TY - GEN
T1 - Grid peeling and the affine curve-shortening flow
AU - Eppstein, David
AU - Har-Peled, Sariel
AU - Nivasch, Gabriel
N1 - Publisher Copyright:
Copyright © 2018 by SIAM.
PY - 2018
Y1 - 2018
N2 - 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 ? Z2 of the integer grid, previously studied for the particular case G = {1, . . ., m}2 by Har-Peled and Lidický (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 = N2 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 N2, (1) the number of grid points removed up to iteration n is ?(n3/2 log n); and (2) the boundary at iteration n is sandwiched between two hyperbolas that are separated from each other by a constant factor.
AB - 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 ? Z2 of the integer grid, previously studied for the particular case G = {1, . . ., m}2 by Har-Peled and Lidický (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 = N2 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 N2, (1) the number of grid points removed up to iteration n is ?(n3/2 log n); and (2) the boundary at iteration n is sandwiched between two hyperbolas that are separated from each other by a constant factor.
KW - Computational geometry
KW - Constant factors;
KW - Differential geometry
KW - L-shaped
KW - Number of Grids
KW - Theoretical arguments
KW - Iterative methods
UR - http://www.scopus.com/inward/record.url?scp=85041428663&partnerID=8YFLogxK
U2 - 10.1137/1.9781611975055.10
DO - 10.1137/1.9781611975055.10
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AN - SCOPUS:85041428663
T3 - Proceedings of the Workshop on Algorithm Engineering and Experiments
SP - 109
EP - 116
BT - 2018 Proceedings of the 20th Workshop on Algorithm Engineering and Experiments, ALENEX 2018
A2 - Pagh, Rasmus
A2 - Venkatasubramanian, Suresh
PB - Society for Industrial and Applied Mathematics Publications
T2 - 20th Workshop on Algorithm Engineering and Experiments, ALENEX 2018
Y2 - 7 January 2018 through 8 January 2018
ER -