Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T20:55:51.790Z Has data issue: false hasContentIssue false
  • English
  • Français

StructuralProperties of Solutions to Total Variation Regularization Problems

Published online by Cambridge University Press:  15 April 2002

Wolfgang Ring
Wolfgang Ring*
Affiliation:
Institut für Mathematik, Universität Graz, Heinrichstrasse 36, 8010 Graz, Austria. e-mail: [email protected]
Get access

Abstract

In dimension one it is proved that the solution to a total variation-regularizedleast-squares problem is always a function which is "constant almost everywhere" ,provided that the data are in a certain sense outside the range of the operatorto be inverted. A similar, but weaker result is derived in dimension two.

Keywords

Total variation regularizationpiecewise constant functionconvex optimizationLebesgue decompositionsingular measures.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 4 , July 2000 , pp. 799 - 810
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acar, R. and Vogel, C.R., Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems 10 (1994) 1217-1229. CrossRef
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Math. Sci. Engrg. 190 (1993).
Chambolle, A. and Lions, P.L., Image recovery via total variation minimization and related problems. Numer. Math. 76 (1997) 167-188. CrossRef
Chavent, G. and Kunisch, K., Regularization of linear least squares problems by total bounded variation. ESAIM Control Optim. Calc. Var. 2 (1997) 359-376. CrossRef
Dobson, D. and Scherzer, O., Analysis of regularized total variation penalty methods for denoising. Inverse Problems 12 (1996) 601-617. CrossRef
Dobson, D.C. and Santosa, F., Recovery of blocky images from noisy and blurred data. SIAM J. Appl. Math. 56 (1996) 1181-1192. CrossRef
I. Ekeland and T. Turnbull, Infinite-Dimensional Optimization and Convexity. Chicago Lectures in Math., The University of Chicago Press, Chicago and London (1983).
L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992).
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren Math. Wiss. 224 (1977).
E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monogr. Math. 80 (1984).
Ito, K. and Kunisch, K., An active set strategy based on the augmented lagrantian formulation for image restauration. RAIRO Modél. Math. Anal. Numér. 33 (1999) 1-21. CrossRef
Ito, K. and Kunisch, K., BV-type regularization methods for convoluted objects with edge-flat-grey scales. Inverse Problems 16 (2000) 909-928. CrossRef
Nashed, M.Z. and Scherzer, O., Least squares and bounded variation regularization with nondifferentiable functionals. Numer. Funct. Anal. Optim. 19 (1998) 873-901. CrossRef
M. Nikolova, Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math. (to appear).
Rudin, L., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithm. Physica D 60 (1992) 259-268. CrossRef
W. Rudin, Real and Complex Analysis, 3rd edn. McGraw-Hill, New York-St Louis-San Francisco (1987).
Vogel, C. and Oman, M., Iterative methods for total variation denoising. SIAM J. Sci. Comp. 17 (1996) 227-238. CrossRef
W.P. Ziemer, Weakly Differentiable Functions. Grad. Texts in Math. 120 (1989).