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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.

MSC:
49K40Sensitivity, stability, well-posedness of optimal solutions
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
47A52Ill-posed problems, regularization
49M30Other numerical methods in calculus of variations
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Full Text: DOI EuDML
References:
[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 · doi:10.1088/0266-5611/10/6/003
[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 · doi:10.1007/s002110050258
[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 · doi:10.1051/cocv:1997113 · http://www.edpsciences.org/articles/cocv/abs/1998/01/cocvEng-Vol2.14.html · eudml:90513
[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 · doi:10.1088/0266-5611/12/5/005
[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 · doi:10.1137/S003613999427560X
[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 · doi:10.1051/m2an:1999102 · http://publish.edpsciences.org/abstract/m2an/v33/p1 · eudml:197600
[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 · doi:10.1088/0266-5611/16/4/303
[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 · doi:10.1080/01630569808816863
[14] M. Nikolova, Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math. (to appear). Zbl0991.94015 · Zbl 0991.94015 · doi:10.1137/S0036139997327794
[15] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm. Physica D60 (1992) 259-268. · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[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 · doi:10.1137/0917016
[18] W.P. Ziemer, Weakly Differentiable Functions. Grad. Texts in Math.120 (1989). Zbl0692.46022 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3