Geometry of the matching distance for 2D filtering functions

TL;DR

This paper investigates the geometric properties of the matching distance for 2D filtering functions on Riemannian manifolds, revealing where the maximum distances are achieved in the parameter space.

Contribution

It introduces the extended Pareto grid to analyze the matching distance, showing it is realized at specific geometric lines or points.

Findings

Matching distance is realized at special values or along lines with slopes 0, 1, or infinity.

The extended Pareto grid effectively characterizes the geometric structure of the matching distance.

Provides insights into the geometric behavior of 2D filtering functions on manifolds.

Abstract

In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for $\mathbb{R}^2$-valued regular functions defined on a Riemannian closed manifold. In particular, we prove that in this case the matching distance is realised either at special values or at values corresponding to vertical, horizontal or slope 1 lines.