{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:29:40Z","timestamp":1787326180808,"version":"build-2736575974"},"reference-count":9,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>An important question about affine skeletons is the existence of differential equations that are related to the \u201caffine distance\u201d and the \u201carea distance\u201d (hence to affine skeletons) in the same way the Eikonal equation is related to the \u201cEuclidean distance\u201d (and the medial axis). We show a surprisingly simple nonlinear second order PDE of Monge\u2013Amp\u00e8re type that relates to the affine skeletons (and extends the Eikonal equation for the medial axis). We also discuss some consequences and ideas that this new PDE formulation suggests.<\/jats:p>","DOI":"10.1137\/080725015","type":"journal-article","created":{"date-parts":[[2009,9,28]],"date-time":"2009-09-28T15:18:51Z","timestamp":1254151131000},"page":"987-1001","source":"Crossref","is-referenced-by-count":1,"title":["Affine Skeletons and Monge\u2013Amp\u00e8re Equations"],"prefix":"10.1137","volume":"2","author":[{"given":"Moacyr Alvim","family":"Silva","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ralph","family":"Teixeira","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luiz","family":"Velho","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,9,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0031-3203(00)00045-5"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1109\/83.661191"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1023\/B:VISI.0000036835.28674.d0"},{"key":"R4","unstructured":"P. J. Olver, G. Sapiro, and A. Tannenbaum,\n                      Affine Invariant Edge Maps and Active Contours\n                      , IMA Preprint Series 1360, Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, 1995."},{"key":"R5","doi-asserted-by":"crossref","unstructured":"P. J. Giblin and G. Sapiro,\n                      Affine invariant symmetry sets\n                      , in Foundations of Computational Mathematics (Rio de Janeiro, 1997), Springer, Berlin, 1997, pp. 152\u2013168.","DOI":"10.1007\/978-3-642-60539-0_11"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005099011913"},{"key":"R7","unstructured":"J. Serra,\n                      Image Analysis and Mathematical Morphology\n                      , Vol. 1, Academic Press, London, 1982."},{"key":"R8","unstructured":"M. A. Silva,\n                      Esqueletos Afins e Equa\u00e7\u00f5es de Propaga\u00e7\u00e3\n                      , Ph.D. thesis, Instituto de Matem\u00e1tica Pura e Aplicada - IMPA, Rio de Janeiro, Brazil, 2005."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-007-0038-1"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080725015","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:27:57Z","timestamp":1787322477000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080725015"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080725015"],"URL":"https:\/\/doi.org\/10.1137\/080725015","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}