{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:45:17Z","timestamp":1740109517877,"version":"3.37.3"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,3,3]],"date-time":"2023-03-03T00:00:00Z","timestamp":1677801600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,3,3]],"date-time":"2023-03-03T00:00:00Z","timestamp":1677801600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["V 662","I 4052","Y1292","F65"],"award-info":[{"award-number":["V 662","I 4052","Y1292","F65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"name":"BMBWF","award":["OeAD-WTZ CZ04\/2019"],"award-info":[{"award-number":["OeAD-WTZ CZ04\/2019"]}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1411646","DMS-1906238"],"award-info":[{"award-number":["DMS-1411646","DMS-1906238"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/N014588\/1"],"award-info":[{"award-number":["EP\/N014588\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A bilevel training scheme is used to introduce a novel class of regularizers, providing a unified approach to standard regularizers<jats:inline-formula><jats:alternatives><jats:tex-math>$$TGV^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>T<\/mml:mi><mml:mi>G<\/mml:mi><mml:msup><mml:mi>V<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:tex-math>$$NsTGV^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>N<\/mml:mi><mml:mi>s<\/mml:mi><mml:mi>T<\/mml:mi><mml:mi>G<\/mml:mi><mml:msup><mml:mi>V<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Optimal parameters and regularizers are identified, and the existence of a solution for any given set of training imaging data is proved by<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-convergence under a conditional uniform bound on the trace constant of the operators and a finite-null-space condition. Some first examples and numerical results are given.<\/jats:p>","DOI":"10.1007\/s00332-023-09902-4","type":"journal-article","created":{"date-parts":[[2023,3,3]],"date-time":"2023-03-03T12:02:37Z","timestamp":1677844957000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Adaptive Image Processing: First Order PDE Constraint Regularizers and a Bilevel Training Scheme"],"prefix":"10.1007","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1715-5004","authenticated-orcid":false,"given":"Elisa","family":"Davoli","sequence":"first","affiliation":[]},{"given":"Irene","family":"Fonseca","sequence":"additional","affiliation":[]},{"given":"Pan","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,3]]},"reference":[{"key":"9902_CR1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198502456.001.0001","volume-title":"Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs","author":"L Ambrosio","year":"2000","unstructured":"Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press\/Oxford University Press, New York (2000)"},{"key":"9902_CR2","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1080\/00207179.2016.1192688","volume":"90","author":"T Barbu","year":"2017","unstructured":"Barbu, T., Marinoschi, G.: Image denoising by a nonlinear control technique. Int. J. Control 90, 1005\u20131017 (2017)","journal-title":"Int. J. Control"},{"key":"9902_CR3","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1515\/jip-2013-0068","volume":"22","author":"K Bredies","year":"1993","unstructured":"Bredies, K., Holler, M.: Regularization of linear inverse problems with total generalized variation. J. Inverse Ill Posed Probl. 22, 871\u2013913 (1993)","journal-title":"J. Inverse Ill Posed Probl."},{"key":"9902_CR4","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/090769521","volume":"3","author":"K Bredies","year":"2010","unstructured":"Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3, 492\u2013526 (2010)","journal-title":"SIAM J. Imaging Sci."},{"key":"9902_CR5","unstructured":"Bredies, K., Valkonen, T.: Inverse problems with second-order total generalized variation constraints. In: Proceedings of SampTA 2011\u20149th International Conference on Sampling Theory and Applications, Singapore (2011)"},{"key":"9902_CR6","unstructured":"Breit, D., Diening, L., Gmeineder, F.: Traces of functions of bounded A-variation and variational problems with linear growth. Preprint arXiv:1707.06804"},{"key":"9902_CR7","doi-asserted-by":"crossref","unstructured":"Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011)","DOI":"10.1007\/978-0-387-70914-7"},{"key":"9902_CR8","doi-asserted-by":"publisher","first-page":"571","DOI":"10.1007\/s10851-018-0861-6","volume":"61","author":"E-M Brinkmann","year":"2019","unstructured":"Brinkmann, E.