{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T03:12:34Z","timestamp":1787368354049,"version":"build-2736575974"},"reference-count":55,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100016838","name":"Berlin Mathematical School","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100016838","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100019239","name":"Berlin Mathematics Research Center MATH+","doi-asserted-by":"crossref","award":["EF1-1"],"award-info":[{"award-number":["EF1-1"]}],"id":[{"id":"10.13039\/501100019239","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100019239","name":"Berlin Mathematics Research Center MATH+","doi-asserted-by":"crossref","award":["EF1-4"],"award-info":[{"award-number":["EF1-4"]}],"id":[{"id":"10.13039\/501100019239","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Swedish Foundation of Strategic Research","award":["AM13-004"],"award-info":[{"award-number":["AM13-004"]}]},{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","doi-asserted-by":"publisher","award":["AP4"],"award-info":[{"award-number":["AP4"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CRC 1114"],"award-info":[{"award-number":["CRC 1114"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CRC\/TR 109"],"award-info":[{"award-number":["CRC\/TR 109"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["C02"],"award-info":[{"award-number":["C02"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["C03"],"award-info":[{"award-number":["C03"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["RTG 2433"],"award-info":[{"award-number":["RTG 2433"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["RTG 2260"],"award-info":[{"award-number":["RTG 2260"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SPP 1798"],"award-info":[{"award-number":["SPP 1798"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006188","name":"Einstein Stiftung Berlin","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006188","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences. Of particular importance is the analysis of wavefront sets and the correct extraction of those. In this paper, we introduce the first algorithmic approach to extract the wavefront set of images, which combines data-based and model-based methods. Based on a celebrated property of the shearlet transform to unravel information on the wavefront set, we extract the wavefront set of an image by first applying a discrete shearlet transform and then feeding local patches of this transform to a deep convolutional neural network trained on labeled data. The resulting algorithm outperforms all competing algorithms in edge-orientation and ramp-orientation detection.<\/jats:p>","DOI":"10.1137\/19m1237594","type":"journal-article","created":{"date-parts":[[2019,11,14]],"date-time":"2019-11-14T10:47:52Z","timestamp":1573728472000},"page":"1936-1966","source":"Crossref","is-referenced-by-count":14,"title":["Extraction of Digital Wavefront Sets Using Applied Harmonic Analysis and Deep Neural Networks"],"prefix":"10.1137","volume":"12","author":[{"given":"Hector","family":"Andrade-Loarca","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gitta","family":"Kutyniok","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ozan","family":"\u00d6ktem","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Philipp C.","family":"Petersen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,11,14]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2018.2799231"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2010.161"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492919000059"},{"key":"atypb4","first-page":"4380","author":"Bertasius G.","year":"2015","journal-title":"Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition"},{"key":"atypb5","first-page":"37","volume":"2008","author":"Bertero M.","year":"2007","journal-title":"Italy"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/0004-3702(81)90025-4"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1145\/356869.356871"},{"key":"atypb8","first-page":"35","author":"Bubba T. A.","year":"2019","journal-title":"Inverse Problems"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/060671139"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2005.02.003"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.1986.4767851"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1109\/34.1000236"},{"key":"atypb13","unstructured":"P. Doll\u00e1r, Z. Tu, and S. Belongie,\n                      Supervised learning of edges and object boundaries\n                      , in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2006."},{"key":"atypb15","first-page":"43","author":"Davison M. E.","year":"1983","journal-title":"SIAM J. Appl. Math."},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"H. W. Engl, M. Hanke, and A. Neubauer,\n                      Regularization of Inverse Problems\n                      , Math. Appl. 375, Springer, New York, 2000.","DOI":"10.1007\/978-94-009-1740-8_3"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1515\/jiip.1995.3.6.429"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-015-9445-7"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1023\/B:VISI.0000022288.19776.77"},{"key":"atypb20","unstructured":"G. B. Folland,\n                      Introduction to Partial Differential Equations\n                      , Princeton University Press, Princeton, NJ, 1995."