{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T22:27:41Z","timestamp":1769725661088,"version":"3.49.0"},"reference-count":16,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T00:00:00Z","timestamp":1739923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>We propose a method for generating a constrained Delaunay triangulation CDT applied to labeled 2D images with high morphological complexity. In the previous paper, Part 1, we established an unbiased planar straight-line graph (PLSG) on image objects of any morphological complexity, using mathematical morphology tools. In the case of labeled images of this paper, the PSLG is defined by the set of local PSLG pieces for each adjacent label. The global PSLG of each label must respect the injective nature of Jordan\u2019s curve.<\/jats:p>","DOI":"10.3390\/computation13020057","type":"journal-article","created":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T05:34:26Z","timestamp":1739943266000},"page":"57","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Unbiased Finite Element Mesh Delaunay Constrained Triangulation Applied to 2D Images with High Morphological Complexity Using Mathematical Morphology Tools Part 2: Labeled Images"],"prefix":"10.3390","volume":"13","author":[{"given":"F.","family":"N\u2019Guyen","sequence":"first","affiliation":[{"name":"Unit\u00e9 de M\u00e9canique de Lille UML J. Boussinesq, Universit\u00e9 de Lille, Villeneuve d\u2019Ascq, 59000 Lille, France"},{"name":"CMAT\u2014Centre des Mat\u00e9riaux, CNRS UMR 7633, Mines-Paris, PSL University, F-91003 Evry, France"}]},{"given":"T.","family":"Kanit","sequence":"additional","affiliation":[{"name":"Unit\u00e9 de M\u00e9canique de Lille UML J. Boussinesq, Universit\u00e9 de Lille, Villeneuve d\u2019Ascq, 59000 Lille, France"}]},{"given":"A.","family":"Imad","sequence":"additional","affiliation":[{"name":"Unit\u00e9 de M\u00e9canique de Lille UML J. Boussinesq, Universit\u00e9 de Lille, Villeneuve d\u2019Ascq, 59000 Lille, France"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.camwa.2022.06.025","article-title":"Anisotropic mesh adaptation for region-based segmentation accounting for image spatial information","volume":"121","author":"Giacomini","year":"2022","journal-title":"Comput. Math. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1145\/37402.37422","article-title":"Marching cubes: A high resolution 3D surface construction algorithm","volume":"21","author":"Lorensen","year":"1987","journal-title":"Comput. Graph."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"103303","DOI":"10.1016\/j.cad.2022.103303","article-title":"Computing the Intersection of Two Rational Surfaces Using Matrix Representations","volume":"150","author":"Jia","year":"2022","journal-title":"Comput.-Aided Des."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1682","DOI":"10.1016\/j.camwa.2014.03.001","article-title":"Adaptively weighted numerical integration over arbitrary domains","volume":"67","author":"Thiagarajan","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_5","unstructured":"Lin, M.C., and Manocha, D. (1996, January 27\u201328). Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. Proceedings of the Applied Computational Geometry Towards Geometric Engineering, Philadelphia, PA, USA."},{"key":"ref_6","first-page":"49:1","article-title":"Restricted Constrained Delaunay Triangulations","volume":"Volume 189","author":"Buchin","year":"2021","journal-title":"Proceedings of the 37th International Symposium on Computational Geometry (SoCG 2021)"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"055014","DOI":"10.1088\/0965-0393\/24\/5\/055014","article-title":"Automatic processing of an orientation map into a finite element mesh that conforms to grain boundaries","volume":"24","author":"Dancette","year":"2016","journal-title":"Model. Simul. Mater. Sci. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2094","DOI":"10.21105\/joss.02094","article-title":"MTEX2Gmsh: A tool for generating 2D meshes from EBSD data","volume":"5","author":"Depriester","year":"2020","journal-title":"J. Open Source Softw."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.actamat.2016.10.030","article-title":"The five-parameter grain boundary curvature distribution in an austenitic and ferritic steel","volume":"123","author":"Zhong","year":"2017","journal-title":"Acta Mater."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1117\/12.159642","article-title":"Morphological gradients","volume":"2","author":"Rivest","year":"1993","journal-title":"J. Electron. Imaging"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Soille, P. (1999). Morphological Image Analysis: Principles and Applications, Springer.","DOI":"10.1007\/978-3-662-03939-7"},{"key":"ref_12","unstructured":"Schmitt, M., and Mattioli, J. (2013). Morphologie Math\u00e9matique, Masson. Logique Math\u00e9matiques Informatique."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/1047-3203(91)90025-B","article-title":"Morphological hit-or-miss transformation for shape recognition","volume":"2","author":"Zhao","year":"1991","journal-title":"J. Vis. Commun. Image Represent."},{"key":"ref_14","unstructured":"Lantuejoul, C. (2002). Geostatistical Simulation: Models and Algorithms, Springer. Collezione legale Pirola."},{"key":"ref_15","unstructured":"Jordan, C. (1887). Cours d\u2019Analyse de l\u2019\u00c9cole Polytechnique, Cambridge University Press."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.cag.2023.10.003","article-title":"Efficient boundary surface reconstruction from multi-label volumetric data with mathematical morphology","volume":"117","author":"NGuyen","year":"2023","journal-title":"Comput. Graph."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/2\/57\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:37:47Z","timestamp":1760027867000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/2\/57"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,19]]},"references-count":16,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["computation13020057"],"URL":"https:\/\/doi.org\/10.3390\/computation13020057","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,19]]}}}