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

G. Nardi, B. Charlier, A. Trouvé

TL;DR

The paper proves a Γ-convergence result for the discrete-to-continuous matching energy of signals on surfaces, under a BV penalty and a varifold-type attachment term. It replaces the original $L^2$ penalty with $BV$ (and allows $H^1$/$L^2$ variants) and uses a standard dual norm instead of an RKHS-based metric, proving existence and convergence of minimizers for fixed geometry. It introduces admissible triangulations and geometric conditions that ensure area convergence, then builds a finite-element discretization (with $P_0$/$P_1$ elements) and defines discrete fvarifold norms. The main results show that discrete minimizers converge to continuous minimizers as triangulation diameter vanishes, providing a rigorous foundation for reliable surface-shape matching computations in a BV-regularized functional-shape framework.

Abstract

This paper proves a $Γ$-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.

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

TL;DR

The paper proves a Γ-convergence result for the discrete-to-continuous matching energy of signals on surfaces, under a BV penalty and a varifold-type attachment term. It replaces the original penalty with (and allows / variants) and uses a standard dual norm instead of an RKHS-based metric, proving existence and convergence of minimizers for fixed geometry. It introduces admissible triangulations and geometric conditions that ensure area convergence, then builds a finite-element discretization (with / elements) and defines discrete fvarifold norms. The main results show that discrete minimizers converge to continuous minimizers as triangulation diameter vanishes, providing a rigorous foundation for reliable surface-shape matching computations in a BV-regularized functional-shape framework.

Abstract

This paper proves a -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 penalty term is used instead of the original norm.

Paper Structure

This paper contains 23 sections, 15 theorems, 124 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context and previous work.
  3. Contributions.
  4. Structure of the paper.
  5. $BV$ functions on manifolds
  6. Functional varifolds
  7. Existence of optimal solutions for the matching problem
  8. Surfaces and triangulations
  9. Triangulations of a surface
  10. Convergence of areas
  11. From the triangulation to the surface
  12. Discretization
  13. Notations
  14. $P_0$ and $P_1$ triangular finite elements
  15. Discrete operators for finite elements
  16. ...and 8 more sections

Key Result

Theorem 2.2

Let $X$ be a smooth manifold and $\{f_h\}_h$ be a sequence of $BV(X)$ such that $\|f_h\|_{BV(X)}$ is uniformly bounded. Then $\{f_h\}_h$ is relatively compact in $BV(X)$ with respect to the weakly-$*$ convergence.

Figures (3)

  • Figure 1: The map $\mathop{\mathrm{Ext}}\nolimits$
  • Figure 2: The surface $X$ is a bent smooth star (solid grey), and two triangulations (black lines) are illustrated. Figure \ref{['tri_inscribed']}: the triangulation $\mathcal{T}_1$ is not in one-to-one correspondence with $X$ through the projection map (for instance, the part of the smooth star in red exceeds the triangulation). Figure \ref{['tri_hadmissible']}: the subset $\mathcal{T}^{\operatorname{in}}_2$ of $\mathcal{T}_2$ is in one-to-one correspondance with $X$
  • Figure 3: Labels of various points in the triangle $T_k$.

Theorems & Definitions (46)

  • Definition 2.1
  • Theorem 2.2: Compactness
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5: $L^2$ and $H^1$ norms
  • Definition 3.1: fshape
  • Definition 3.2: fvarifolds
  • ...and 36 more