{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T13:34:14Z","timestamp":1772717654277,"version":"3.50.1"},"reference-count":50,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Model. Simul. Sci. Comput."],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:p> A fractional operator for optical flow estimation is presented in this research paper. The formulation of this operator is carried out by the Caputo\u2013Fabrizio derivative and the Horn\u2013Schunck method. In this way, our operator generalizes the traditional Horn\u2013Schunck technique. The Caputo\u2013Fabrizio fractional-order derivative is introduced for its high capability to enhance and preserve both texture detail and image edges, making it more suitable for optical flow estimation in images with weak edges caused by low contrast or noise. A fractional-order multiscale strategy is also considered to deal with large displacements. This fractional-order approach produces a dense flow and greatly increases robustness at outliers. Experimental comparisons with the conventional multiscale Horn\u2013Schunck algorithm and some integer- and fractional-order methods existing on the MPI Sintel and Middlebury Stereo datasets demonstrate that our operator provides significantly lower average endpoint error and average angular error. The firefly algorithm is used to get the optimal order for all the stereoscopic image sequences. <\/jats:p>","DOI":"10.1142\/s1793962325500539","type":"journal-article","created":{"date-parts":[[2025,7,18]],"date-time":"2025-07-18T15:56:08Z","timestamp":1752854168000},"source":"Crossref","is-referenced-by-count":1,"title":["On the estimation of optical flow: A novel robust estimation based in fractional derivatives with kernel exponential"],"prefix":"10.1142","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3632-3373","authenticated-orcid":false,"given":"J. E.","family":"Lav\u00edn-Delgado","sequence":"first","affiliation":[{"name":"Direcci\u00f3n de Ingenier\u00eda en Tecnolog\u00edas de la Informaci\u00f3n e Innovaci\u00f3n Digital, Universidad Polit\u00e9cnica del Estado de Guerrero (UPEGro), Carretera Federal Iguala\u2013Taxco Km 105, Puente Campuzano, 40321 Taxco de Alarc\u00f3n, Guerrero, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9403-3767","authenticated-orcid":false,"given":"J. F.","family":"G\u00f3mez-Aguilar","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n en Ingenier\u00eda y Ciencias Aplicadas (CIICAp-IICBA), Universidad Aut\u00f3noma del Estado de Morelos (UAEM), 62210 Cuernavaca, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-8774-6012","authenticated-orcid":false,"given":"E.","family":"P\u00e9rez-Careta","sequence":"additional","affiliation":[{"name":"Departamento de Electr\u00f3nica, Universidad de Guanajuato, Carretera Salamanca \u2013 Valle de Santiago km 3  1.8, 36885 Salamanca, Guanajuato, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5214-4984","authenticated-orcid":false,"given":"V. H.","family":"Olivares-Peregrino","sequence":"additional","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico\/CENIDET, Interior Internado Palmira S\/N, Col. Palmira, 62490 Cuernavaca, Morelos, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8006-6397","authenticated-orcid":false,"given":"J.","family":"Torres-Jim\u00e9nez","sequence":"additional","affiliation":[{"name":"Divisi\u00f3n de Ingenier\u00eda El\u00e9ctrica, Tecnol\u00f3gico Nacional de M\u00e9xico\/Instituto, Tecnol\u00f3gico Superior de Huauchinango, Av. Tecnol\u00f3gico No. 80, Col. 5 de octubre, 73173 Huauchinango, Puebla, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2025,7,31]]},"reference":[{"key":"S1793962325500539BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/s10462-022-10159-8"},{"key":"S1793962325500539BIB002","doi-asserted-by":"publisher","DOI":"10.1109\/TITS.2024.3382495"},{"key":"S1793962325500539BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/0004-3702(81)90024-2"},{"key":"S1793962325500539BIB004","first-page":"674","volume-title":"IJCAI\u201981: Proc. 7th Int. Joint Conf. Artificial Intelligence","author":"Lucas B. D.","year":"1981"},{"key":"S1793962325500539BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/S0734-189X(83)80030-9"},{"key":"S1793962325500539BIB006","doi-asserted-by":"publisher","DOI":"10.1006\/cviu.1996.0006"},{"key":"S1793962325500539BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-24673-2_3"},{"key":"S1793962325500539BIB008","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-03641-5_16"},{"issue":"9","key":"S1793962325500539BIB009","doi-asserted-by":"crossref","first-page":"1744","DOI":"10.1109\/TPAMI.2011.236","volume":"34","author":"Xu L.","year":"2011","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"S1793962325500539BIB010","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2013.316"},{"key":"S1793962325500539BIB011","doi-asserted-by":"publisher","DOI":"10.1109\/LRA.2016.2592513"},{"key":"S1793962325500539BIB012","series-title":"Dover Books on Mathematics","volume-title":"The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order","volume":"111","author":"Oldham K.","year":"2006","edition":"1"},{"key":"S1793962325500539BIB013","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2012.0148"},{"key":"S1793962325500539BIB014","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijleo.2016.05.118"},{"key":"S1793962325500539BIB015","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/6278719"},{"key":"S1793962325500539BIB016","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijleo.2022.169136"},{"key":"S1793962325500539BIB017","doi-asserted-by":"crossref","first-page":"1","DOI":"10.70891\/JSE.2024.100001","volume":"1","author":"Singh B.","year":"2024","journal-title":"Arab. J. Sci. Eng."