Grid Peeling and the Affine Curve-Shortening Flow

ABSTRACT

In this article, 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, …, m}² by Har-Peled and Lidický. The second one is the affine curve-shortening flow (ACSF), first studied by Alvarez et al. and Sapiro and Tannenbaum. 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 Θ(n^3/2log n); and (2) the boundary at iteration n is sandwiched between two hyperbolas that are separated from each other by a constant factor.

KEYWORDS:

• convex-layer decomposition
• onion decomposition
• curve-shortening flow
• affine curve-shortening flow
• integer grid

2010 AMS SUBJECT CLASSIFICATION:

• 53C44
• 68U05
• 52A10
• 11H06
• 11P21

Acknowledgements

The third author would like to thank Franck Assous and Elad Horev for useful conversations.

Notes

1 Note that G_{n − 1} might contain points which lie on the boundary of H_n but are not vertices. These points will still be present in G_n.

2 This seems to be the case empirically.

3 Actually, for R₅ we did not simulate ACSF. We simply used the closed-form solution given by r(t) = (r(0)^4/3 − 4t/3)^3/4, where r(t) is the radius of the circle at time t.

4 We computed the Hausdorff distances using a simple brute-force approach, using the fact that for convex polygonal chains the maximum distance is attained by a vertex (Atallah [Atallah 83]). (Atallah [Atallah 83] also presented a more efficent Hausdorff-distance algorithm for convex polygonal chains, which we did not use.)

5 The techniques of [Har-Peled and Lidický 13] are also presented in Section 5 below.

6 The existence of a unique solution was proven for doubly-differentiable curves, without the assumption of closedness, by Angenent et al. [Angenent et al. 98, Section 6] and stated for closed curves without the assumption of smoothness in [Cao 03, Theorem 3.28]. In the case here, the uniqueness of the solution can be proven by applying the result for closed curves to the boundary of a large square: if the quarterplane had multiple solutions, they could be approximated arbitrarily well by the solution near the corner of a large enough square, which would necessarily also have multiple solutions, violating [Cao 03, Theorem 3.28].

Additional information

Funding

Supported in part by the National Science Foundation under Grants CCF-1228639, CCF-1618301, and CCF-1616248. Work on this paper was partially supported by a NSF AF awards CCF-1421231 and CCF-1217462.