Invariants for Discrete Structures – An Extension of Haar Integrals over Transformation Groups to Dirac Delta Functions

Abstract

Due to the increasing interest in 3D models in various applications there is a growing need to support e.g. the automatic search or the classification in such databases. As the description of 3D objects is not canonical it is attractive to use invariants for their representation. We recently published a methodology to calculate invariants for continuous 3D objects defined in the real domain ${\mathbb R}^3$ by integrating over the group of Euclidean motion with monomials of a local neighborhood of voxels as kernel functions and we applied it successfully for the classification of scanned pollen in 3D. In this paper we are going to extend this idea to derive invariants from discrete structures, like polygons or 3D-meshes by summing over monomials of discrete features of local support. This novel result for a space-invariant description of discrete structures can be derived by extending Haar integrals over the Euclidean transformation group to Dirac delta functions.