{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:32:29Z","timestamp":1787322749299,"version":"build-2736575974"},"reference-count":62,"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":[[2011,1]]},"abstract":"<jats:p>Image inpainting in wavelet domains refers to the recovery of an image from incomplete and\/or inaccurate wavelet coefficients. To reconstruct the image, total variation (TV) models have been widely used in the literature, and they produce high-quality reconstructed images. In this paper, we consider an unconstrained, TV-regularized, $\\ell_2$-data-fitting model to recover the image. The model is solved by the alternating direction method (ADM). At each iteration, the ADM needs to solve three subproblems, all of which have closed-form solutions. The per-iteration computational cost of the ADM is dominated by two Fourier transforms and two wavelet transforms, all of which admit fast computation. Convergence of the ADM iterative scheme is readily obtained. We also discuss extensions of this ADM scheme to solving two closely related constrained models. We present numerical results to show the efficiency and stability of the ADM for solving wavelet domain image inpainting problems. Numerical results comparing the ADM with some recent algorithms are also reported.<\/jats:p>","DOI":"10.1137\/100807247","type":"journal-article","created":{"date-parts":[[2011,9,2]],"date-time":"2011-09-02T10:26:02Z","timestamp":1314959162000},"page":"807-826","source":"Crossref","is-referenced-by-count":76,"title":["Alternating Direction Method for Image Inpainting in Wavelet Domains"],"prefix":"10.1137","volume":"4","author":[{"given":"Raymond H.","family":"Chan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junfeng","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoming","family":"Yuan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,9,1]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/10\/6\/003"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2010.2047910"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"M. V. Afonso, J. Bioucas-Dias, and M. Figueiredo,\n                      A Fast Algorithm for the Constrained Formulation of Compressive Image Reconstruction and Other Linear Inverse Problems\n                      , in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing ICASSP'2010, Dallas, TX, 2010.","DOI":"10.1109\/ICASSP.2010.5495758"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1109\/83.136597"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1109\/83.935036"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester,\n                      Image inpainting\n                      , in SIGGRAPH 2000, Computer Graphics Proceedings, K. Akeley, ed., ACM Press\/ACM SIGGRAPH\/Addison Wesley Longman, New York, 2000, pp. 417\u2013424.","DOI":"10.1145\/344779.344972"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2007.909319"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(67)90040-7"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2007.10.002"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2010.01.005"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050258"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1023\/B:JMIV.0000011325.36760.1e"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-010-0251-1"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2009.2019806"},{"key":"R15","unstructured":"T. Chan, S. Esedoglu, F. Park, and A. Yip,\n                      Recent Developments in Total Variation Image Restoration\n                      , CAM Report TR05\u201301, Los Angeles, CA, 2005."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139901390088"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1006\/jvci.2001.0487"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139900368844"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"T. F. Chan and J. Shen,\n                      Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods\n                      , SIAM, Philadelphia, 2005.","DOI":"10.1137\/1.9780898717877"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-006-5257-3"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160450502"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/050626090"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20042"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/313\/05370"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1956-0084194-4"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S003613999427560X"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"J. Eckstein and D. Bertsekas,\n                      On the Douglas-Rachford Splitting Method and the Proximal Point Algorithm for Maximal Monotone Operators\n                      , Math. Program. 