{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:29:50Z","timestamp":1772296190876,"version":"3.50.1"},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2017,6,9]],"date-time":"2017-06-09T00:00:00Z","timestamp":1496966400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Deutsche Forschungsgemeinschaft (DE)","award":["HI 1466\/7-1"],"award-info":[{"award-number":["HI 1466\/7-1"]}]},{"name":"MATHEON through Einstein Center for Mathematics Berlin","award":["OT1, SE5"],"award-info":[{"award-number":["OT1, SE5"]}]},{"name":"MATH EON through Einstein Center for Mathematics Berlin","award":["SE15"],"award-info":[{"award-number":["SE15"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Math Imaging Vis"],"published-print":{"date-parts":[[2017,11]]},"DOI":"10.1007\/s10851-017-0736-2","type":"journal-article","created":{"date-parts":[[2017,6,9]],"date-time":"2017-06-09T10:30:00Z","timestamp":1497004200000},"page":"515-533","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part II: Algorithm, Its Analysis and Numerical Tests"],"prefix":"10.1007","volume":"59","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9471-2479","authenticated-orcid":false,"given":"Michael","family":"Hinterm\u00fcller","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carlos N.","family":"Rautenberg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tao","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andreas","family":"Langer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2017,6,9]]},"reference":[{"key":"736_CR1","unstructured":"Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Volume 140 of Pure and Applied Mathematics. Elsevier\/Academic Press, Amsterdam (2003)"},{"issue":"3","key":"736_CR2","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s10915-007-9160-x","volume":"34","author":"A Almansa","year":"2008","unstructured":"Almansa, A., Ballester, C., Caselles, V., Haro, G.: A TV based restoration model with local constraints. J. Sci. Comput. 34(3), 209\u2013236 (2008)","journal-title":"J. Sci. Comput."},{"key":"736_CR3","unstructured":"Athavale, P., Jerrard, R., Novaga, M., Orlandi, G.: Weighted TV minimization and applications to vortex density models. Technical report, University of Pisa, Department of Mathematics, (2015)"},{"key":"736_CR4","doi-asserted-by":"crossref","unstructured":"Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces. MPS-SIAM, (2006)","DOI":"10.1137\/1.9780898718782"},{"key":"736_CR5","unstructured":"Barbu, V.: Optimal control of variational inequalities. Res, vol. 100. Notes Math. Pitman, London, United Kingdom (1984)"},{"key":"736_CR6","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1023\/A:1025391506181","volume":"19","author":"M Bertalmio","year":"2003","unstructured":"Bertalmio, M., Caselles, V., Roug\u00e9, B., Sol\u00e9, A.: TV based image restoration with local constraints. J. Sci. Comput. 19, 95\u2013122 (2003)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"736_CR7","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1109\/TAC.1976.1101194","volume":"AC\u201321","author":"DP Bertsekas","year":"1976","unstructured":"Bertsekas, D.P.: On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans. Autom. Control AC\u201321(2), 174\u2013184 (1976)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"2","key":"736_CR8","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1137\/0320018","volume":"20","author":"DP Bertsekas","year":"1982","unstructured":"Bertsekas, D.P.: Projected Newton methods for optimization problems with simple constraints. SIAM J. Control Optim. 20(2), 221\u2013246 (1982)","journal-title":"SIAM J. Control Optim."},{"key":"736_CR9","unstructured":"Bertsekas, D.P. Gafni, E.M.: Convergence of a gradient projection method. Report P-121, Laboratory for Information and Decision Systems Report, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, (1982)"},{"key":"736_CR10","unstructured":"Br\u00e9zis, H.: Probl\u00e8mes Unilat\u00e9raux. PhD thesis, Sc. math. Paris VI. 1971., (1972)"},{"issue":"6","key":"736_CR11","doi-asserted-by":"crossref","first-page":"3270","DOI":"10.1137\/070694855","volume":"30","author":"T Bui-Thanh","year":"2008","unstructured":"Bui-Thanh, T., Willcox, K., Ghattas, O.: Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput. 30(6), 3270\u20133288 (2008)","journal-title":"SIAM J. Sci. Comput."