Structural properties of solutions to total variation regularization problems. (English) Zbl 1018.49021

Summary: In dimension one it is proved that the solution to a total variation-regularized least-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 operator to be inverted. A similar, but weaker result is derived in dimension two.


49K40 Sensitivity, stability, well-posedness
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47A52 Linear operators and ill-posed problems, regularization
49M30 Other numerical methods in calculus of variations (MSC2010)
Full Text: DOI EuDML


[1] R. Acar and C.R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems10 (1994) 1217-1229. · Zbl 0809.35151
[2] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Math. Sci. Engrg.190 (1993). Zbl0776.49005 · Zbl 0776.49005
[3] A. Chambolle and P.L. Lions, Image recovery via total variation minimization and related problems. Numer. Math.76 (1997) 167-188. · Zbl 0874.68299
[4] G. Chavent and K. Kunisch, Regularization of linear least squares problems by total bounded variation. ESAIM Control Optim. Calc. Var.2 (1997) 359-376. · Zbl 0890.49010
[5] D. Dobson and O. Scherzer, Analysis of regularized total variation penalty methods for denoising. Inverse Problems12 (1996) 601-617. Zbl0866.65041 · Zbl 0866.65041
[6] D.C. Dobson and F. Santosa, Recovery of blocky images from noisy and blurred data. SIAM J. Appl. Math.56 (1996) 1181-1192. Zbl0858.68121 · Zbl 0858.68121
[7] I. Ekeland and T. Turnbull, Infinite-Dimensional Optimization and Convexity. Chicago Lectures in Math., The University of Chicago Press, Chicago and London (1983). · Zbl 0565.49003
[8] L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). · Zbl 0804.28001
[9] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren Math. Wiss.224 (1977). Zbl0361.35003 · Zbl 0361.35003
[10] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monogr. Math.80 (1984). · Zbl 0545.49018
[11] K. Ito and K. Kunisch, An active set strategy based on the augmented lagrantian formulation for image restauration. RAIRO Modél. Math. Anal. Numér.33 (1999) 1-21. · Zbl 0918.65050
[12] K. Ito and K. Kunisch, BV-type regularization methods for convoluted objects with edge-flat-grey scales. Inverse Problems16 (2000) 909-928. · Zbl 0981.65149
[13] M.Z. Nashed and O. Scherzer, Least squares and bounded variation regularization with nondifferentiable functionals. Numer. Funct. Anal. Optim.19 (1998) 873-901. · Zbl 0914.65067
[14] M. Nikolova, Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math. (to appear). Zbl0991.94015 · Zbl 0991.94015
[15] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm. Physica D60 (1992) 259-268. · Zbl 0780.49028
[16] W. Rudin, Real and Complex Analysis, 3rd edn. McGraw-Hill, New York-St Louis-San Francisco (1987). · Zbl 0925.00005
[17] C. Vogel and M. Oman, Iterative methods for total variation denoising. SIAM J. Sci. Comp.17 (1996) 227-238. · Zbl 0847.65083
[18] W.P. Ziemer, Weakly Differentiable Functions. Grad. Texts in Math.120 (1989). Zbl0692.46022 · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.