{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:56Z","timestamp":1787337296985,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>Segmentation with depth is the challenging problem of obtaining three dimensional information from a single two dimensional image. Unlike the standard segmentation problem, the goal of segmentation with depth is to determine not only the boundaries of objects that appear in the image, but also their relative distances to the observer by making use of occlusion relations. Nitzberg, Mumford, and Shiota [The 2.1D sketch, in Proceedings of the Third International Conference on Computer Vision, Osaka, Japan, 1990; Filtering, Segmentation, and Depth, Lecture Notes in Comput. Sci. 662, Springer\u2010Verlag, Berlin, 1993] proposed a variational formulation of this problem; according to their model, called the 2.1D sketch model, the regions that the objects occupy and their relative depth are to be extracted from the two dimensional image by minimizing a curvature based functional. Numerically, this is a highly nontrivial problem as the functional involves curvatures of the unknown contours. In this paper, we develop a level set based procedure for minimizing the Nitzberg\u2013Mumford\u2013Shiota energy. Unlike the minimization technique of Nitzberg\u2013Mumford\u2013Shiota [Filtering, Segmentation, and Depth, Lecture Notes in Comput. Sci. 662, Springer\u2010Verlag, Berlin, 1993], our technique represents contours implicitly and thus allows for connections between T\u2010junctions to take place automatically.<\/jats:p>","DOI":"10.1137\/050622213","type":"journal-article","created":{"date-parts":[[2006,11,28]],"date-time":"2006-11-28T18:05:07Z","timestamp":1164737107000},"page":"1957-1973","source":"Crossref","is-referenced-by-count":37,"title":["Segmentation with Depth: A Level Set Approach"],"prefix":"10.1137","volume":"28","author":[{"given":"Wei","family":"Zhu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tony","family":"Chan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Selim","family":"Esedolu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,11,24]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160430805"},{"key":"R2","first-page":"91","volume":"4","author":"Bellettini G.","year":"1997","journal-title":"J. Convex Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0944-6532","issn-type":"print"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"G. Bellettini and R. March,\n                      Variational properties of a model for image segmentation with overlapping regions\n                      , in Variational Methods for Discontinuous Structures, G. Dal Maso and F. Tomarelli, eds., Prog. 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Vese,\n                      A level set algorithm for minimizing the Mumford\u2013Shah functional in image processing\n                      , in the IEEE Workshop on Variational and Level Set Methods, Vancouver, Canada, 2001, pp. 161\u2013168.","DOI":"10.1109\/VLSM.2001.938895"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021837026373"},{"key":"R12","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1017\/S0956792502004904","volume":"13","author":"Esedolu S.","year":"2002","journal-title":"European J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0956-7925","issn-type":"print"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Y. Giga and K. Ito,\n                      Loss of convexity of simple closed curves moved by surface diffusion\n                      , in Topics in Nonlinear Analysis, The Herbert Amann Anniversary volume, J. Escher and G. Simonett, eds., Prog. 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Solimini,\n                      Variational Methods in Image Segmentation\n                      , Birkh\u00e4user Boston, Boston, 1995.","DOI":"10.1007\/978-1-4684-0567-5"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"M. Nitzberg and D. Mumford,\n                      The 2.1D sketch\n                      , in Proceedings of the Third International Conference on Computer Vision, Osaka, Japan, 1990, pp. 138\u2013144.","DOI":"10.1109\/ICCV.1990.139511"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"M. Nitzberg, D. Mumford, and T. Shiota,\n                      Filtering, Segmentation, and Depth\n                      , Lecture Notes in Comput. Sci. 662, Springer\u2010Verlag, Berlin, 1993.","DOI":"10.1007\/3-540-56484-5"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90002-2"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6636"},{"key":"R29","doi-asserted-by":"crossref","unstructured":"S. Osher and R. Fedkiw,\n                      Level Set Methods and Dynamic Implicit Surfaces\n                      , Springer\u2010Verlag, New York, 2002.","DOI":"10.1007\/b98879"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1023\/A:1025324613450"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.0167"},{"key":"R32","unstructured":"W. Zhu and T. 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