{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:34:54Z","timestamp":1772296494216,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T00:00:00Z","timestamp":1517356800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Program of research support in King Khalid University","award":["G.R.P-92-38"],"award-info":[{"award-number":["G.R.P-92-38"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>Total variation (TV) is widely used in many image processing problems including the regularization of optical flow estimation. In order to deal with non differentiability of the TV regularization term, smooth approximations have been considered in the literature. In this paper, we investigate the use of three known smooth TV approximations, namely: the Charbonnier, Huber and Green functions. We establish the maximum theoretical error of these approximations and discuss their performance evaluation when applied to the optical flow problem.<\/jats:p>","DOI":"10.3390\/jimaging4020031","type":"journal-article","created":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T12:41:24Z","timestamp":1517402484000},"page":"31","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["An Investigation of Smooth TV-Like Regularization in the Context of the Optical Flow Problem"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1434-2541","authenticated-orcid":false,"given":"El","family":"Kalmoun","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61421, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,31]]},"reference":[{"key":"ref_1","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_2","unstructured":"Shulman, D., and Herve, J.-Y. 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