We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
L. Alvarez, Y. Gousseau, and J.M. Morel, “Scales in natural images and a consequence on their bounded variation norm,” in Scale-Space ‘99, volume 1682 of Lecture Notes in Computer Science, 1999.
J.F. Aujol, G. Aubert, L. Blanc-Féraud, and Antonin Chambolle, “Decomposing an image: Application to textured images and sar images,” 2003. INRIA Research Report 4704.
A. Chambolle, “An algorithm for total variation minimization and applications,” JMIV, Vol. 20, p. 89, 2004.
A. Chambolle and P.L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik, Vol. 76, No. 3, pp. 167–188, 1997.
I. Ekeland and R. Temam, “Analyse convexe et problemes variationnels,” Grundlehren der mathematischen Wissenschaften, 2nd edn, Dunod, Vol. 224, 1983.
J.B. Hiriart-Urruty and C. Lemarechal, “Convex analysis ans minimisation algorithms I,” Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Vol. 305, 1993.
Henderson Lewis, “Principle and applications of Imaging Radar,” Manual of Remote Sensing, 3rd edn., J. Wiley and Sons, Vol. 2, 1998.
Yves Meyer, “Oscillating patterns in image processing and in some nonlinear evolution equations,” The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures, March 2001.
S.J. Osher, A. Sole, and L.A. Vese, “Image decomposition and restoration using total variation minimization and the H−1 norm,” SIAM Journal on Multiscale Modeling and Simulation, Vol. 1, No. 3, pp. 349–370, 2002.
T. Rockafellar, “Convex analysis,” Etudes Mathematiques Princeton University Press, 1974.
L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, pp. 259–268, 1992.
Luminita A. Vese and Stanley J. Osher, “Modeling textures with total variation minimization and oscillating patterns in image processing,” Journal of Scientific Computing, Vol. 15, pp. 553–572, 2003.
Jean-François Aujol graduated from “1’ Ecole Normale Supérieure de Cachan” in 2001. He was a PH’D student in Mathematics at the University of Nice-Sophia-Antipolis (France). He was a member of the J.A. Dieudonné Laboratory at Nice, and also a member of the Ariana research group (CNRS/INRIA/UNSA) at Sophia-Antipolis (France). His research interests are calculus of variations, nonlinear partial differential equations, numerical analysis and mathematical image processing (and in particular classification, texture, decomposition model, restoration). He is Assistant Researcher at UCLA (Math Department).
Gilles Aubert received the These d’Etat es-sciences Mathematiques from the University of Paris 6, France, in 1986. He is currently professor of mathematics at the University of Nice-Sophia Antipolis and member of the J.A. Dieudonne Laboratory at Nice, France. His research interests are calculus of variations, nonlinear partial differential equations and numerical analysis; fields of applications including image processing and, in particular, restoration, segmentation, optical flow and reconstruction in medical imaging.
Laure Blanc-Féraud received the Ph.D. degree in image restoration in 1989 and the “Habilitation á Diriger des Recherches” on inverse problems in image processing in 2000, from the University of Nice-Sophia Antipolis, France. She is currently director of research at CNRS in Sophia Antipolis. Her research interests are inverse problems in image processing by deterministic approach using calculus of variation and PDEs. She is also interested in stochastic models for parameter estimation and their relationship with the deterministic approach. She is currently working in the Ariana research group (I3S/INRIA) which is focussed on Earth observation.
Antonin Chambolle studied mathematics and physics at the Ecole normale Supérieure in Paris and received the Ph.D. degree in applied mathematics from the Université de Paris-Dauphine in 1993. Since then he has been a CNRS researcher at the CEREMADE, Université de Paris-Dauphine, and, for a short period, a researcher at the SISSA, Trieste, Italy. His research interest include calculus of variations, with applications to shape optimization, mechanics and image processing.
About this article
Cite this article
Aujol, JF., Aubert, G., Blanc-Féraud, L. et al. Image Decomposition into a Bounded Variation Component and an Oscillating Component. J Math Imaging Vis 22, 71–88 (2005). https://doi.org/10.1007/s10851-005-4783-8
- total variation minimization
- SAR images