Affine-invariant geodesic geometry of deformable 3D shapes

TL;DR

This paper introduces an affine-invariant geodesic framework for deformable 3D shape analysis, enabling classical Euclidean tools to be applied to affine-invariant surface representations, improving invariance, efficiency, and accuracy.

Contribution

It redefines surface metrics as equi-affine, extending geodesic computation methods to affine-invariant shape analysis, a novel approach in the field.

Findings

Framework demonstrates invariance to affine transformations.

Enhanced efficiency and accuracy in shape analysis tasks.

Applicable to various non-rigid shape analysis applications.

Abstract

Natural objects can be subject to various transformations yet still preserve properties that we refer to as invariants. Here, we use definitions of affine invariant arclength for surfaces in R^3 in order to extend the set of existing non-rigid shape analysis tools. In fact, we show that by re-defining the surface metric as its equi-affine version, the surface with its modified metric tensor can be treated as a canonical Euclidean object on which most classical Euclidean processing and analysis tools can be applied. The new definition of a metric is used to extend the fast marching method technique for computing geodesic distances on surfaces, where now, the distances are defined with respect to an affine invariant arclength. Applications of the proposed framework demonstrate its invariance, efficiency, and accuracy in shape analysis.