# 2D-Shape Analysis Using Conformal Mapping

 dc.contributor.author Sharon, E. dc.contributor.author Mumford, David Bryant dc.date.accessioned 2010-03-18T15:21:45Z dc.date.issued 2006 dc.identifier.citation Sharon, E., and David Bryant Mumford. 2006. 2D-shape analysis using conformal mapping. International Journal of Computer Vision 70(1): 55-75. en_US dc.identifier.issn 0920-5691 en_US dc.identifier.issn 1573-1405 en_US dc.identifier.uri http://nrs.harvard.edu/urn-3:HUL.InstRepos:3720034 dc.description.abstract The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this space every simple closed curve in the plane (a “shape”) is represented by a ‘fingerprint’ which is a diffeomorphism of the unit circle to itself (a differentiable and invertible, periodic function). More precisely, every shape defines to a unique equivalence class of such diffeomorphisms up to right multiplication by a Möbius map. The fingerprint does not change if the shape is varied by translations and scaling and any such equivalence class comes from some shape. This coset space, equipped with the infinitesimal Weil-Petersson (WP) Riemannian norm is a metric space. In this space, the shortest path between each two shapes is unique, and is given by a geodesic connecting them. Their distance from each other is given by integrating the WP-norm along that geodesic. In this paper we concentrate on solving the “welding” problem of “sewing” together conformally the interior and exterior of the unit circle, glued on the unit circle by a given diffeomorphism, to obtain the unique 2D shape associated with this diffeomorphism. This will allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this “space of shapes”. We then present an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes. The group of diffeomorphisms of S^1 acts as a group of isometries on the space of shapes and we show how this can be used to define shape transformations, like for instance ‘adding a protruding limb’ to any shape. en_US dc.description.sponsorship Mathematics en_US dc.language.iso en_US en_US dc.publisher Springer Verlag en_US dc.relation.isversionof doi:10.1007/s11263-006-6121-z en_US dc.relation.hasversion http://www.dam.brown.edu/people/mumford/Papers/DigitizedVisionPapers--forNonCommercialUse/x06b--Conformal-Sharon.pdf en_US dash.license LAA dc.subject group of diffeomorphisms en_US dc.subject group of shape transformations en_US dc.subject shape representation en_US dc.subject metrics between shapes en_US dc.subject Riemann mapping theorem en_US dc.subject Weil-Petersson metric en_US dc.subject fingerprints of shapes en_US dc.title 2D-Shape Analysis Using Conformal Mapping en_US dc.type Journal Article en_US dc.description.version Version of Record en_US dc.relation.journal International Journal of Computer Vision en_US dash.depositing.author Mumford, David Bryant dc.date.available 2010-03-18T15:21:45Z

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