TY - JOUR

T1 - Maximizing the overlap of two planar convex sets under rigid motions

AU - Ahn, Hee Kap

AU - Cheong, Otfried

AU - Park, Chong Dae

AU - Shin, Chan Su

AU - Vigneron, Antoine

N1 - Funding Information:
✩ Work by the first three authors was supported by the Brain Korea 21 Project, The School of Information Technology, KAIST, in 2005. Work by the fourth author was supported by the Hankuk University of Foreign Studies Research Fund of 2005. Work by the fifth author was supported by the National University of Singapore under grant R-252-000-166-112. * Corresponding author. E-mail addresses: [email protected] (H.-K. Ahn), [email protected] (O. Cheong), [email protected] (C.-D. Park), [email protected] (C.-S. Shin), [email protected] (A. Vigneron).

PY - 2007/5

Y1 - 2007/5

N2 - Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any >0, we compute a rigid motion such that the area of overlap is at least 1- times the maximum possible overlap. Our algorithm uses O(1/) extreme point and line intersection queries on P and Q, plus O((1/ 2)log(1/)) running time. If only translations are allowed, the extra running time reduces to O((1/)log(1/)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/)logn+(1/2)log(1/)) for rigid motions and O((1/)logn+(1/)log(1/)) for translations.

AB - Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any >0, we compute a rigid motion such that the area of overlap is at least 1- times the maximum possible overlap. Our algorithm uses O(1/) extreme point and line intersection queries on P and Q, plus O((1/ 2)log(1/)) running time. If only translations are allowed, the extra running time reduces to O((1/)log(1/)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/)logn+(1/2)log(1/)) for rigid motions and O((1/)logn+(1/)log(1/)) for translations.

KW - Approximation algorithm

KW - Convex shape

KW - Geometric pattern matching

KW - Sublinear algorithm

UR - http://www.scopus.com/inward/record.url?scp=33847121346&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2006.01.005

DO - 10.1016/j.comgeo.2006.01.005

M3 - Article

AN - SCOPUS:33847121346

SN - 0925-7721

VL - 37

SP - 3

EP - 15

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 1 SPEC. ISS.

ER -