{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T17:10:16Z","timestamp":1787677816333,"version":"build-2784847793"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>In this paper we study finite-difference approximations to the variational problem using the bounded variation (BV) smoothness penalty that was introduced in an image smoothing context by Rudin, Osher, and Fatemi. We give a dual formulation for an upwind finite-difference approximation for the BV seminorm; this formulation is in the same spirit as one popularized by the first author for a simpler, less isotropic, finite-difference approximation to the (isotropic) BV seminorm. We introduce a multiscale method for speeding up the approximation of both Chambolle's original method and of the new formulation of the upwind scheme. We demonstrate numerically that the multiscale method is effective, and we provide numerical examples that illustrate both the qualitative and quantitative behavior of the solutions of the numerical formulations.<\/jats:p>","DOI":"10.1137\/090752754","type":"journal-article","created":{"date-parts":[[2011,3,17]],"date-time":"2011-03-17T18:18:20Z","timestamp":1300385900000},"page":"277-299","source":"Crossref","is-referenced-by-count":49,"title":["An Upwind Finite-Difference Method for Total Variation\u2013Based Image Smoothing"],"prefix":"10.1137","volume":"4","author":[{"given":"Antonin","family":"Chambolle","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stacey E.","family":"Levine","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bradley J.","family":"Lucier","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,3,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/060662617"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.4171\/IFB\/112"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-005-4968-1"},{"key":"R4","unstructured":"J.F. Aujol,\n                      Some Algorithms for Total Variation Based Image Restoration\n                      , CMLA preprint 2008-05, CMLA, Cachan, France, 2008."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-005-4783-8"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.2001.4150"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/23.159774"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"A. Braides,\n                      $\\Gamma$-Convergence for Beginners\n                      , Oxford Lecture Series in Mathematics and Its Applications 22, Oxford University Press, Oxford, 2002.","DOI":"10.1093\/acprof:oso\/9780198507840.001.0001"},{"key":"R9","unstructured":"X. Bresson and T. Chan,\n                      Fast Minimization of the Vectorial Total Variation Norm and Applications to Color Image Processing\n                      , UCLA CAM report 07-25, UCLA, Los Angeles, CA, 2007."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2006.v4.n1.a7"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/070683003"},{"key":"R12","doi-asserted-by":"crossref","first-page":"233","DOI":"10.4171\/rmi\/634","volume":"27","author":"Caselles V.","year":"2010","journal-title":"Rev. Mat. Iberoamericana","ISSN":"https:\/\/id.crossref.org\/issn\/0213-2230","issn-type":"print"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1023\/B:JMIV.0000011325.36760.1e"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"A. Chambolle,\n                      Total variation minimization and a class of binary mrf models\n                      , in Proceedings of Energy Minimization Methods in Computer Vision and Pattern Recognition: 5th International Workshop (EMMCVPR 2005, St. Augustine, FL), A. Rangarajan, B. Vemuri, and A. L. Yuille, eds., Lecture Notes in Comput. Sci. 3757, Springer-Verlag, New York, 2005, pp. 136\u2013152.","DOI":"10.1007\/11585978_10"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050258"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1080\/02331930412331327157"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/050626090"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"J. Darbon and M. Sigelle,\n                      Exact optimization of discrete constrained total variation minimization problems\n                      , in Combinatorial Image Analysis, Lecture Notes in Comput. Sci. 3322, Springer, Berlin, 2004, pp. 548\u2013557.","DOI":"10.1007\/978-3-540-30503-3_40"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1109\/18.119733"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1080\/01630569208816489"},{"key":"R21","unstructured":"I. Ekeland and R. Temam,\n                      Convex Analysis and Variational Problems\n                      , Stud. Math. Appl. 1, North\u2013Holland, Amsterdam, 1976."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/040608982"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"Y. Meyer,\n                      Oscillating Patterns in Image Processing and Nonlinear Evolution Equations\n                      , University Lecture Ser. 22, AMS, Providence, RI, 2001.","DOI":"10.1090\/ulect\/022"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1967-11761-0"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/040605412"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90002-2"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2000104"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(92)90242-F"},{"key":"R29","unstructured":"J. Wang and B. J. Lucier,\n                      Error bounds for finite difference methods for Rudin\u2013Osher\u2013Fatemi image smoothing\n                      , SIAM J. Numer. Anal., to appear."}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090752754","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:43:30Z","timestamp":1787316210000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090752754"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/090752754"],"URL":"https:\/\/doi.org\/10.1137\/090752754","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}