Generalized heat kernel signatures

Abstract

In this work we propose a generalization of the Heat Kernel Signature (HKS). The HKS is a point signature derived from the heat kernel of the Laplace-Beltrami operator of a surface. In the theory of exterior calculus on a Riemannian manifold, the Laplace-Beltrami operator of a surface is a special case of the Hodge Laplacian which acts on r-forms, i. e. the Hodge Laplacian on 0-forms (functions) is the Laplace-Beltrami operator. We investigate the usefulness of the heat kernel of the Hodge Laplacian on 1-forms (which can be seen as the vector Laplacian) to derive new point signatures which are invariant under isometric mappings. A similar approach used to obtain the HKS yields a symmetric tensor field of second order; for easier comparability we consider several scalar tensor invariants. Computed examples show that these new point signatures are especially interesting for surfaces with boundary.