Basis restricted elastic shape analysis on the space of unregistered surfaces

TL;DR

This paper develops a finite-dimensional, basis-restricted elastic shape analysis framework for unregistered surfaces, enabling effective shape comparison and manipulation without pre-registered meshes, demonstrated on human body and face data.

Contribution

It introduces a novel basis-restricted elastic shape analysis method that simplifies shape space representation while preserving a Riemannian metric, improving surface analysis tasks.

Findings

Outperforms state-of-the-art methods in shape registration and interpolation

Effectively handles unregistered and non-uniform mesh surfaces

Enables shape analysis without pre-registered meshes or consistent mesh structures

Abstract

This paper introduces a new mathematical and numerical framework for surface analysis derived from the general setting of elastic Riemannian metrics on shape spaces. Traditionally, those metrics are defined over the infinite dimensional manifold of immersed surfaces and satisfy specific invariance properties enabling the comparison of surfaces modulo shape preserving transformations such as reparametrizations. The specificity of the approach we develop is to restrict the space of allowable transformations to predefined finite dimensional bases of deformation fields. These are estimated in a data-driven way so as to emulate specific types of surface transformations observed in a training set. The use of such bases allows to simplify the representation of the corresponding shape space to a finite dimensional latent space. However, in sharp contrast with methods involving e.g. mesh autoencoders, the latent space is here equipped with a non-Euclidean Riemannian metric precisely inherited from the family of aforementioned elastic metrics. We demonstrate how this basis restricted model can be then effectively implemented to perform a variety of tasks on surface meshes which, importantly, does not assume these to be pre-registered (i.e. with given point correspondences) or to even have a consistent mesh structure. We specifically validate our approach on human body shape and pose data as well as human face scans, and show how it generally outperforms state-of-the-art methods on problems such as shape registration, interpolation, motion transfer or random pose generation.