Affine Invariant Geometry for Non-rigid Shapes

Abstract

Shape recognition deals with the study geometric structures. Modern surface processing methods can cope with non-rigidity—by measuring the lack of isometry, deal with similarity or scaling—by multiplying the Euclidean arc-length by the Gaussian curvature, and manage equi-affine transformations—by resorting to the special affine arc-length definition in classical equi-affine differential geometry. Here, we propose a computational framework that is invariant to the full affine group of transformations (similarity and equi-affine). Thus, by construction, it can handle non-rigid shapes. Technically, we add the similarity invariant property to an equi-affine invariant one and establish an affine invariant pseudo-metric. As an example, we show how diffusion geometry can encapsulate the proposed measure to provide robust signatures and other analysis tools for affine invariant surface matching and comparison.