Grid peeling and the affine curve-shortening flow.

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 callgrid peeling) is the convex-layer decomposition of subsetsG subset of Z2of the integer grid, previously studied for the particular caseG= {1, horizontal ellipsis ,m}(2)by Har-Peled and Lidicky. 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 whereG=N2is 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 timest> 0. We prove that, in the grid peeling ofN2, (1) the number of grid points removed up to iterationnis Theta(n(3/2)log n); and (2) the boundary at iterationnis sandwiched between two hyperbolas that are separated from each other by a constant factor.