Abstract
We investigate the problem of decomposing an image into three parts, namely a cartoon part, a texture part and a noise part. We argue that norms originating in the theory of optimal transport should have the ability to distinguish certain noise types from textures. Hence, we present a brief introduction to optimal transport metrics and show their relation to previously proposed texture norms. We propose different variational models and investigate their performance.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22(University Lecture Series). AMS (2001)
Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computing 19(1–3), December 2003
Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(\mathit{H}^{-1}\) norm. SIAM Multiscale Modeling and Simulation 1(3), 349–370 (2003)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40, 120–145 (2011)
Lorenz, D., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. Journal of Mathematical Imaging and Vision (2014)
Lellmann, J., Lorenz, D.A., Schönlieb, C.B., Valkonen, T.: Imaging with Kantorovich-Rubinstein discrepancy. SIAM Journal on Imaging Sciences, July 2014 (to appear). arXiv
Kaipio, J., Somersalo, E.: Statistical and computational inverse problems. Springer-Verlag, New York (2005)
Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \(L^1\) function approximation. SIAM Journal on Applied Mathematics 65(5), 1817–1837 (2005)
Clason, C., Jin, B., Kunisch, K.: A semismooth Newton method for \(L^1\) data fitting with automatic choice of regularization parameters and noise calibration. SIAM Journal on Imaging Sciences 3(2), 199–231 (2010)
Clason, C.: \(L^\infty \) fitting for inverse problems with uniform noise. Inverse Problems 28(10), 104007 (2012)
Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM Journal on Imaging Sciences 3(3), 492–526 (2010)
Monge, G.: Mémoire sur la théorie des déblais et des remblais. De l’Imprimerie Royale (1781)
Kantorovič, L.V.: On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37, 199–201 (1942)
Vasershtein, L.N.: Markov processes over denumerable products of spaces describing large system of automata. Problemy Peredači Informacii 5(3), 64–72 (1969)
Kantorovič, L.V., Rubinšteĭn, G.Š.: On a functional space and certain extremum problems. Doklady Akademii Nauk SSSR 115, 1058–1061 (1957)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Brauer, C., Lorenz, D. (2015). Cartoon-Texture-Noise Decomposition with Transport Norms. In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-18461-6_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18460-9
Online ISBN: 978-3-319-18461-6
eBook Packages: Computer ScienceComputer Science (R0)