},{"key":"736_CR12","unstructured":"Cao, V.C., De los Reyes, J. C., Schoenlieb, C.B.: Learning optimal spatially-dependent regularization parameters in total variation image restoration. ArXiv e-prints, Mar. (2016)"},{"key":"736_CR13","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1137\/100807247","volume":"4","author":"RH Chan","year":"2011","unstructured":"Chan, R.H., Yang, J., Yuan, X.: Alternating direction method for image inpainting in wavelet domain. SIAM J. Imaging Sci. 4, 807\u2013826 (2011)","journal-title":"SIAM J. Imaging Sci."},{"key":"736_CR14","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s10851-006-5257-3","volume":"25","author":"TF Chan","year":"2006","unstructured":"Chan, T.F., Shen, J., Zhou, H.-M.: Total variation wavelet inpainting. J. Math. Imaging Vis. 25, 107\u2013125 (2006)","journal-title":"J. Math. Imaging Vis."},{"issue":"2","key":"736_CR15","doi-asserted-by":"crossref","first-page":"323","DOI":"10.3934\/ipi.2011.5.323","volume":"5","author":"K Chen","year":"2011","unstructured":"Chen, K., Dong, Y., Hinterm\u00fcller, M.: A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration. Inverse Probl. Imaging 5(2), 323\u2013339 (2011)","journal-title":"Inverse Probl. Imaging"},{"key":"736_CR16","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1120-4","volume-title":"Variational Inequalities and Flow in Porous Media","author":"M Chipot","year":"1984","unstructured":"Chipot, M.: Variational Inequalities and Flow in Porous Media. Springer, New York (1984)"},{"key":"736_CR17","doi-asserted-by":"crossref","unstructured":"Combettes, P.L., Pesquet, J.-C.: Proximal splitting methods in signal processing. In: Fixed-point algorithms for inverse problems in science and engineering, volume\u00a049 of Springer Optim. Appl., pp.185\u2013212. Springer, New York, (2011)","DOI":"10.1007\/978-1-4419-9569-8_10"},{"key":"736_CR18","doi-asserted-by":"crossref","unstructured":"De\u00a0los Reyes, J.C., Sch\u00f6nlieb, C.-B., Valkonen, T.: Bilevel parameter learning for higher-order total variation regularisation models. Journal of Mathematical Imaging and Vision, pages 1\u201325, (2016)","DOI":"10.1007\/s10851-016-0662-8"},{"issue":"1","key":"736_CR19","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1016\/j.jmaa.2015.09.023","volume":"434","author":"JC Los Reyes De","year":"2016","unstructured":"De Los Reyes, J.C., Sch\u00f6nlieb, C.-B., Valkonen, T.: The structure of optimal parameters for image restoration problems. J. Math. Anal. Appl. 434(1), 464\u2013500 (2016)","journal-title":"J. Math. Anal. Appl."},{"issue":"4","key":"736_CR20","doi-asserted-by":"crossref","first-page":"2448","DOI":"10.1137\/140968045","volume":"7","author":"C-A Deledalle","year":"2014","unstructured":"Deledalle, C.-A., Vaiter, S., Fadili, J., Peyr\u00e9, G.: Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection. SIAM J. Imaging Sci. 7(4), 2448\u20132487 (2014)","journal-title":"SIAM J. Imaging Sci."},{"issue":"1","key":"736_CR21","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1007\/s10851-010-0248-9","volume":"40","author":"Y Dong","year":"2011","unstructured":"Dong, Y., Hinterm\u00fcller, M., Rincon-Camacho, M.: Automated regularization parameter selection in multi-scale total variation models for image restoration. J. Math. Imaging Vis. 40(1), 82\u2013104 (2011)","journal-title":"J. Math. Imaging Vis."},{"issue":"3","key":"736_CR22","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1007\/s11263-010-0359-1","volume":"92","author":"Y Dong","year":"2011","unstructured":"Dong, Y., Hinterm\u00fcller, M., Rincon-Camacho, M.: A multi-scale vectorial l $$^{\\tau }$$ \u03c4 -TV framework for color image restoration. Int. J. Comput. Vis. 92(3), 296\u2013307 (2011)","journal-title":"Int. J. Comput. Vis."},{"issue":"1","key":"736_CR23","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1007\/s10851-010-0248-9","volume":"40","author":"Y Dong","year":"2011","unstructured":"Dong, Y., Hinterm\u00fcller, M., Rincon-Camacho, M.M.: Automated regularization parameter selection in multi-scale total variation models for image restoration. J. Math. Imaging Vis. 40(1), 82\u2013104 (2011)","journal-title":"J. Math. Imaging Vis."},{"key":"736_CR24","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1214\/12-EJS671","volume":"6","author":"K Frick","year":"2012","unstructured":"Frick, K., Marnitz, P., Munk, A.: Statistical multiresolution Dantzig estimation in imaging: fundamental concepts and algorithmic framework. Electron. J. Stat. 6, 231\u2013268 (2012)","journal-title":"Electron. J. Stat."