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Axioms 2014, 3(3), 300-319; doi:10.3390/axioms3030300

Matching the LBO Eigenspace of Non-Rigid Shapes via High Order Statistics

*  and
Technion, Israel Institution of Technologies, Haifa 32000, Israel
* Author to whom correspondence should be addressed.
Received: 17 October 2013 / Revised: 23 February 2014 / Accepted: 23 June 2014 / Published: 15 July 2014
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A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes into flat domains, while preserving the distances measured on the manifold. Recently, attention has been given to embedding shapes into the eigenspace of the Laplace–Beltrami operator. The Laplace–Beltrami eigenspace preserves the diffusion distance and is invariant under isometric transformations. However, Laplace–Beltrami eigenfunctions computed independently for different shapes are often incompatible with each other. Applications involving multiple shapes, such as pointwise correspondence, would greatly benefit if their respective eigenfunctions were somehow matched. Here, we introduce a statistical approach for matching eigenfunctions. We consider the values of the eigenfunctions over the manifold as the sampling of random variables and try to match their multivariate distributions. Comparing distributions is done indirectly, using high order statistics. We show that the permutation and sign ambiguities of low order eigenfunctions can be inferred by minimizing the difference of their third order moments. The sign ambiguities of antisymmetric eigenfunctions can be resolved by exploiting isometric invariant relations between the gradients of the eigenfunctions and the surface normal. We present experiments demonstrating the success of the proposed method applied to feature point correspondence.
Keywords: embedding; Laplace–Beltrami operator; high order statistics embedding; Laplace–Beltrami operator; high order statistics
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Shtern, A.; Kimmel, R. Matching the LBO Eigenspace of Non-Rigid Shapes via High Order Statistics. Axioms 2014, 3, 300-319.

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