  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 in flat domains
while preserving distances measured on the manifold. Recently, attention has
been given to embedding shapes into the eigenspace of the Lapalce-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 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.
