Document Zbl 1242.90160

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(ed.) et al., Fixed-point algorithms for inverse problems in science and engineering. Based on the presentations at the interdisciplinary workshop, BIRS, Banff, Canada, November 1–6, 2009. New York, NY: Springer (ISBN 978-1-4419-9568-1/hbk; 978-1-4419-9569-8/ebook). Springer Optimization and Its Applications 49, 185-212 (2011). </div> <div class="abstract">Summary: The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1217.00018">Zbl 1217.00018</a>].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6054038">Cited in <strong>458</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C25" title="MSC2020">90C25</a> </td> <td class="space"> Convex programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65K05" title="MSC2020">65K05</a> </td> <td class="space"> Numerical mathematical programming methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C90" title="MSC2020">90C90</a> </td> <td class="space"> Applications of mathematical programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A08" title="MSC2020">94A08</a> </td> <td class="space"> Image processing (compression, reconstruction, etc.) in information and communication theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aalternating-direction+method+of+multipliers">alternating-direction method of multipliers</a>; <a href="/?q=ut%3Abackward-backward+algorithm">backward-backward algorithm</a>; <a href="/?q=ut%3Aconvex+optimization">convex optimization</a>; <a href="/?q=ut%3Adenoising">denoising</a>; <a href="/?q=ut%3ADouglas-Rachford+algorithm">Douglas-Rachford algorithm</a>; <a href="/?q=ut%3Aforward-backward+algorithm">forward-backward algorithm</a>; <a href="/?q=ut%3Aframe">frame</a>; <a href="/?q=ut%3ALandweber+method">Landweber method</a>; <a href="/?q=ut%3Aiterative+thresholding">iterative thresholding</a>; <a href="/?q=ut%3Aparallel+computing">parallel computing</a>; <a href="/?q=ut%3APeaceman-Rachford+algorithm">Peaceman-Rachford algorithm</a>; <a href="/?q=ut%3Aproximal+algorithm">proximal algorithm</a>; <a href="/?q=ut%3Arestoration+and+reconstruction">restoration and reconstruction</a>; <a href="/?q=ut%3Asparsity">sparsity</a>; <a href="/?q=ut%3Asplitting">splitting</a></div> <div class="software"> <h3>Software:</h3><a href="/software/7521">UNLocBoX</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1242.90160" data-ciurl="/ci/06054038" data-biburl="/bibtex/06054038.bib" data-amsurl="/amsrefs/06054038.bib" data-xmlurl="/xml/06054038.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06054038.pdf" title="Zbl 1242.90160 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/978-1-4419-9569-8_10" aria-label="DOI for “Proximal splitting methods in signal processing”" title="10.1007/978-1-4419-9569-8_10">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/0912.3522"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Acker, F.; Prestel, MA, Convergence d’un schéma de minimisation alternée, Ann. 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