{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T18:33:40Z","timestamp":1791225220191,"version":"4.1.0"},"reference-count":61,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>This article proposes a new method for image separation into a linear combination of morphological components. Sparsity in fixed dictionaries is used to extract the cartoon and oscillating content of the image. Complicated texture patterns are extracted by learning adapted local dictionaries that sparsify patches in the image. These fixed and learned sparsity priors define a nonconvex energy, and the separation is obtained as a stationary point of this energy. This variational optimization is extended to solve more general inverse problems such as inpainting. A new adaptive morphological component analysis algorithm is derived to find a stationary point of the energy. Using adapted dictionaries learned from data allows one to circumvent some difficulties faced by fixed dictionaries. Numerical results demonstrate that this adaptivity is indeed crucial in capturing complex texture patterns.<\/jats:p>","DOI":"10.1137\/090770783","type":"journal-article","created":{"date-parts":[[2010,10,5]],"date-time":"2010-10-05T18:10:31Z","timestamp":1286302231000},"page":"646-669","source":"Crossref","is-referenced-by-count":49,"title":["Learning the Morphological Diversity"],"prefix":"10.1137","volume":"3","author":[{"given":"Gabriel","family":"Peyr\u00e9","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jalal","family":"Fadili","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Luc","family":"Starck","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,9,29]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2006.881199"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-005-4783-8"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1109\/83.935036"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0042-6989(97)00121-1"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"M. Bertalm\u00eco, G. Sapiro, V. Caselles, and C. Ballester,\n                      Image inpainting\n                      , in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press, Addison\u2013Wesley, New York, 2000, pp. 417\u2013424.","DOI":"10.1145\/344779.344972"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2005.863972"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2007.906256"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2007.907073"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.10116"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1109\/83.661188"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596304010"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/050626090"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2004.833105"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20042"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2007.03.003"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1018031261"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/81.3.425"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1214\/009053604000000067"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"A. A. Efros and T. K. Leung,\n                      Texture synthesis by non-parametric sampling\n                      , in Proceedings of the International Conference on Computer Vision (ICCV '99), Vol. 2, IEEE Computer Society, Washington, DC, 1999, pp. 1033\u20131038.","DOI":"10.1109\/ICCV.1999.790383"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2006.881969"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/23\/3\/007"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2005.03.005"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"K. Engan, S. O. Aase, and J. H. Husoy,\n                      Method of optimal directions for frame design\n                      , in Proceedings of the 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '99), IEEE Computer Society, Washington, DC, 1999, pp. 2443\u20132446.","DOI":"10.1109\/ICASSP.1999.760624"},{"key":"R24","unstructured":"M. J. Fadili and J.L. Starck,\n                      Sparse representation-based image deconvolution by iterative thresholding\n                      , in Proceedings of the Astronomical Data Analysis Conference IV, F. Murtagh and J.L. Starck, eds., Marseille, France, CEA Saclay, Gif-sur-Yvette, France, 2006."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/bxm055"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2003.814255"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"M. Figueiredo and R. Nowak,\n                      A bound optimization approach to wavelet-based image deconvolution\n                      , in Proceedings of the IEEE International Conference on Image Processing (ICIP), 2005, pp. 782\u2013785.","DOI":"10.1109\/ICIP.2005.1530172"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1162\/089976603762552951"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/040619454"},{"key":"R30","doi-asserted-by":"crossref","unstructured":"S. Lesage, R. Gribonval, F. Bimbot, and L. Benaroya,\n                      Learning unions of orthonormal bases with thresholded singular value decomposition\n                      , in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2005, pp. 