},{"issue":"11","key":"S1793962325500539BIB018","first-page":"3489","volume":"27","author":"Liang X.","year":"2024","journal-title":"J. Appl. Sci. Eng."},{"key":"S1793962325500539BIB019","doi-asserted-by":"publisher","DOI":"10.1007\/s12046-023-02318-6"},{"key":"S1793962325500539BIB020","doi-asserted-by":"publisher","DOI":"10.3390\/fractalfract8040179"},{"key":"S1793962325500539BIB021","series-title":"Mathematics in Science and Engineering","volume-title":"Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications","volume":"198","author":"Podlubny I.","year":"1998"},{"issue":"2","key":"S1793962325500539BIB022","first-page":"1","volume":"1","author":"Caputo M.","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"S1793962325500539BIB023","doi-asserted-by":"publisher","DOI":"10.5201\/ipol.2013.20"},{"key":"S1793962325500539BIB024","doi-asserted-by":"publisher","DOI":"10.1051\/mmnp\/2018010"},{"key":"S1793962325500539BIB025","doi-asserted-by":"publisher","DOI":"10.1140\/epjp\/i2017-11717-0"},{"key":"S1793962325500539BIB026","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-33783-3_44"},{"key":"S1793962325500539BIB027","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-57956-7_5"},{"key":"S1793962325500539BIB028","doi-asserted-by":"publisher","DOI":"10.1007\/BF01420984"},{"issue":"1","key":"S1793962325500539BIB029","first-page":"9444803","volume":"2021","author":"Abu-Shady M.","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"S1793962325500539BIB030","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-020-03191-x"},{"key":"S1793962325500539BIB031","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2018.07.023"},{"key":"S1793962325500539BIB032","first-page":"23","volume":"1","author":"Pinto A. M.","year":"2013","journal-title":"J. Comput. Eng. Inf."},{"key":"S1793962325500539BIB033","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.1998.710828"},{"key":"S1793962325500539BIB034","doi-asserted-by":"publisher","DOI":"10.1007\/BF00202895"},{"key":"S1793962325500539BIB035","doi-asserted-by":"publisher","DOI":"10.1016\/j.cja.2015.07.005"},{"key":"S1793962325500539BIB036","first-page":"7","volume-title":"Int. Workshop Computer Vision and Image Analysis, IWCVIA03","author":"Papenberg N.","year":"2003"},{"key":"S1793962325500539BIB037","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.1993.378214"},{"key":"S1793962325500539BIB038","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-021-00955-5"},{"key":"S1793962325500539BIB039","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-010-0390-2"},{"key":"S1793962325500539BIB040","doi-asserted-by":"publisher","DOI":"10.1109\/AVSS.2018.8639113"},{"key":"S1793962325500539BIB041","doi-asserted-by":"publisher","DOI":"10.1006\/cviu.2001.0930"},{"key":"S1793962325500539BIB042","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2007.383248"},{"key":"S1793962325500539BIB043","doi-asserted-by":"publisher","DOI":"10.5201\/ipol.2013.26"},{"key":"S1793962325500539BIB044","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-013-0644-x"},{"key":"S1793962325500539BIB045","doi-asserted-by":"publisher","DOI":"10.1016\/j.cviu.2015.12.003"},{"key":"S1793962325500539BIB046","doi-asserted-by":"publisher","DOI":"10.5201\/ipol.2013.21"},{"key":"S1793962325500539BIB047","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2016.2526903"},{"key":"S1793962325500539BIB048","doi-asserted-by":"publisher","DOI":"10.1016\/j.imavis.2016.06.004"},{"key":"S1793962325500539BIB049","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2010.5539939"},{"key":"S1793962325500539BIB050","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2010.143"}],"container-title":["International Journal of Modeling, Simulation, and Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793962325500539","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T03:18:17Z","timestamp":1754968697000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S1793962325500539"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,31]]},"references-count":50,"journal-issue":{"issue":"04","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["10.1142\/S1793962325500539"],"URL":"https:\/\/doi.org\/10.1142\/s1793962325500539","relation":{},"ISSN":["1793-9623","1793-9615"],"issn-type":[{"value":"1793-9623","type":"print"},{"value":"1793-9615","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,31]]},"article-number":"2550053"}}