Maximum overlap and minimum convex hull of two convex polyhedra under translations

Authors:
Hee-Kap Ahn;Peter Brass;Chan-Su Shin
Affiliations:
Department of Computer Science and Engineering, POSTECH, Pohang, South Korea;Department of Computer Science, City College, New York, USA;School of Electrical and Information Engineering, Hankuk University of Foreign Studies, Yongin, South Korea
Venue:
Computational Geometry: Theory and Applications
Year:
2008

Citing 6
Cited 4

An optimal algorithm for intersecting three-dimensional convex polyhedra

SIAM Journal on Computing
Cutting hyperplanes for divide-and-conquer

Discrete & Computational Geometry
On the area of overlap of translated polygons

Computer Vision and Image Understanding
Convex Optimization

Convex Optimization
Maximizing the overlap of two planar convex sets under rigid motions

Computational Geometry: Theory and Applications
Stacking and bundling two convex polygons

ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation

ROSy+: 3D Object Pose Normalization Based on PCA and Reflective Object Symmetry with Application in 3D Object Retrieval

International Journal of Computer Vision
Volume matching with application in medical treatment planning

Proceedings of the 2nd ACM Conference on Bioinformatics, Computational Biology and Biomedicine
Maximum overlap of convex polytopes under translation

Computational Geometry: Theory and Applications
Overlap of convex polytopes under rigid motion

Computational Geometry: Theory and Applications

Quantified Score

Hi-index	0.00

Visualization

Abstract

Given two convex polyhedra P and Q in three-dimensional space, we consider two related problems of shape matching: (1) finding a translation t"1@?R^3 of Q that maximizes the volume of their overlap P@?(Q+t"1), and (2) finding a translation t"2@?R^3 that minimizes the volume of the convex hull of P@?(Q+t"2). For the maximum overlap problem, we observe that the dth root of the objective function is concave and present an algorithm that computes the optimal translation in expected time O(n^3log^4n). This method generalizes to higher dimensions d3 with expected running time O(n^d^+^1^-^3^d(logn)^d^+^1). For the minimum convex hull problem, we show that the objective function is convex. The same method used for the maximum overlap problem can be applied to this problem and the optimal translation can be computed in the same time bound.