. Springer, Berlin (2011)
Chan, T., Osher, S., Shen, J.: The digital TV filter and nonlinear denoising. IEEE Trans. Image Process. 10(2), 231–241 (2001)
Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20(6), 1964–1977 (1999)
Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, vol. 49, pp. 185–212. Springer, Berlin (2011)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)
Daubechies, I., Fornasier, M., Loris, I.: Accelerated projected gradient method for linear inverse problems with sparsity constraints. J. Fourier Anal. Appl. 14, 764–792 (2008)
Elmoataz, A., Lézoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17(7), 1047–1060 (2008)
Fadili, J., Peyré, G.: Total variation projection with first order schemes. IEEE Trans. Image Process. 20(3), 657–669 (2011)
Getreuer, P.: Image zooming with contour stencils. In: Proceedings of SPIE, vol. 7246 (2009)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)
Hidane, M., Lézoray, O., Ta, V.T., Elmoataz, A.: Nonlocal multiscale hierarchical decomposition on graphs. In: Computer Vision ECCV 2010. LNCS, vol. 6314, pp. 638–650 (2010)
Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)
Lézoray, O., Ta, V.T., Elmoataz, A.: Partial differences as tools for filtering data on graphs. Pattern Recognit. Lett. 31(14), 2201–2213 (2010)
von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17, 395–416 (2007)
Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series. American Mathematical Society, Providence (2001)
Mohar, B.: The Laplacian spectrum of graphs. Graph Theory Comb. Appl. 2(6), 871–898 (1991)
Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. Paris Sér. A Math. 255, 2897–2899 (1962)
Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)
Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)
Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H −1 norm. Multiscale Model. Simul. 1(3), 349–370 (2003)
Pauly, M., Kobbelt, L.P., Gross, M.: Point-based multiscale surface representation. ACM Trans. Graph. 25(2), 177–193 (2006)
Peyré, G., Bougleux, S., Cohen, L.: Non-local regularization of inverse problems. In: Computer Vision ECCV 2008. LNCS, vol. 5304, pp. 57–68 (2008)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)
Scherzer, O., Groetsch, C.: Inverse scale space theory for inverse problems. In: Scale-Space and Morphology in Computer Vision. LNCS, vol. 2106, pp. 317–325 (2006)
Strong, D.M., Aujol, J.F., Chan, T.F.: Scale recognition, regularization parameter selection, and Meyer’s g norm in total variation regularization. Multiscale Model. Simul. 5(1), 273–303 (2006)
Tadmor, E., Nezzar, S., Vese, L.: A multiscale image representation using hierarchical (BV,L 2) decompositions. Multiscale Model. Simul. 2(4), 554–579 (2004)
Tadmor, E., Nezzar, S., Vese, L.: Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation. Commun. Math. Sci. 6(2), 281–307 (2008)
Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19, 553–572 (2003)
Weickert, J.: Anisotropic Diffusion in Image Processing. ECMI Series. Teubner, Leipzig (1998)
Yun, S., Woo, H.: Linearized proximal alternating minimization algorithm for motion deblurring by nonlocal regularization. Pattern Recognit. 44(6), 1312–1326 (2011)
Zhou, D., Schölkopf, B.: Regularization on discrete spaces. In: Pattern Recognition. LNCS, vol. 3663, pp. 361–368 (2005)
Acknowledgements
The authors would like to thank Jalal Fadili for advices and fruitful discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hidane, M., Lézoray, O. & Elmoataz, A. Nonlinear Multilayered Representation of Graph-Signals. J Math Imaging Vis 45, 114–137 (2013). https://doi.org/10.1007/s10851-012-0348-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-012-0348-9