{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T04:46:35Z","timestamp":1767156395188,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,23]],"date-time":"2021-12-23T00:00:00Z","timestamp":1640217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>The effectiveness of variational methods for restoring images corrupted by Poisson noise strongly depends on the suitable selection of the regularization parameter balancing the effect of the regulation term(s) and the generalized Kullback\u2013Liebler divergence data term. One of the approaches still commonly used today for choosing the parameter is the discrepancy principle proposed by Zanella et al. in a seminal work. It relies on imposing a value of the data term approximately equal to its expected value and works well for mid- and high-count Poisson noise corruptions. However, the series truncation approximation used in the theoretical derivation of the expected value leads to poor performance for low-count Poisson noise. In this paper, we highlight the theoretical limits of the approach and then propose a nearly exact version of it based on Monte Carlo simulation and weighted least-square fitting. Several numerical experiments are presented, proving beyond doubt that in the low-count Poisson regime, the proposed modified, nearly exact discrepancy principle performs far better than the original, approximated one by Zanella et al., whereas it works similarly or slightly better in the mid- and high-count regimes.<\/jats:p>","DOI":"10.3390\/jimaging8010001","type":"journal-article","created":{"date-parts":[[2021,12,23]],"date-time":"2021-12-23T10:16:18Z","timestamp":1640254578000},"page":"1","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Nearly Exact Discrepancy Principle for Low-Count Poisson Image Restoration"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8244-3355","authenticated-orcid":false,"given":"Francesca","family":"Bevilacqua","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alessandro","family":"Lanza","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3074-1550","authenticated-orcid":false,"given":"Monica","family":"Pragliola","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fiorella","family":"Sgallari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bertero, M., Boccacci, P., and Ruggiero, V. (2018). Inverse Imaging with Poisson Data, IOP Publishing.","DOI":"10.1088\/2053-2563\/aae109"},{"key":"ref_2","unstructured":"Calvetti, D., and Somersalo, E. (2007). Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing (Surveys and Tutorials in the Applied Mathematical Sciences), Springer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1109\/TPAMI.1984.4767596","article-title":"Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images","volume":"6","author":"Geman","year":"1984","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_4","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":"Physica D"},{"key":"ref_5","first-page":"0450100","article-title":"Efficient gradient projection methods for edge-preserving removal of Poisson noise","volume":"25","author":"Zanella","year":"2013","journal-title":"Inverse Probl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1834","DOI":"10.1109\/TIP.2011.2175934","article-title":"Sparse Poisson Noisy Image Deblurring","volume":"21","author":"Carlavan","year":"2012","journal-title":"IEEE Trans. Image Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"105004","DOI":"10.1088\/0266-5611\/26\/10\/105004","article-title":"A discrepancy principle for Poisson data","volume":"26","author":"Bertero","year":"2010","journal-title":"Inverse Probl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"055004","DOI":"10.1088\/0266-5611\/30\/5\/055004","article-title":"Accelerated gradient methods for the X-ray imaging of solar flares","volume":"30","author":"Bonettini","year":"2014","journal-title":"Inverse Probl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2187","DOI":"10.1137\/15M1049051","article-title":"A study on regularization for discrete inverse problems with model-dependent noise","volume":"55","author":"Benvenuto","year":"2017","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"0150104","DOI":"10.1088\/1361-6420\/abcd26","article-title":"A mathematical model for image saturation with an application to the restoration of solar images via adaptive sparse deconvolution","volume":"37","author":"Guastavino","year":"2021","journal-title":"Inverse Probl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/s10851-014-0553-9","article-title":"Numerical methods for parameter estimation in Poisson data inversion","volume":"52","author":"Zanni","year":"2015","journal-title":"J. Math. Imaging Vis."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"607","DOI":"10.3934\/ipi.2018026","article-title":"Morozov principle for Kullback-Leibler residual term and Poisson noise","volume":"12","author":"Sixou","year":"2018","journal-title":"Inverse Probl. Imaging"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Johnson, N.L., Kemps, A.W., and Kotz, S. (2005). Univariate Discrete Distributions, Wiley.","DOI":"10.1002\/0471715816"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"035007","DOI":"10.1088\/0266-5611\/29\/3\/035007","article-title":"Minimization and parameter estimation for seminorm regularization models with I-divergence constraints","volume":"29","author":"Teuber","year":"2013","journal-title":"Inverse Probl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/2200000016","article-title":"Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers","volume":"3","author":"Boyd","year":"2011","journal-title":"Found. Trends Mach. Learn."},{"key":"ref_16","first-page":"41","article-title":"Sur l\u2019approximation, par \u00e9l\u00e9ments finis d\u2019ordre un, et la r\u00e9solution, par p\u00e9nalisation-dualit\u00e9 d\u2019une classe de probl\u00e8mes de Dirichlet non lin\u00e9aires","volume":"9","author":"Glowinski","year":"1975","journal-title":"Math. Model. Numer. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: From error visibility to structural similarity","volume":"13","author":"Wang","year":"2004","journal-title":"IEEE Trans. Image Process."}],"container-title":["Journal of Imaging"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2313-433X\/8\/1\/1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:51:40Z","timestamp":1760169100000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2313-433X\/8\/1\/1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,23]]},"references-count":17,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["jimaging8010001"],"URL":"https:\/\/doi.org\/10.3390\/jimaging8010001","relation":{},"ISSN":["2313-433X"],"issn-type":[{"type":"electronic","value":"2313-433X"}],"subject":[],"published":{"date-parts":[[2021,12,23]]}}}