Grid peeling and the affine curve-shortening flow

TL;DR

This paper explores the connection between grid peeling, a computational geometry process, and the affine curve-shortening flow from differential geometry, providing empirical and theoretical insights into their relationship, especially for infinite grids.

Contribution

It demonstrates that grid peeling approximates affine curve-shortening flow on convex curves and provides rigorous results for the case of the infinite grid, linking discrete and continuous geometric processes.

Findings

Grid peeling behaves like ACSF on convex curves at the limit.

Number of points removed in grid peeling of N^2 is Θ(n^{3/2} log n).

Boundary of the peeling process is bounded between two hyperbolas.

Abstract

In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call "grid peeling") is the convex-layer decomposition of subsets $G\subset \mathbb Z^2$ of the integer grid, previously studied for the particular case $G=\{1,\ldots,m\}^2$ by Har-Peled and Lidick\'y (2013). The second one is the affine curve-shortening flow (ACSF), first studied by Alvarez et al. (1993) and Sapiro and Tannenbaum (1993). We present empirical evidence that, in a certain well-defined sense, grid peeling behaves at the limit like ACSF on convex curves. We offer some theoretical arguments in favor of this conjecture. We also pay closer attention to the simple case where $G=\mathbb N^2$ is a quarter-infinite grid. This case corresponds to ACSF starting with an infinite L-shaped curve, which when transformed using the ACSF becomes a hyperbola for all times $t>0$. We prove that, in the grid peeling of $\mathbb N^2$, (1) the number of grid points removed up to iteration $n$ is $\Theta(n^{3/2}\log n)$; and (2) the boundary at iteration $n$ is sandwiched between two hyperbolas that are separated from each other by a constant factor.