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One Point Isometric Matching with the Heat Kernel
Author(s) -
Ovsjanikov Maks,
Mérigot Quentin,
Mémoli Facundo,
Guibas Leonidas
Publication year - 2010
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/j.1467-8659.2010.01764.x
Subject(s) - geodesic , isometry (riemannian geometry) , homogeneous space , mathematics , dimension (graph theory) , heat kernel signature , kernel (algebra) , matching (statistics) , isometric exercise , point (geometry) , parameterized complexity , pure mathematics , computer science , algorithm , geometry , artificial intelligence , active shape model , medicine , statistics , segmentation , physical therapy
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.