Total Variation Isoperimetric Profiles

TL;DR

This paper introduces a convex relaxation of the isoperimetric profile using total variation, providing theoretical guarantees and practical algorithms for shape analysis, with applications in political redistricting.

Contribution

It proposes a novel convex Eulerian relaxation of the isoperimetric profile based on total variation, with proven theoretical properties and practical optimization methods.

Findings

The relaxation satisfies an isoperimetric inequality.

It results in a convex function of the prescribed area.

Experiments demonstrate the effectiveness of the relaxation.

Abstract

Applications such as political redistricting demand quantitative measures of geometric compactness to distinguish between simple and contorted shapes. While the isoperimetric quotient, or ratio of area to perimeter squared, is commonly used in practice, it is sensitive to noisy data and irrelevant geographic features like coastline. These issues are addressed in theory by the isoperimetric profile, which plots the minimum perimeter needed to inscribe regions of different prescribed areas within the boundary of a shape. Efficient algorithms for computing this profile, however, are not known in practice. Hence, in this paper, we propose a convex Eulerian relaxation of the isoperimetric profile using total variation. We prove theoretical properties of our relaxation, showing that it still satisfies an isoperimetric inequality and yields a convex function of the prescribed area. Furthermore, we provide a discretization of the problem, an optimization technique, and experiments demonstrating the value of our relaxation.