A Metric Approach to Elastic Reformations

TL;DR

This paper introduces a variational framework using optimal transport theory to compare shapes modeled as measures, quantifying their differences from isometric transformations through a metric-based approach.

Contribution

It develops a novel metric approach to shape comparison using reformations, weak deformations, and Wasserstein distances within a variational framework.

Findings

Introduces a new shape comparison method based on Wasserstein distances.

Provides a variational framework for shape reformation analysis.

Links shape deformations to optimal transport theory.

Abstract

We study a variational framework to compare shapes, modeled as Radon measures on R^N, in order to quantify how they differ from isometric copies. To this purpose we discuss some notions of weak deformations termed reformations as well as integral functionals having some kind of isometries as minimizers. The approach pursued is based on the notion of pointwise Lipschitz constant leading to a space metric framework. In particular, to compare general shapes, we study this reformation problem by using the notion of transport plan and of Wasserstein distances as in optimal mass transportation theory.