Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$(BV,L^p)$-decomposition, $p=1,2$, of Functions in Metric Random Walk Spaces

J. M. Mazon, M. Solera, J. Toledo

Abstract

In this paper we study the $(BV,L^p)$-decomposition, $p=1,2$, of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case $p=1$ we also study the associated geometric problem and the thresholding parameters.

$(BV,L^p)$-decomposition, $p=1,2$, of Functions in Metric Random Walk Spaces

Abstract

In this paper we study the -decomposition, , of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case we also study the associated geometric problem and the thresholding parameters.

Paper Structure

This paper contains 12 sections, 45 theorems, 331 equations, 1 figure.

Table of Contents

  1. Introduction and Preliminaries
  2. Metric Random Walk Spaces
  3. $m$-Perimeter
  4. $m$-Total Variation
  5. The $1$-Laplacian in MRW Spaces
  6. The Rudin-Osher-Fatemi Model in MRW Spaces
  7. The $m$-ROF Model
  8. The Gradient Descent Method
  9. The $m$-ROF-Model with $L^1$-fidelity term
  10. The Geometric Problem
  11. Thresholding Parameters
  12. The Gradient Descent Method

Key Result

Proposition \oldthetheorem

1. Let $A,\ B \subset X$ be $\nu$-measurable sets with finite $m$-perimeter such that $\nu(A \cap B) = 0$. Then, 2. Let $A,\ B,\ C$ be $\nu$-measurable sets in $X$ with pairwise $\nu$-null intersections. Then

Figures (1)

  • Figure 1: The point $(0,0)$ is labelled in the graphs, and the adjacent points are represented by the dots.

Theorems & Definitions (96)

  • Example \oldthetheorem
  • Example \oldthetheorem
  • Proposition \oldthetheorem: MST1
  • Proposition \oldthetheorem: Submodularity
  • proof
  • Example \oldthetheorem
  • Lemma \oldthetheorem: MST1
  • Theorem \oldthetheorem: Coarea formula, MST1
  • Theorem \oldthetheorem: MST1
  • Remark \oldthetheorem
  • ...and 86 more