One Point Isometric Matching with the Heat Kernel

Abstract
A common operation in many geometry processing algorithms consists of finding correspondences between pairs of shapes by finding structure-preserving maps between them. A particularly useful case of such maps is isometries, which preserve geodesic distances between points on each shape. Although several algorithms have been proposed to find approximately isometric maps between a pair of shapes, the structure of the space of isometries is not well understood. In this paper, we show that under mild genericity conditions, a single correspondence can be used to recover an isometry defined on entire shapes, and thus the space of all isometries can be parameterized by one correspondence between a pair of points. Perhaps surprisingly, this result is general, and does not depend on the dimensionality or the genus, and is valid for compact manifolds in any dimension. Moreover, we show that both the initial correspondence and the isometry can be recovered efficiently in practice. This allows us to devise an algorithm to find intrinsic symmetries of shapes, match shapes undergoing isometric deformations, as well as match partial and incomplete models efficiently. Journal compilation © 2010 The Eurographics Association and Blackwell Publishing Ltd.

Citation
Ovsjanikov M, Mérigot Q, Mémoli F, Guibas L (2010) One Point Isometric Matching with the Heat Kernel. Computer Graphics Forum 29: 1555–1564. Available: http://dx.doi.org/10.1111/j.1467-8659.2010.01764.x.

Acknowledgements
This work was supported by NSF grants0808515 and 0808515, NIH grant GM-072970, ONR grant N00014-09-1-0783, a grant form the King Abdullah University of Scienceand Technology, and a gift from Google, Inc. The authors wouldalso like to thank the anonymous reviewers for the helpful suggestions and comments.

Publisher
Wiley

Journal
Computer Graphics Forum

DOI
10.1111/j.1467-8659.2010.01764.x

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