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Shape Analysis of Planar Multiply-connected Objects using Conformal Welding
PrePrint
ISSN: 0162-8828
Lok Ming Lui, The Chinese University of Hong Kong, Hong Kong
Wei Zeng, Florida International University, Miami
Shing-Tung Yau, Harvard University, Cambridge
Xianfeng Gu, Stony Brook University at Stony Brook, Stony Brook
This paper proposes a shape representation scheme for planar objects with arbitrary topologies. Although the study of 2D simply-connected shapes has been subject to a corpus of literatures, the analysis of multiply-connected shapes is comparatively less studied. In this work, we propose a representation for general 2D multiply-connected domains with arbitrary topologies using conformal welding. A natural metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains, using holomorphic 1-forms. A set of diffeomorphisms of the unit circle can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. The proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the shape signatures. Experiments results on real images demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme.
Index Terms:
Computational models of vision,Numerical algorithms,Applications,Vision and Scene Understanding,Representations,data structures,and transforms,Image Processing and Computer Vision,Reconstruction,Shape
Citation:
Lok Ming Lui, Wei Zeng, Shing-Tung Yau, Xianfeng Gu, "Shape Analysis of Planar Multiply-connected Objects using Conformal Welding," IEEE Transactions on Pattern Analysis and Machine Intelligence, 21 Nov. 2013. IEEE computer Society Digital Library. IEEE Computer Society, <http://doi.ieeecomputersociety.org/10.1109/TPAMI.2013.215>
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