},{"key":"736_CR25","unstructured":"Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Volume 24 of Monographs and Studies in Mathematics. Pitman. Advanced Publishing Program, Boston, MA (1985)"},{"issue":"2","key":"736_CR26","first-page":"115","volume":"5","author":"E Gumbel","year":"1935","unstructured":"Gumbel, E.: Les valeurs extr\u00eames des distributions statistiques. Ann. Inst. H. Poincar\u00e9 5(2), 115\u2013158 (1935)","journal-title":"Ann. Inst. H. Poincar\u00e9"},{"key":"736_CR27","unstructured":"Gumbel, E.J.: Statistics of extremes. Dover Publications, Inc., Mineola, NY, 2004. Reprint of the 1958 original [Columbia University Press, New York; MR0096342]"},{"issue":"3","key":"736_CR28","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1088\/0266-5611\/19\/3\/309","volume":"19","author":"E Haber","year":"2003","unstructured":"Haber, E., Tenorio, L.: Learning regularization functionals\u2014a supervised training approach. Inverse Probl. 19(3), 611\u2013626 (2003)","journal-title":"Inverse Probl."},{"issue":"2","key":"736_CR29","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1137\/080720681","volume":"20","author":"M Hinterm\u00fcller","year":"2009","unstructured":"Hinterm\u00fcller, M., Kopacka, I.: Mathematical programs with complementarity constraints in function space: $$C$$ C - and strong stationarity and a path-following algorithm. SIAM J. Optim. 20(2), 868\u2013902 (2009)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"736_CR30","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1137\/040611598","volume":"17","author":"M Hinterm\u00fcller","year":"2006","unstructured":"Hinterm\u00fcller, M., Kunisch, K.: Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1), 159\u2013187 (2006). (electronic)","journal-title":"SIAM J. Optim."},{"key":"736_CR31","unstructured":"Hinterm\u00fcller, M., Rautenberg, C.N.: Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory. WIAS Preprint No. 2235, (2016)"},{"issue":"1","key":"736_CR32","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.jmaa.2015.01.060","volume":"426","author":"M Hinterm\u00fcller","year":"2015","unstructured":"Hinterm\u00fcller, M., Rautenberg, C.N.: On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces. J. Math. Anal. Appl. 426(1), 585\u2013593 (2015)","journal-title":"J. Math. Anal. Appl."},{"issue":"8","key":"736_CR33","doi-asserted-by":"crossref","first-page":"085005","DOI":"10.1088\/0266-5611\/26\/8\/085005","volume":"26","author":"M Hinterm\u00fcller","year":"2010","unstructured":"Hinterm\u00fcller, M., Rincon-Camacho, M.: Expected absolute value estimators for a spatially adapted regularization parameter choice rule in L1-TV-based image restoration. Inverse Probl. 26(8), 085005 (2010)","journal-title":"Inverse Probl."},{"issue":"1\u20132","key":"736_CR34","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/s10107-013-0704-6","volume":"146","author":"M Hinterm\u00fcller","year":"2014","unstructured":"Hinterm\u00fcller, M., Surowiec, T.M., Mordukhovich, B.S.: Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints. Math. Program. 146(1\u20132), 555\u2013582 (2014)","journal-title":"Math. Program."},{"issue":"4","key":"736_CR35","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.3934\/ipi.2015.9.1139","volume":"9","author":"M Hinterm\u00fcller","year":"2015","unstructured":"Hinterm\u00fcller, M., Wu, T.: Bilevel optimization for calibrating point spread functions in blind deconvolution. Inverse Probl. Imaging 9(4), 1139\u20131169 (2015)","journal-title":"Inverse Probl. Imaging"},{"key":"736_CR36","volume-title":"Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, volume 23","author":"M Hinze","year":"2009","unstructured":"Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, volume 23. Springer, New York (2009)"},{"issue":"3","key":"736_CR37","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/j.csda.2011.08.018","volume":"56","author":"T Hotz","year":"2012","unstructured":"Hotz, T., Marnitz, P., Stichtenroth, R., Davies, L., Kabluchko, Z., Munk, A.: Locally adaptive image denoising by a statistical multiresolution criterion. Comput. Stat. Data Anal. 56(3), 543\u2013558 (2012)","journal-title":"Comput. Stat. Data Anal."},{"key":"736_CR38","unstructured":"Jalalzai, K.: Regularization of inverse problems in image processing. Ph.D. thesis, Ecole Polytechnique (2012)"},{"key":"736_CR39","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719451","volume-title":"An introduction to Variational Inequalities and Their Applications","author":"D Kinderlehrer","year":"2000","unstructured":"Kinderlehrer, D., Stampacchia, G.: An introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (2000)"},{"key":"736_CR40","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1137\/120882706","volume":"6","author":"K Kunisch","year":"2012","unstructured":"Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM J. Imaging Sci. 6, 938\u2013983 (2012)","journal-title":"SIAM J. Imaging Sci."