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Closed-Form Solutions for Continuous PCA and Bounding Box Algorithms

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Computer Vision and Computer Graphics. Theory and Applications (VISIGRAPP 2008)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 24))

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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(n 3) 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 proved that the continuous version of PCA overcomes these problems.

The contribution of this paper is two-fold. First, we study the impact of the recent theoretical results on applications of several PCA variants in practice. We analyze the advantages and disadvantages of the different variants on realistic inputs, randomly generated inputs, and specially constructed (worst case) instances. Second, we evaluate and compare the performances of several existing bounding box algorithms.

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Author information

Authors and Affiliations

  1. Institute of Computer Science, Freie Universität Berlin, Takustrasse 9, D-14195, Berlin, Germany

    Darko Dimitrov, Christian Knauer & Klaus Kriegel

  2. Institute of Computer Science, Universität Rostock, Albert Einstein Str. 21, D-18059, Rostock, Germany

    Mathias Holst

Authors
  1. Darko Dimitrov
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  2. Mathias Holst
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  3. Christian Knauer
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  4. Klaus Kriegel
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Editor information

Editors and Affiliations

  1. INSTICC, Avenida D. Manuel I, 2910, Setúbal, Portugal

    AlpeshKumar Ranchordas

  2. Institute for Systems and Robotics, Department of Electrical and Computer Engineering Polo II, University of Coimbra, 3030-290, Coimbra, Portugal

    Hélder J. Araújo

  3. Departamento de Engenharia Informática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    João Madeiras Pereira

  4. Departamento de Sistemas e Informatica, Escola Superior de Tecnologia do IPS, Rua do Vale de Chaves Estefanilha, 2910, Setúbal, Portugal

    José Braz

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© 2009 Springer-Verlag Berlin Heidelberg

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Dimitrov, D., Holst, M., Knauer, C., Kriegel, K. (2009). Closed-Form Solutions for Continuous PCA and Bounding Box Algorithms. In: Ranchordas, A., Araújo, H.J., Pereira, J.M., Braz, J. (eds) Computer Vision and Computer Graphics. Theory and Applications. VISIGRAPP 2008. Communications in Computer and Information Science, vol 24. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10226-4_3

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  • DOI: https://doi.org/10.1007/978-3-642-10226-4_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10225-7

  • Online ISBN: 978-3-642-10226-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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