-M., Burger, M., Grah, J.S.: Unified models for second-Order TV-type regularisation in imaging\u2014a new perspective based on vector operators. J. Math. Imaging Vis. 61, 571\u2013601 (2019)","journal-title":"J. Math. Imaging Vis."},{"key":"9902_CR9","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/s10851-010-0251-1","volume":"40","author":"A Chambolle","year":"2011","unstructured":"Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120\u2013145 (2011)","journal-title":"J. Math. Imaging Vis."},{"key":"9902_CR10","doi-asserted-by":"crossref","unstructured":"Chen, Y., Pock, T., Ranftl, R., Bischof, H.: Revisiting loss-specific training of filter-based MRFs for image restoration. In: Pattern Recognition, pp. 271\u2013281. Springer (2013)","DOI":"10.1007\/978-3-642-40602-7_30"},{"key":"9902_CR11","doi-asserted-by":"publisher","first-page":"1060","DOI":"10.1109\/TIP.2014.2299065","volume":"23","author":"Y Chen","year":"2014","unstructured":"Chen, Y., Ranftl, R., Pock, T.: Insights into analysis operator learning: from patch-based sparse models to higher order MRFs. IEEE Trans. Image Process. 23, 1060\u20131072 (2014)","journal-title":"IEEE Trans. Image Process."},{"key":"9902_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0327-8","volume-title":"An introduction to $$\\Gamma $$-convergence","author":"G Dal Maso","year":"1993","unstructured":"Dal Maso, G.: An introduction to $$\\Gamma $$-convergence. Birkh\u00e4user, Boston (1993)"},{"key":"9902_CR13","doi-asserted-by":"publisher","first-page":"213","DOI":"10.4310\/CMS.2018.v16.n1.a10","volume":"16","author":"E Davoli","year":"2018","unstructured":"Davoli, E., Liu, P.: One dimensional fractional order $$TGV$$: Gamma-convergence and bilevel training scheme. Commun. Math. Sci. 16, 213\u2013237 (2018)","journal-title":"Commun. Math. Sci."},{"key":"9902_CR14","doi-asserted-by":"publisher","first-page":"464","DOI":"10.1016\/j.jmaa.2015.09.023","volume":"434","author":"JC De Los Reyes","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, 464\u2013500 (2016)","journal-title":"J. Math. Anal. Appl."},{"key":"9902_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10851-016-0662-8","volume":"57","author":"JC De los Reyes","year":"2017","unstructured":"De los Reyes, J.C., Sch\u00f6nlieb, C.-B., Valkonen, T.: Bilevel parameter learning for higher-order total variation regularisation models. J. Math. Imaging Vis. 57, 1\u201325 (2017)","journal-title":"J. Math. Imaging Vis."},{"key":"9902_CR16","first-page":"318","volume":"22","author":"J Domke","year":"2012","unstructured":"Domke, J.: Generic methods for optimization-based modeling. AISTATS 22, 318\u2013326 (2012)","journal-title":"AISTATS"},{"key":"9902_CR17","doi-asserted-by":"publisher","first-page":"779","DOI":"10.4007\/annals.2007.166.779","volume":"166","author":"C Fefferman","year":"2007","unstructured":"Fefferman, C.: $$C^m$$ extension by linear operators. Ann. Math. (2) 166, 779\u2013835 (2007)","journal-title":"Ann. Math. (2)"},{"key":"9902_CR18","doi-asserted-by":"publisher","first-page":"4491","DOI":"10.1137\/16M1088661","volume":"49","author":"I Fonseca","year":"2017","unstructured":"Fonseca, I., Liu, P.: The weighted Ambrosio\u2013Tortorelli approximation scheme. SIAM J. Math. Anal. 49, 4491\u20134520 (2017)","journal-title":"SIAM J. Math. Anal."},{"key":"9902_CR19","first-page":"284","volume":"27","author":"E Gagliardo","year":"1957","unstructured":"Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $$n$$ variabili. Rend. Sem. Mat. Univ. Padova 27, 284\u2013305 (1957)","journal-title":"Rend. Sem. Mat. Univ. Padova"},{"key":"9902_CR20","doi-asserted-by":"publisher","first-page":"108278","DOI":"10.1016\/j.jfa.2019.108278","volume":"277","author":"F Gmeineder","year":"2019","unstructured":"Gmeineder, F., Rai\u0163\u0103, B.: Embeddings for $${{\\mathbb{A} }}$$-weakly differentiable functions on domains. J. Funct. Anal. 277, 108278 (2019)","journal-title":"J. Funct. Anal."