Abstract
We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.
Similar content being viewed by others
References
A. Braides, Gamma-Convergence for Beginners, No. 22 in Oxford Lecture Series in Mathematics and Its Applications.Oxford University Press, 2002.
K.A. Brakke, The Motion of a Surface by its Mean Curva-ture, Vol. 20 of Mathematical Notes. Princeton University Press: Princeton, NJ, 1978.
J.L. Carter, "Dual methods for total variation-Based image restoration," Ph.D. thesis, U.C.L.A. (Advisor: T. F. Chan), 2001.
A. Chambolle, "An algorithm for mean curvature motion," to appear in Interfaces Free Bound.
A. Chambolle and P.-L. Lions, "Image recovery via total vari-ation minimization and related problems," Numer. Math.,Vol. 76, No. 2, pp. 167–188, 1997.
T.F. Chan, G.H. Golub, and P. Mulet, "A nonlinear primal-dual method for total variation-based image restoration," SIAM J. Sci. Comput.,Vol. 20, No.6, pp. 1964–1977, 1999 (electronic).
P.G. Ciarlet, EsIntroduction à l'analyse numèrique matricielle et à l'optimisation, Collection Mathèmatiques Appliquèes pour la Mâýtrise. [Collection of Applied Mathematics for the Master's Degree]. Masson: Paris, 1982.
P.L. Combettes and J. Luo, "An adaptive level set method for nondifferentiable constrained image recovery," IEEE Trans.Image Process.,Vol. 11, 2002.
F. Dibos and G. Koepfler, "Global total variation minimiza-tion," SIAM J. Numer. Anal.,Vol. 37, No. 2, pp. 646–664, 2000 (electronic).
D.C. Dobson and C.R. Vogel, "Convergence of an iterative method for total variation denoising," SIAM J. Numer. Anal., Vol. 34, No. 5, pp. 1779–1791, 1997.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems. Amsterdam: North Holland, 1976.
E. Giusti, Minimal Surfaces and Functions of Bounded Varia-tion. Birkhäuser Verlag: Basel, 1984.
F. Guichard and F. Malgouyres, "Total variation based inter-polation," in Proceedings of the European Signal Processing Conference,Vol. 3, pp. 1741–1744, 1998.
J.-B. Hiriart-Urruty and C. Lemarèchal, Convex Analysis and Minimization Algorithms. I, II,Vol. 305–306 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag: Berlin, 1993 (two volumes).
Y. Li and F. Santosa, "Acomputational algorithm for minimizing total variation in image restoration," IEEE Trans. Image Process-ing, Vol. 5, pp. 987–995, 1996.
F. Malgouyres and F. Guichard, "Edge direction preserving im-age zooming: A mathematical and numerical analysis," SIAM J. Numer. Anal.,Vol. 39, No. 1, pp. 1–37, 2001 (electronic).
L.I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D,Vol. 60, pp. 259–268, 1992.
J.A. Sethian, "Fast marching methods," SIAM Rev., Vol. 41, No. 2, pp. 199–235, 1999 (electronic).
C.R. Vogel and M.E. Oman, "Iterative methods for total varia-tion denoising," SIAM J. Sci. Comput.,Vol. 17, No. 1, pp. 227–238, 1996. Special issue on iterative methods in numerical linear algebra (Breckenridge, CO, 1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chambolle, A. An Algorithm for Total Variation Minimization and Applications. Journal of Mathematical Imaging and Vision 20, 89–97 (2004). https://doi.org/10.1023/B:JMIV.0000011325.36760.1e
Issue Date:
DOI: https://doi.org/10.1023/B:JMIV.0000011325.36760.1e