NDLI: 2D-shape analysis using conformal mapping

Content Provider	IEEE Xplore Digital Library
Author	Sharon, E. Mumford, D.
Copyright Year	2004
Description	Author affiliation: Div. of Appl. Math., Brown Univ., Providence, RI, USA (Sharon, E.; Mumford, D.)
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 Teichmuller 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 Mobius 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, it appears very likely to be true that 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. These allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this "space of shapes".
Sponsorship	IEEE Comput. Soc
File Size	768734
File Format	PDF
ISBN	0769521584
ISSN	10636919
DOI	10.1109/CVPR.2004.1315185
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2004-06-27
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Conformal mapping Extraterrestrial measurements Shape measurement Mathematics Clustering algorithms Fingerprint recognition Joining processes Humans Mathematical analysis Spatial databases
Content Type	Text
Resource Type	Article
Subject	Computer Vision and Pattern Recognition Software

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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