},{"key":"9902_CR21","doi-asserted-by":"crossref","unstructured":"Gmeineder, F., Rai\u0163\u0103, B., Van Schaftingen, J.: On limiting trace inequalities for vectorial differential operators. Indiana Univ. Math. J. 70, 2133\u20132176 (2021)","DOI":"10.1512\/iumj.2021.70.8682"},{"key":"9902_CR22","doi-asserted-by":"crossref","unstructured":"Guerra, A., Rai\u0163\u0103, B.: On the necessity of the constant rank condition for $$L^p$$ estimates. Comptes Rendus. Math\u00e9matique, 358(9\u201310), 1091\u20131095 (2020)","DOI":"10.5802\/crmath.105"},{"key":"9902_CR23","doi-asserted-by":"crossref","unstructured":"Guerra, A., Rai\u0163\u0103, B., Schrecker, M.R.I.: Compensated compactness: continuity in optimal weak topologies. J. Funct. Anal. 283, 109596 (2022). Preprint arXiv:2007.00564","DOI":"10.1016\/j.jfa.2022.109596"},{"key":"9902_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-68545-6","volume-title":"Boundary Integral Equations","author":"GC Hsiao","year":"2008","unstructured":"Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, New York (2008)"},{"key":"9902_CR25","doi-asserted-by":"crossref","unstructured":"Kristensen, J., Rai\u0163\u0103, B.: Oscillation and concentration in sequences of PDE constrained measures. Arch. Rational Mech. Anal. 246, 823\u2013875 (2022). Preprint arXiv:1912.09190","DOI":"10.1007\/s00205-022-01828-7"},{"key":"9902_CR26","doi-asserted-by":"crossref","unstructured":"Leoni, G.: A first course in Sobolev spaces. In: Graduate Studies in Mathematics, vol. 105. American Mathematical Society, Providence (2009)","DOI":"10.1090\/gsm\/105"},{"key":"9902_CR27","unstructured":"Rai\u0163\u0103, B.: $$L^1$$-estimates for constant rank operators (2018). Preprint arXiv:1811.10057"},{"key":"9902_CR28","doi-asserted-by":"crossref","unstructured":"Rai\u0163\u0103, B.: Potentials for $${\\mathscr {A}}$$-quasiconvexity. Calc. Var. PDEs 58, (2019). article 105","DOI":"10.1007\/s00526-019-1544-x"},{"key":"9902_CR29","doi-asserted-by":"publisher","first-page":"7297","DOI":"10.1090\/tran\/7878","volume":"372","author":"B Rai\u0163\u0103","year":"2019","unstructured":"Rai\u0163\u0103, B.: Critical $$\\text{ L}^p$$-differentiability of $$\\text{ BV}^{{\\mathbb{A} }}$$-maps and canceling operators Trans. Am. Math. Soc. 372, 7297\u20137326 (2019)","journal-title":"Am. Math. Soc."},{"key":"9902_CR30","doi-asserted-by":"crossref","unstructured":"Rai\u0163\u0103, B., Skorobogatova, A.: Continuity and canceling operators of order $$n$$ on $${\\mathbb{R}}^n$$. Calc. Var. PDEs 59, (2020). article 85","DOI":"10.1007\/s00526-020-01739-z"},{"key":"9902_CR31","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1016\/0167-2789(92)90242-F","volume":"60","author":"LI Rudin","year":"1992","unstructured":"Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259\u2013268 (1992)","journal-title":"Phys. D"},{"key":"9902_CR32","doi-asserted-by":"crossref","unstructured":"Tappen, M.F., Liu, C., Adelson, E.H., Freeman, W.T.: Learning gaussian conditional random fields for low-level vision. In: 2007 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1\u20138 (June 2007)","DOI":"10.1109\/CVPR.2007.382979"},{"key":"9902_CR33","unstructured":"Temam, R.: Probl\u00e8mes math\u00e9matiques en plasticit\u00e9. M\u00e9thodes Math\u00e9matiques de l\u2019Informatique [Mathematical Methods of Information Science], vol. 12. Gauthier-Villars, Montrouge (1983)"},{"key":"9902_CR34","doi-asserted-by":"publisher","first-page":"1044","DOI":"10.1016\/j.jmaa.2017.04.037","volume":"453","author":"T Valkonen","year":"2017","unstructured":"Valkonen, T.: The jump set under geometric regularisation. Part 2: higher-order approaches. J. Math. Anal. Appl. 453, 1044\u20131085 (2017)","journal-title":"J. Math. Anal. Appl."},{"key":"9902_CR35","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1137\/120867172","volume":"6","author":"T Valkonen","year":"2013","unstructured":"Valkonen, T., Bredies, K., Knoll, F.: Total generalized variation in diffusion tensor imaging. SIAM J. Imaging Sci. 6, 487\u2013525 (2013)","journal-title":"SIAM J. Imaging Sci."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-023-09902-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-023-09902-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-023-09902-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,8]],"date-time":"2023-12-08T06:39:31Z","timestamp":1702017571000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-023-09902-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,3]]},"references-count":35,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["9902"],"URL":"https:\/\/doi.org\/10.1007\/s00332-023-09902-4","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"type":"print","value":"0938-8974"},{"type":"electronic","value":"1432-1467"}],"subject":[],"published":{"date-parts":[[2023,3,3]]},"assertion":[{"value":"13 November 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The Authors confirm hereby that the only funding sources for this research are those summarized in the Acknowledgements section above and that they have no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"41"}}