293\u2013296.","DOI":"10.1109\/ICASSP.2005.1416298"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1162\/089976600300015826"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-008-9047-8"},{"key":"R33","first-page":"19","volume":"11","author":"Mairal J.","year":"2010","journal-title":"J. Machine Learning Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"key":"R34","doi-asserted-by":"crossref","unstructured":"J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman,\n                      Discriminative learned dictionaries for local image analysis\n                      , in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2008, pp. 1\u20138.","DOI":"10.1109\/CVPR.2008.4587652"},{"key":"R35","unstructured":"J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman,\n                      Supervised dictionary learning\n                      , in Advances in Neural Information Processing Systems 21 (Vancouver, Canada, 2008), D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, eds., MIT Press, Cambridge, MA, 2009, pp. 1033\u20131040."},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2007.911828"},{"key":"R37","doi-asserted-by":"crossref","unstructured":"S. Mallat,\n                      A Wavelet Tour of Signal Processing\n                      , Academic Press, San Diego, 1998.","DOI":"10.1016\/B978-012466606-1\/50008-8"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2008.03.004"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20242"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1109\/83.982815"},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1006\/acha.1997.0208"},{"key":"R42","doi-asserted-by":"crossref","unstructured":"Y. Meyer,\n                      Oscillating Patterns in Image Processing and Nonlinear Evolution Equations\n                      , American Mathematical Society, Boston, 2001.","DOI":"10.1090\/ulect\/022"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-004-0552-5"},{"key":"R44","unstructured":"Yu. Nesterov,\n                      Gradient Methods for Minimizing Composite Objective Function\n                      , CORE Discussion Papers 2007\/76, Center for Operations Research and Econometrics (CORE), Universit\u00e9 catholique de Louvain, Louvain-la-Neuve, Belgium, 2007."},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1038\/381607a0"},{"key":"R46","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/20.3.389"},{"key":"R47","doi-asserted-by":"publisher","DOI":"10.1109\/TASL.2009.2031510"},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-008-0120-3"},{"key":"R49","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2009.54"},{"key":"R50","doi-asserted-by":"crossref","unstructured":"G. Peyr\u00e9, J. Fadili, and J. Starck,\n                      Learning dictionaries for geometry and texture separation\n                      , in Proceedings of SPIE, Wavelets XII, SPIE, Bellingham, WA, 2007.","DOI":"10.1117\/12.731244"},{"key":"R51","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-008-0197-6"},{"key":"R52","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(92)90242-F"},{"key":"R53","doi-asserted-by":"crossref","unstructured":"N. Shoham and M. Elad,\n                      Algorithms for signal separation exploiting sparse representations, with application to texture image separation\n                      , in Proceedings of the IEEE 25th Convention of Electrical and Electronics Engineers in Israel, IEEE Computer Society, Washington, DC, 2008, pp. 538\u2013542.","DOI":"10.1109\/EEEI.2008.4736587"},{"key":"R54","doi-asserted-by":"publisher","DOI":"10.1016\/S1076-5670(04)32006-9"},{"key":"R55","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2005.852206"},{"key":"R56","doi-asserted-by":"crossref","unstructured":"D. Tschumperl\u00e9, Y. Bentolila, J. Martinot, and M. Fadili,\n                      Fast time-space tracking of smoothly moving fine structures in image sequences\n                      , in Proceedings of the IEEE International Conference on Imaging (ICIP), San Antonio, TX, 2007, 317\u2013320.","DOI":"10.1109\/ICIP.2007.4379585"},{"key":"R57","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2005.87"},{"key":"R58","doi-asserted-by":"crossref","unstructured":"D. Tschumperl\u00e9, J. Fadili, and Y. Bentolila,\n                      Wire structure pattern extraction and tracking from X-ray images of composite mechanisms\n                      , in Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06), New York, 2006, pp. 2461\u20132466.","DOI":"10.1109\/CVPR.2006.335"},{"key":"R59","doi-asserted-by":"publisher","DOI":"10.1023\/A:1017501703105"},{"key":"R60","doi-asserted-by":"publisher","DOI":"10.1023\/A:1025384832106"},{"key":"R61","doi-asserted-by":"publisher","DOI":"10.1162\/089976601300014385"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090770783","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:30:41Z","timestamp":1787322641000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090770783"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":61,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090770783"],"URL":"https:\/\/doi.org\/10.1137\/090770783","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}