A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves
IEEE Transactions on Pattern Analysis and Machine Intelligence
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International Journal of Computer Vision
Shape Matching and Object Recognition Using Shape Contexts
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Shock-Based Indexing into Large Shape Databases
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part III
An affine invariant deformable shape representation for general curves
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
2D-Shape Analysis Using Conformal Mapping
International Journal of Computer Vision
A Shape Representation for Planar Curves by Shape Signature Harmonic Embedding
CVPR '06 Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Volume 2
3D Non-rigid Surface Matching and Registration Based on Holomorphic Differentials
ECCV '08 Proceedings of the 10th European Conference on Computer Vision: Part III
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Generalized Koebe's method for conformal mapping multiply connected domains
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SMI 2012: Full Canonical conformal mapping for high genus surfaces with boundaries
Computers and Graphics
Surface- and volume-based techniques for shape modeling and analysis
SIGGRAPH Asia 2013 Courses
Teichmüller Shape Descriptor and Its Application to Alzheimer's Disease Study
International Journal of Computer Vision
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The study of 2D shapes is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial and yet difficult. It usually involves an efficient representation of 2D shape space with natural metric, so that its mathematical structure can be used for further analysis. Although significant progress has been made for the study of 2D simply-connected shapes, very few works have been done on the study of 2D objects with arbitrary topologies. In this work, we propose a representation of general 2D domains with arbitrary topologies using conformal geometry. 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. Aset of diffeomorphisms from the unit circle S1 to itself can be obtained, which together with the conformal modules are used to define the shape signature. We prove mathematically that our proposed signature uniquely represents shapes with arbitrary topologies. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us tomove back and forth between shapes and signatures. Experiments show the efficacy of our proposed algorithm as a stable shape representation scheme.