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

Abstract

In this paper we study a variant of the matching model between functional shapes introduced in \cite{ABN}. Such a model allows to compare surfaces equipped with a signal and the matching energy is defined by the $L^2$-norm of the signal on the surface and a varifold-type attachment term. In this work we study the problem with fixed geometry which means that we optimize the initial signal (supported on the initial surface) with respect to a target signal supported on a different surface. In particular, we consider a $BV$ or $H^1$-penalty for the signal instead of its $L^2$-norm. Several numerical examples are shown in order to prove that the $BV$-penalty improves the quality of the matching. Moreover, we prove a $\Gamma$-convergence result for the discrete matching energy towards the continuous-one