We extend some recent results on the Hausdorff convergence of level-sets for total variation regularized linear inverse problems. Dimensions higher than two and measurements in Banach spaces are considered. We investigate the relation between the dimension and the assumed integrability of the solution that makes such an extension possible. We also give some counterexamples of practical application scenarios where the natural choice of fidelity term makes such a convergence fail.
Mots-clés : Inverse problems, total variation, Hausdorff convergence, level-sets, density estimates
@article{COCV_2020__26_1_A52_0, author = {Iglesias, Jos\'e A. and Mercier, Gwenael}, title = {Influence of dimension on the convergence of level-sets in total variation regularization}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {26}, year = {2020}, doi = {10.1051/cocv/2019035}, mrnumber = {4145244}, zbl = {1458.49030}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2019035/} }
TY - JOUR AU - Iglesias, José A. AU - Mercier, Gwenael TI - Influence of dimension on the convergence of level-sets in total variation regularization JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019035/ DO - 10.1051/cocv/2019035 LA - en ID - COCV_2020__26_1_A52_0 ER -
%0 Journal Article %A Iglesias, José A. %A Mercier, Gwenael %T Influence of dimension on the convergence of level-sets in total variation regularization %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019035/ %R 10.1051/cocv/2019035 %G en %F COCV_2020__26_1_A52_0
Iglesias, José A.; Mercier, Gwenael. Influence of dimension on the convergence of level-sets in total variation regularization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 52. doi : 10.1051/cocv/2019035. http://www.numdam.org/articles/10.1051/cocv/2019035/
[1] Analysis of bounded variation penalty methods for ill-posed problems. Inverse Prob. 10 (1994) 1217–1229. | DOI | MR | Zbl
and ,[2] Sobolev spaces. In Vol. 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition (2003). | MR | Zbl
and ,[3] Metric and generalized projection operators in Banach spaces: properties and applications. Theory and applications of nonlinear operators of accretive and monotone type, In Vol. 178 of Lecture Notes in Pure and Appl. Math. Dekker, New York (1996) 15–50. | MR | Zbl
,[4] On some estimates for projection operators in Banach spaces. Commun. Appl. Nonlinear Anal. 2 (1995) 47–55. | MR | Zbl
and ,[5] Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000). | MR | Zbl
, and ,[6] Parabolic quasilinear equations minimizing linear growth functionals. In Vol. 223 of Progress in Mathematics. Birkhäuser Verlag, Basel (2004). | MR | Zbl
, and ,[7] Morozov’s discrepancy principle for Tikhonov-type functionals with nonlinear operators. Inverse Prob. 26 (2010) 025001. | DOI | MR | Zbl
and ,[8] Mathematical problems in image processing. Second edition in Vol. 147 of Applied Mathematical Sciences. Springer, New York (2006). | DOI | MR | Zbl
and ,[9] The curvature of a set with finite area. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 5 (1994) 149–159. | MR | Zbl
,[10] Regularity of minimal boundaries with obstacles. Rend. Sem. Mat. Univ. Padova 66 (1982) 129–135. | Numdam | MR | Zbl
and ,[11] Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies. Appl. Opt. 51 (2012) 8216–8227. | DOI
, , and ,[12] Morozov’s discrepancy principle and Tikhonov-type functionals. Inverse Prob. 25 (2009) 015015. | DOI | MR | Zbl
,[13] Convex functions: constructions, characterizations and counterexamples. In Vol. 109 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2010). | MR | Zbl
and ,[14] Techniques of variational analysis. In Vol. 20 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer-Verlag, New York (2005). | MR | Zbl
and ,[15] Functional analysis, Sobolev spaces and partial differential equations. Second edition Universitext. Springer, New York (2011). | DOI | MR | Zbl
,[16] Convergence rates of convex variational regularization. Inverse Prob. 20 (2004) 1411–1421. | DOI | MR | Zbl
and ,[17] Total variation regularization in measurement and image space for PET reconstruction. Inverse Prob. 30 (2014) 105003. | DOI | MR | Zbl
, , and ,[18] Convergence rates and structure of solutions of inverse problems with imperfect forward models. Inverse Prob. 35 (2019) 024006. | DOI | MR | Zbl
, and ,[19] Long-time behavior of the mean curvature flow with periodic forcing. Commun. Part. Differ. Equ. 38 (2013) 780–801. | DOI | MR | Zbl
and ,[20] Geometric properties of solutions to the total variation denoising problem. Inverse Prob. 33 (2017) 015002. | DOI | MR | Zbl
, , and ,[21] Convex analysis and variational problems. In Vol. 28 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, english edition (1999). | MR | Zbl
and ,[22] Variational mean curvatures. Partial differential equations, II (Turin, 1993). Rend. Sem. Mat. Univ. Politec. Torino 52 (1994) 1–28. | MR | Zbl
and ,[23] On the uniform convexity of and . Ark. Mat. 3 (1956) 239–244. | DOI | MR | Zbl
,[24] A note on convergence of solutions of total variation regularized linear inverse problems. Inverse Probl. 34 (2018) 055011. | DOI | MR | Zbl
, and ,[25] Sets of finite perimeter and geometric variational problems. In Vol. 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press (2012). | MR | Zbl
,[26] An introduction to Banach space theory. In Vol. 183 of Graduate Texts in Mathematics. Springer-Verlag, New York (1998). | DOI | MR | Zbl
,[27] The mathematics of computerized tomography. Reprint ofthe 1986 original. In Vol. 32 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001). | MR | Zbl
,[28] A necessary criterion for the existence of certain minimal surfaces. J. Math. Mech. 13 (1964) 659–666. | MR | Zbl
,[29] A supplement to the condition of J. Douglas. Rend. Circ. Mat. Palermo 13 (1964) 192–198. | DOI | MR | Zbl
,[30] Lp − Lq mapping properties of the Radon transform. Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981). In Vol. 995 of Lecture Notes in Math. Springer, Berlin (1983) 95–102. | MR | Zbl
,[31] Mapping properties of the Radon transform. Indiana Univ. Math. J. 31 (1982) 641–650. | DOI | MR | Zbl
and ,[32] Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, NJ (1970). | MR | Zbl
,[33] Minimal surfaces with planar boundary curves. Kyushu J. Math. 52 (1998) 209–225. | DOI | MR | Zbl
,[34] Nonlinear total variation based noise removal algorithms. Phys. D 60 (1992) 259–268. | DOI | MR | Zbl
, and ,[35] Variational methods in imaging. Number 167 in Applied Mathematical Sciences. Springer, New York (2009). | MR | Zbl
, , , and ,[36] Regularization methods in Banach spaces. In Vol. 10 Radon Series on Computational and Applied Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin (2012). | MR | Zbl
, , and ,[37] Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators. Springer (1990). | MR | Zbl
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We would like to thank Otmar Scherzer for encouragement to work on the interplay between the space dimension and convergence of level-sets.