55, North\u2013Holland, Amsterdam, 1992.","DOI":"10.1007\/BF01581204"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.885522"},{"key":"R29","unstructured":"E. Esser,\n                      Applications of Lagrangian-based Alternating Direction Methods and Connections to Split Bregman\n                      , CAM Report 09\u201331, UCLA, Los Angeles, CA, 2009."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2003.814255"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(76)90003-1"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1109\/83.392335"},{"key":"R33","doi-asserted-by":"crossref","unstructured":"R. Glowinski and P. Le Tallec,\n                      Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics\n                      , SIAM Stud. Appl. Math. 9, SIAM, Philadelphia, 1989.","DOI":"10.1137\/1.9781611970838"},{"key":"R34","first-page":"41","volume":"2","author":"Glowinski R.","year":"1975","journal-title":"Rev. Francaise Automat. Inform. Rech. Oper."},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1137\/080725891"},{"key":"R36","first-page":"495","volume":"21","author":"He B. S.","year":"2003","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1007\/BF00927673"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598341384"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1137\/090774823"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1137\/040605412"},{"key":"R41","unstructured":"M. J. D. Powell,\n                      A method for nonlinear constraints in minimization problems\n                      , in Optimization, R. Fletcher, ed., Academic Press, New York, 1969, pp. 283\u2013298."},{"key":"R42","doi-asserted-by":"crossref","unstructured":"S. Rane, J. Remus, and G. Sapiro,\n                      Wavelet-domain reconstruction of lost blocks in wireless image transmission and packet-switched networks\n                      , in Proceedings of the IEEE International Conference in Image Processing, Rochester, NY, 2002, pp. 309\u2013312.","DOI":"10.1109\/ICIP.2002.1038021"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2002.804264"},{"key":"R44","doi-asserted-by":"crossref","unstructured":"L. Rudin and S. Osher,\n                      Total variation based image restoration with free local constraints\n                      , in Proceedings of the 1st IEEE International Conference on Image Processing, Austin, TX, 1994, pp. 31\u201345.","DOI":"10.1109\/ICIP.1994.413269"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(92)90242-F"},{"key":"R46","doi-asserted-by":"crossref","unstructured":"S. Setzer,\n                      Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage\n                      , in Proceedings of the 2nd International Conference on Scale Space Methods and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. 5567, Springer, New York, 2009, pp. 464\u2013476.","DOI":"10.1007\/978-3-642-02256-2_39"},{"key":"R47","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-1684(03)00150-6"},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-007-0105-9"},{"key":"R49","doi-asserted-by":"publisher","DOI":"10.1137\/100781894"},{"key":"R50","unstructured":"A. Tikhonov and V. Arsenin,\n                      Solution of Ill-Posed Problems\n                      , Winston, Washington, DC, 1977."},{"key":"R51","doi-asserted-by":"publisher","DOI":"10.1137\/0917016"},{"key":"R52","doi-asserted-by":"publisher","DOI":"10.1109\/83.679423"},{"key":"R53","doi-asserted-by":"publisher","DOI":"10.1137\/080724265"},{"key":"R54","doi-asserted-by":"publisher","DOI":"10.1007\/s12532-010-0017-1"},{"key":"R55","unstructured":"Z. Lin, M. Chen, L. Wu, and Y. Ma,\n                      The augmented Lagrange multiplier method for exact recovery of a corrupted low-rank matrices\n                      , Math. Prog., submitted."},{"key":"R56","doi-asserted-by":"publisher","DOI":"10.1137\/080730421"},{"key":"R57","doi-asserted-by":"publisher","DOI":"10.1137\/090777761"},{"key":"R58","doi-asserted-by":"publisher","DOI":"10.1137\/080732894"},{"key":"R59","doi-asserted-by":"publisher","DOI":"10.1109\/JSTSP.2010.2042333"},{"key":"R60","doi-asserted-by":"publisher","DOI":"10.3934\/ipi.2010.4.191"},{"key":"R61","doi-asserted-by":"publisher","DOI":"10.1137\/090746379"},{"key":"R62","unstructured":"R. Baraniuk, H. Choi, R. Neelamani, V. Ribeiro, J. Romberg, H. Guo, F. Fernandes, B. Hendricks, R. Gopinath, M. Long, J. E. Odegard, and D. Wei,\n                      Rice Wavelet Toolbox\n                      , Version 2.4, 2002, http:\/\/dsp.rice.edu\/software\/rwt.shtml."}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100807247","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:25:10Z","timestamp":1787318710000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100807247"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":62,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/100807247"],"URL":"https:\/\/doi.org\/10.1137\/100807247","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}