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s00605-010-0264-2"},{"key":"atypb22","unstructured":"P. Grohs and Z. Kereta,\n                      Continuous Parabolic Molecules\n                      , Research report, ETH Zurich, 2015."},{"key":"atypb23","first-page":"189","author":"Guo K.","year":"2005","journal-title":"TN"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2008.10.004"},{"key":"atypb25","first-page":"991","author":"Hariharan B.","year":"2011","journal-title":"DC, IEEE"},{"key":"atypb26","first-page":"147","author":"Harris C.","year":"1988","journal-title":"Proceedings of the Fourth Alvey Vision Conference"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392052"},{"key":"atypb28","unstructured":"L. Hormander,\n                      The Analysis of Linear Partial Differential OperatorsI: Distribution Theory and Fourier Analysis\n                      , Grundlehren Math. Wiss. 256, Springer, Berlin, 1990."},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.sigpro.2011.04.025"},{"key":"atypb30","first-page":"153","author":"Koh J. Y.","year":"2017","journal-title":"Berlin"},{"key":"atypb31","doi-asserted-by":"crossref","unstructured":"V. P. Krishnan and E. T. Quinto,\n                      Microlocal Analysis in Tomography\n                      , Handbook Math. Methods Imaging, Springer, New York, 2015.","DOI":"10.1007\/978-1-4939-0790-8_36"},{"key":"atypb32","first-page":"1097","author":"Krizhevsky A.","year":"2012","journal-title":"Proceedings of Advances in Neural Information Processing Systems"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-08-04700-4"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1145\/2740960"},{"key":"atypb35","first-page":"239","author":"Kutyniok G.","year":"2012","journal-title":"New York"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2015.08.006"},{"key":"atypb37","unstructured":"R. J. I. Marks,\n                      Introduction to Shannon Sampling and Interpolation Theory\n                      , Springer, New York, 2012."},{"key":"atypb38","first-page":"187","volume":"207","author":"Marr D.","year":"1980","journal-title":"Proc. B"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.1990.139492"},{"key":"atypb40","first-page":"15","volume":"10","author":"Prewitt J. M.","year":"1970","journal-title":"Pict. Process. Psychopictorics"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1137\/0524069"},{"key":"atypb42","doi-asserted-by":"crossref","unstructured":"E. T. Quinto,\n                      An Introduction to X-Ray Tomography and Radon Transforms\n                      , Proc. Sympos. Appl. Math., 63, AMS, Providence, RI, 2006.","DOI":"10.1090\/psapm\/063\/2208234"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1137\/07068326X"},{"key":"atypb44","first-page":"57","author":"Reisenhofer R.","year":"2015","journal-title":"Exp. Fluids"},{"key":"atypb45","unstructured":"L. G. Roberts,\n                      Machine Perception of Three-Dimensional Solids\n                      , Ph.D. thesis, Massachusetts Institute of Technology, 1963."},{"key":"atypb46","first-page":"1","volume":"1","author":"Sasaki Y.","year":"2007","journal-title":"Teach Tutor Mater"},{"key":"atypb47","first-page":"785","author":"Sato M.","year":"1971","journal-title":"Paris"},{"key":"atypb48","doi-asserted-by":"crossref","unstructured":"O. Scherzer, M. Grasmair, and H. Grossauer,\n                      Variational Methods in Imaging\n                      , Appl. Math. Sci., 167, Springer, New York, 2009.","DOI":"10.1007\/978-0-387-69277-7"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1109\/34.868688"},{"key":"atypb50","unstructured":"I. Sobel,\n                      An Isotropic 3x3 Image Gradient Operator\n                      , presented at Stanford A.I. Project, 1968."},{"key":"atypb51","unstructured":"M. E. Taylor,\n                      Pseudodifferential Operators\n                      , Princeton University Press, Princeton, NJ, 1981."},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.1986.4767769"},{"key":"atypb53","first-page":"205","author":"Uhlmann G.","year":"2012","journal-title":"Invent. Math."},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2009.2013082"},{"key":"atypb55","first-page":"21","author":"Yu Z.","year":"2017","journal-title":"Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI"},{"key":"atypb56","doi-asserted-by":"crossref","unstructured":"Z. Yu, W. Liu, Y. Zou, C. Feng, S. Ramalingam, B. V. Kumar, and J. Kautz,\n                      Simultaneous Edge Alignment and Learning\n                      , preprint,arXiv:1808.01992, 2018.","DOI":"10.1007\/978-3-030-01219-9_24"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1237594","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:28:52Z","timestamp":1787322532000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1237594"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":55,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/19M1237594"],"URL":"https:\/\/doi.org\/10.1137\/19m1237594","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}