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

Abstract

The matching problem often arises in image processing and involves finding a correspondence between similar objects. In particular, variational matching models optimize suitable energies that evaluate the dissimilarity between the current shape and the relative template. A penalty term often appears in the energy to constrain the regularity of the solution. To perform numerical computation, a discrete version of the energy is defined. Then, the question of consistency between the continuous and discrete solutions arises. This paper proves a $\Gamma$-convergence result for the discrete energy to the continuous one. In particular, we highlight some geometric properties that must be guaranteed in the discretization process to ensure the convergence of minimizers. We prove the result in the framework introduced in the 2017 paper of Charlier et al., which studies the matching problem between geometric structures carrying on a signal (fshapes). The matching energy is defined for $L^2$ signals and evaluates the difference between fshapes in terms of the varifold norm. This paper maintains a dual attachment term, but we consider a $BV$ penalty term in place of the original $L^2$ norm.