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Approximation of symmetric total variation on point clouds

Stefano Almi, Anna Kubin, Emanuele Tasso

Abstract

The paper investigates the approximation of the symmetric Total Variation functional on graphs. Such an approximation is given in terms of a discrete and symmetric finite difference model defined on point clouds obtained by randomly sampling a reference probability measure. We identify suitable scalings of the point distribution that guarantee an almost surely $Γ$-convergence to an anisotropic weighted symmetric Total Variation.

Approximation of symmetric total variation on point clouds

Abstract

The paper investigates the approximation of the symmetric Total Variation functional on graphs. Such an approximation is given in terms of a discrete and symmetric finite difference model defined on point clouds obtained by randomly sampling a reference probability measure. We identify suitable scalings of the point distribution that guarantee an almost surely -convergence to an anisotropic weighted symmetric Total Variation.

Paper Structure

This paper contains 4 sections, 6 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Energy and assumptions
  3. Gamma liminf inequality
  4. Construction of a recovery sequence

Key Result

Theorem 1.1

Assume $({\rm K1})$--$({\rm K3})$, $(\rho1)$--$(\rho2)$, and eq:asssumption_epsilon--e:second-Tn. Then, the sequence of functionals $GTV_{n}$ almost surely $\Gamma$-converges with respect to the $TL^{1}$-convergence to where the function $\phi_{\eta} \colon \mathbb{M}^{d}_{sym} \to [0, + \infty)$ is defined as $\phi_{\eta}(A):= \int_{\mathbb{R}^d} \eta(\xi) |A\xi \cdot \xi|\, \textnormal{d} \xi$

Theorems & Definitions (12)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof : Proof of Theorem \ref{['thm:liminf']}
  • ...and 2 more