The matching problem between functional shapes via a BV penalty term: a $\Gamma$-convergence result

TL;DR

This paper establishes a $$-convergence result for a surface signal matching problem using a BV penalty, ensuring discrete-to-continuous consistency and highlighting geometric discretization properties.

Contribution

It introduces a $$-convergence proof for the discrete energy of surface signal matching with a BV penalty, extending the functional shapes framework.

Findings

Proves $$-convergence of discrete energies to continuous energies.

Identifies geometric discretization properties necessary for convergence.

Demonstrates the effectiveness of BV penalty in surface signal matching.

Abstract

This paper proves a $\Gamma$-convergence result for the discrete energy (to the continuous one) of the matching problem for signals defined on surfaces. In particular, we highlight some geometric properties that must be guaranteed in the discretization process to ensure the convergence of minimizers. The proof is given in the framework of functional shapes introduced in \cite{ABN}. In particular, we consider a varifold-type attachment term, and a $BV$ penalty term is used instead of the original $L^2$ norm.