{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T11:20:46Z","timestamp":1784892046532,"version":"3.55.0"},"reference-count":53,"publisher":"American Mathematical Society (AMS)","issue":"282","license":[{"start":{"date-parts":[[2013,8,15]],"date-time":"2013-08-15T00:00:00Z","timestamp":1376524800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach and analysis based approach, is due to their capability of sparsely approximating piecewise smooth functions like images. Motivated by the balanced approach and analysis based approach, we shall propose a wavelet frame based\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell _0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    minimization model, where the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell _0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u201cnorm\u201d of the frame coefficients is penalized. We adapt the penalty decomposition (PD) method of Lu and Zhang to solve the proposed optimization problem. Some convergence analysis of the adapted PD method will also be provided. Numerical results showed that the proposed model solved by the PD method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2012-02631-7","type":"journal-article","created":{"date-parts":[[2012,8,15]],"date-time":"2012-08-15T10:10:50Z","timestamp":1345025450000},"page":"995-1015","source":"Crossref","is-referenced-by-count":84,"title":["\u2113\u2080 Minimization for wavelet frame based image restoration"],"prefix":"10.1090","volume":"82","author":[{"given":"Yong","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bin","family":"Dong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhaosong","family":"Lu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2012,8,15]]},"reference":[{"issue":"455","key":"1","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1198\/016214501753208942","article-title":"Regularization of wavelet approximations","volume":"96","author":"Antoniadis, Anestis","year":"2001","journal-title":"J. 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