},{"key":"736_CR41","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511983658","volume-title":"Mathematical Programs with Equilibrum Constraints","author":"T Luo","year":"1996","unstructured":"Luo, T., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrum Constraints. Cambridge University Press, Cambridge (1996)"},{"key":"736_CR42","unstructured":"Nittka, R.: Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains. Ph.D. thesis, Universit\u00e4t Ulm (2010)"},{"issue":"3","key":"736_CR43","doi-asserted-by":"crossref","first-page":"1125","DOI":"10.1007\/s00030-012-0201-2","volume":"20","author":"R Nittka","year":"2013","unstructured":"Nittka, R.: Quasilinear elliptic and parabolic Robin problems on Lipschitz domains. NoDEA Nonlinear Differ. Equ. Appl. 20(3), 1125\u20131155 (2013)","journal-title":"NoDEA Nonlinear Differ. Equ. Appl."},{"key":"736_CR44","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2825-5","volume-title":"Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and its Applications","author":"J Outrata","year":"1998","unstructured":"Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and its Applications, vol. 28. Kluwer Academic, Dordrecht (1998)"},{"issue":"12","key":"736_CR45","doi-asserted-by":"crossref","first-page":"4616","DOI":"10.1109\/TSP.2009.2026077","volume":"57","author":"J-C Pesquet","year":"2009","unstructured":"Pesquet, J.-C., Benazza-Benyahia, A., Chaux, C.: A SURE approach for digital signal\/image deconvolution problems. IEEE Trans. Signal Process. 57(12), 4616\u20134632 (2009)","journal-title":"IEEE Trans. Signal Process."},{"key":"736_CR46","volume-title":"Obstacle Problems in Mathematical Physics","author":"JF Rodrigues","year":"1987","unstructured":"Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam (1987)"},{"issue":"4","key":"736_CR47","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.3934\/ipi.2013.7.1183","volume":"7","author":"C Sch\u00f6nlieb","year":"2013","unstructured":"Sch\u00f6nlieb, C., De Los Reyes, J.C.: Image denoising: learning noise distribution via PDE-constrained optimisation. Inverse Probl. Imaging 7(4), 1183\u20131214 (2013)","journal-title":"Inverse Probl. Imaging"},{"key":"736_CR48","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/BF02391014","volume":"111","author":"J Serrin","year":"1964","unstructured":"Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247\u2013302 (1964)","journal-title":"Acta Math."},{"key":"736_CR49","volume-title":"Hilbert Space Methods for Partial Differential Equations. (Monographs and Studies in Mathematics.)","author":"RE Showalter","year":"1977","unstructured":"Showalter, R.E.: Hilbert Space Methods for Partial Differential Equations. (Monographs and Studies in Mathematics.). Pitman, London (1977)"},{"key":"736_CR50","volume-title":"Monotone Operators in Banach Space and Nonlinear Partial Differential Equations","author":"RE Showalter","year":"1997","unstructured":"Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society, Providence (1997)"},{"key":"736_CR51","volume-title":"Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Volume 112 of Graduate Studies in Mathematics","author":"F Tr\u00f6ltzsch","year":"2010","unstructured":"Tr\u00f6ltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2010). Translated from the 2005 German original by J\u00fcrgen Sprekels"},{"issue":"1","key":"736_CR52","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/BF01442543","volume":"5","author":"J Zowe","year":"1979","unstructured":"Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5(1), 49\u201362 (1979)","journal-title":"Appl. Math. Optim."}],"container-title":["Journal of Mathematical Imaging and Vision"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10851-017-0736-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-017-0736-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-017-0736-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,29]],"date-time":"2022-07-29T11:44:00Z","timestamp":1659095040000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10851-017-0736-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,9]]},"references-count":52,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,11]]}},"alternative-id":["736"],"URL":"https:\/\/doi.org\/10.1007\/s10851-017-0736-2","relation":{},"ISSN":["0924-9907","1573-7683"],"issn-type":[{"value":"0924-9907","type":"print"},{"value":"1573-7683","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,6,9]]}}}