Authors:
Darko Dimitrov
1
;
Mathias Holst
2
;
Christian Knauer
1
and
Klaus Kriegel
1
Affiliations:
1
Freie Universität Berlin, Institute of Computer Science, Germany
;
2
Universität Rostock, Institute of Computer Science, Germany
Keyword(s):
Bounding Box Algorithms, Principal Component Analysis, Approximation Algorithms.
Related
Ontology
Subjects/Areas/Topics:
Collision Detection
;
Computer Vision, Visualization and Computer Graphics
;
Fundamental Methods and Algorithms
;
Geometric Computing
;
Geometry and Modeling
;
Interactive Environments
;
Modeling and Algorithms
Abstract:
The computation of the minimum-volume bounding box of a point set in R3 is a hard problem. The best known exact algorithm requires O(n3 ) time, so several approximation algorithms and heuristics are preferred in practice. Among them, the algorithm based on PCA (Principal Component Analysis) plays an important role. Recently, it has been shown that the discrete PCA algorithm may fail to approximate the minimum-volume bounding box even for a large constant factor. Moreover, this happens only for some very special examples with point clusters. As an alternative, it has been proven that the continuous version of PCA overcomes these problems. Here, we study the impact of the recent theoretical results on applications of several PCA variants in practice. We give the closed form solutions for the case when the point set is a polyhedron or a polyhedral surface. To the best of our knowledge, the continuous PCA over the volume of a 3D body is considered for the first time. We analyze the advan
tages and disadvantages of the different variants on realistic inputs, randomly generated inputs, and specially constructed (worst case) instances. The results reveal that for most of the realistic inputs the qualities of the discrete PCA and the continuous PCA bounding boxes are comparable. As it was expected the discrete PCA versions are much faster, but behave bad on the clustered inputs. In addition, we evaluate and compare the performances of several existing bounding box algorithms.
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