Traditional models of bendable surfaces are based on the exact or approximate invariance to deformations that do not tear or stretch the shape, leaving intact an intrinsic geometry associated with it.
Intrinsic geometries are typically defined by shortest paths also known as geodesic distances, or diffusion processes on the surface like diffusion distances. Both methods are implicitly derived from the metric induced by the ambient Euclidean space.
Here, we depart from this restrictive assumption by observing that a different choice of the metric results in a richer set of geometric invariants. We extend the classic equi-affine arclength, defined on hyperbolic surfaces, to arbitrary shapes with non-vanishing Gaussian curvature. As a result, a family of affine-invariant intrinsic geometries is obtained.
We propose a computational framework that is invariant to the affine group of transformations (similarity and equi-affine) and thus, by construction, can handle
non-rigid shapes. Diffusion geometry encapsulates the resulting measure to robustly provide signatures and computational tools for affine invariant analysis.