The Tree of shapes of an image

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• dc.contributor.author Ballester, Coloma
• dc.contributor.author Caselles, Vicente
• dc.contributor.author Monasse, Pascal
• dc.date.accessioned 2018-12-19T16:35:53Z
• dc.date.available 2018-12-19T16:35:53Z
• dc.date.issued 2003
• dc.description.abstract In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem in mathematical morphology: to represent an image in such a way that maxima and minima can be computationally dealt with simultaneously. We prove the finiteness of the tree when the image is the result of applying any extrema killer (a classical denoising filter in image processing). The shape tree also yields an easy mathematical definition of adaptive image quantization.en
• dc.description.sponsorship We acknowledge partial support by the TMR European project \Viscosity solutions and their applications", reference FMRX-CT98-0234 and the CNRS through a PICS project. The first two authors also acknowledge partial support by the PNPGC project, reference BFM2000-0962-C02-01.
• dc.format.mimetype application/pdf
• dc.identifier.citation Ballester C, Caselles V, Monasse P. The Tree of shapes of an image. ESAIM: Control, Optimisation and Calculus of Variations. 2003 Sep 15;9:1-18. DOI: 10.1051/cocv:2002069
• dc.identifier.doi http://dx.doi.org/10.1051/cocv:2002069
• dc.identifier.issn 1292-8119
• dc.identifier.uri http://hdl.handle.net/10230/36153
• dc.language.iso eng
• dc.publisher EDP Sciences
• dc.relation.ispartof ESAIM: Control, Optimisation and Calculus of Variations. 2003 Sep 15;9:1-18
• dc.rights © 2003 EDP Sciences. Tous droits de reproduction par tous procédés réservés pour tous pays. La publication originale est disponible sur le site https://doi.org/10.1051/cocv:2002069
• dc.rights.accessRights info:eu-repo/semantics/openAccess
• dc.subject.keyword Image representation
• dc.subject.keyword Mathematical morphology
• dc.subject.keyword Tree structure
• dc.subject.keyword Level sets
• dc.title The Tree of shapes of an image
• dc.type info:eu-repo/semantics/article
• dc.type.version info:eu-repo/semantics/publishedVersion