{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:40:07Z","timestamp":1760150407728,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,15]],"date-time":"2023-11-15T00:00:00Z","timestamp":1700006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Istituto Nazionale di Alta Matematica, GruppoNazionale per il Calcolo Scientifico (INdAM-GNCS)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>This work tackles the problem of image restoration, a crucial task in many fields of applied sciences, focusing on removing degradation caused by blur and noise during the acquisition process. Drawing inspiration from the multi-penalty approach based on the Uniform Penalty principle, discussed in previous work, here we develop a new image restoration model and an iterative algorithm for its effective solution. The model incorporates pixel-wise regularization terms and establishes a rule for parameter selection, aiming to restore images through the solution of a sequence of constrained optimization problems. To achieve this, we present a modified version of the Newton Projection method, adapted to multi-penalty scenarios, and prove its convergence. Numerical experiments demonstrate the efficacy of the method in eliminating noise and blur while preserving the image edges.<\/jats:p>","DOI":"10.3390\/jimaging9110249","type":"journal-article","created":{"date-parts":[[2023,11,15]],"date-time":"2023-11-15T10:57:46Z","timestamp":1700045866000},"page":"249","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Automatic Pixel-Wise Multi-Penalty Approach to Image Restoration"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9419-1965","authenticated-orcid":false,"given":"Villiam","family":"Bortolotti","sequence":"first","affiliation":[{"name":"Department of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna, 40131 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1250-8218","authenticated-orcid":false,"given":"Germana","family":"Landi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3961-4037","authenticated-orcid":false,"given":"Fabiana","family":"Zama","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hansen, P.C., Nagy, J.G., and O\u2019leary, D.P. (2006). Deblurring Images: Matrices, Spectra, and Filtering, SIAM.","DOI":"10.1137\/1.9780898718874"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/0167-2789(92)90242-F","article-title":"Nonlinear total variation based noise removal algorithms","volume":"60","author":"Rudin","year":"1992","journal-title":"Phys. D"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1137\/090769521","article-title":"Total generalized variation","volume":"3","author":"Bredies","year":"2010","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00211-010-0318-3","article-title":"Multi-parameter regularization and its numerical realization","volume":"118","author":"Lu","year":"2011","journal-title":"Numer. 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