{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T17:11:35Z","timestamp":1765041095095,"version":"3.40.5"},"reference-count":33,"publisher":"Optica Publishing Group","issue":"33","license":[{"start":{"date-parts":[[2020,11,17]],"date-time":"2020-11-17T00:00:00Z","timestamp":1605571200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v1#VOR"},{"start":{"date-parts":[[2020,11,17]],"date-time":"2020-11-17T00:00:00Z","timestamp":1605571200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["Appl. Opt."],"published-print":{"date-parts":[[2020,11,20]]},"abstract":"<jats:p>We introduce a technique for obtaining three-dimensional (3D) profiles of objects captured in two-dimensional (2D) flat images (FI). This technique performs a numerical approximation of the object\u2019s topography from their image gray tones, analyzing topological concepts such as algebraic bases construction, metric functions in terms of such bases, as well as modeling and development of an isomorphism to project masks (fringe patterns) in the FI, allowing us to use the optimal 3D profilometry techniques. Among these techniques, phase shifting (four steps) is applied in the pattern shifts of the projected masks, but with a 2D shift from left to right and from top to bottom, simultaneously. Moreover, the fringe patterns in the masks are binary and with superposed periods in <jats:inline-formula>\n      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n\t<mml:mi>x<\/mml:mi>\n      <\/mml:math>\n    <\/jats:inline-formula>, <jats:inline-formula>\n      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n\t<mml:mi>y<\/mml:mi>\n      <\/mml:math>\n    <\/jats:inline-formula> (bi-Ronchi). We show the results of the construction of the masks, as well as their projection into the FIs. We also show the 3D profilometry of the objects after the projection and phase-shifting application and a simple segmentation to observe a particular object. Subsequently, we perform the analysis of a single Mg crystal\u2019s micrography.<\/jats:p>","DOI":"10.1364\/ao.401316","type":"journal-article","created":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T19:00:07Z","timestamp":1602615607000},"page":"10464","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Optical-topological concepts in isomorphisms projecting bi-Ronchi masks to obtain 3D profiles from objects in 2D images"],"prefix":"10.1364","volume":"59","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8059-9399","authenticated-orcid":true,"given":"J. A. Arriaga","family":"Hern\u00e1ndez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1046-1500","authenticated-orcid":true,"given":"B. T. Cuevas","family":"Otahola","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1995-9743","authenticated-orcid":true,"given":"A. Jaramillo","family":"N\u00fa\u00f1ez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1715-3631","authenticated-orcid":true,"given":"J. Oliveros","family":"Oliveros","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9121-5917","authenticated-orcid":true,"given":"M. Mor\u00edn","family":"Castillo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"285","published-online":{"date-parts":[[2020,11,17]]},"reference":[{"key":"ao-59-33-10464-R1","doi-asserted-by":"crossref","DOI":"10.1002\/9780470135976","volume-title":"Optical Shop Testing","author":"Malacara","year":"2007"},{"key":"ao-59-33-10464-R2","doi-asserted-by":"publisher","first-page":"4922","DOI":"10.1364\/AO.32.004922","volume":"32","author":"O\u2019Donnell","year":"1993","journal-title":"Appl. Opt."},{"key":"ao-59-33-10464-R3","doi-asserted-by":"publisher","first-page":"2930","DOI":"10.1364\/AO.17.002930","volume":"17","author":"Indebetouw","year":"1978","journal-title":"Appl. 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