Functional characterization of intrinsic and extrinsic geometry

Abstract

We propose a novel way to capture and characterize distortion between pairs of shapes by extending the recently proposed framework of shape differences built on functional maps. We modify the original definition of shape differences slightly and prove that, after this change, the discrete metric is fully encoded in two shape difference operators and can be recovered by solving two linear systems of equations. Then, we introduce an extension of the shape difference operators using offset surfaces to capture extrinsic or embedding-dependent distortion, complementing the purely intrinsic nature of the original shape differences. Finally, we demonstrate that a set of four operators is complete, capturing intrinsic and extrinsic structure and fully encoding a shape up to rigid motion in both discrete and continuous settings. We highlight the usefulness of our constructions by showing the complementary nature of our extrinsic shape differences in capturing distortion ignored by previous approaches. We additionally provide examples where we recover local shape structure from the shape difference operators, suggesting shape editing and analysis tools based on manipulating shape differences.

A Unified Deep Metric Representation for Mesh Saliency Detection and Non-Rigid Shape Matching

“…To learn a metric from a mesh, we use its spatial rather than spectral raw features, as the former capture both intrinsic and extrinsic geometry for shape analysis [56]. Specifically, we use the spherical harmonic (SH) descriptors of [47], which are derived from a raw distance field and have a theoretical guarantee of minimal information loss.…”

Automatic unpaired shape deformation transfer

“…These methods similarly require input for correspondences. Recent work [Corman et al 2017;Rustamov et al 2013] develops effective approaches to measuring shape differences, which are used for embedding shape collections. Given a shape in the first collection, these methods can be used to find a similar shape in the second collection without known correspondence.…”

Principal Geodesic Analysis in the Space of Discrete Shells

“…Boscaini et al [BEKB15] showed how to reconstruct shapes from these shape difference operators, enabling shape analogy synthesis and style transfer. While the original shape differences operator captures only intrinsic distortion, Cormen et al [CSBC*17] use offset surfaces to capture extrinsic distortion.…”