{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:48:44Z","timestamp":1787323724534,"version":"build-2736575974"},"reference-count":29,"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":[[2008,1]]},"abstract":"<jats:p>We are motivated by a recently developed nonlinear inverse scale space method for image denoising [M. Burger, G. Gilboa, S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179\u2013212; M. Burger, S. Osher, J. Xu, and G. Gilboa, in Variational, Geometric, and Level Set Methods in Computer Vision, Lecture Notes in Comput. Sci. 3752, Springer, Berlin, 2005, pp. 25\u201336], whereby noise can be removed with minimal degradation. The additive noise model has been studied extensively, using the Rudin\u2013Osher\u2013Fatemi model [L. I. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259\u2013268], an iterative regularization method [S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, Multiscale Model. Simul., 4 (2005), pp. 460\u2013489], and the inverse scale space flow [M. Burger, G. Gilboa, S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179\u2013212; M. Burger, S. Osher, J. Xu, and G. Gilboa, in Variational, Geometric, and Level Set Methods in Computer Vision, Lecture Notes in Comput. Sci. 3752, Springer, Berlin, 2005, pp. 25\u201336]. However, the multiplicative noise model has not yet been studied thoroughly. Earlier total variation models for the multiplicative noise cannot easily be extended to the inverse scale space, due to the lack of global convexity. In this paper, we review existing multiplicative models and present a new total variation framework for the multiplicative noise model, which is globally strictly convex. We extend this convex model to the nonlinear inverse scale space flow and its corresponding relaxed inverse scale space flow. We demonstrate the convergence of the flow for the multiplicative noise model, as well as its regularization effect and its relation to the Bregman distance. We investigate the properties of the flow and study the dependence on flow parameters. The numerical results show an excellent denoising effect and significant improvement over earlier multiplicative models.<\/jats:p>","DOI":"10.1137\/070689954","type":"journal-article","created":{"date-parts":[[2008,10,29]],"date-time":"2008-10-29T13:45:01Z","timestamp":1225287901000},"page":"294-321","source":"Crossref","is-referenced-by-count":256,"title":["A Nonlinear Inverse Scale Space Method for a Convex Multiplicative Noise Model"],"prefix":"10.1137","volume":"1","author":[{"given":"Jianing","family":"Shi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stanley","family":"Osher","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,9,4]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1155\/S1110865703211136"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/060671814"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"L.M. Bregman,\n                      The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming\n                      , USSR Comput. Math. Math. Phys., 7 (1967),pp. 200\u2013217.","DOI":"10.1016\/0041-5553(67)90040-7"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/060660564"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2006.v4.n1.a7"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"M. Burger, S. Osher, J. Xu, and G. Gilboa,\n                      Nonlinear inverse scale space methods for image restoration\n                      , in Variational, Geometric, and Level Set Methods in Computer Vision, Lecture Notes in Comput. 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Tupin,\n                      A Note on Nice-Levelable MRFs for SAR Image Denoising with Contrast Preservation\n                      , Technical report ENST 2006D006, Ecole Nationale Sup\u00e9rieure des T\u00e9l\u00e9communications, Paris, 2006; available online from http:\/\/www.telecom-paristech.fr\/data\/files\/docs\/id6191159280203271.pdf http:\/\/www.telecom-paristech.fr\/data\/files\/docs\/id6191159280203271.pdf."},{"key":"R13","unstructured":"I. Ekeland and R. Temam,\n                      Convex analysis and variational problems\n                      , Stud. Math. Appl. 1, North-Holland, Amsterdam, Oxford, 1976."},{"key":"R14","unstructured":"A. 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Lions, and S. Osher,\n                      Multiplicative denoising and deblurring: Theory and algorithms\n                      , in Geometric Level Set Methods in Imaging, Vision, and Graphics, S. Osher and N. Paragios, eds., Springer, New York, 2003, pp. 103\u2013119.","DOI":"10.1007\/0-387-21810-6_6"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(92)90242-F"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"O. Scherzer,\n                      Explicit versus implicit relative error regularization on the space of functions of bounded variation\n                      , in Inverse Problems, Image Analysis, and Medical Imaging, Contemp. Math. 313, AMS, Providence, RI, 2002, pp. 171\u2013198.","DOI":"10.1090\/conm\/313\/05376"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"O. Scherzer and C. 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