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Optimal Beam Penetrations in Two and Three Dimensions

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Algorithms and Computation (ISAAC 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1969))

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Abstract

The problem of computing an optimal beam among weighted regions (called the optimal beam problem) arises in several applied areas such as radiation therapy, stereotactic brain surgery, medical surgery, geological exploration, manufacturing, and environmental engineering. In this paper, we present computational geometry techniques that enable us to develop efficient algorithms for solving various optimal beam problems among weighted regions in two and three dimensions. In particular, we consider two types of problems: the covering problems (seeking an optimal beam to contain a specified target region), and the piercing problems (seeking an optimal beam of a fixed shape to piercethe target region). We investigate several versions of these problems, with a variety of beam shapes and target region shapes in 2-D and 3-D. Our algorithms are based on interesting combinations of computational geometry techniques and optimization methods, and transform the optimal beam problems to solving a collection of instances of certain special non-linear optimization problems. Our approach makes use of interesting geometric observations, such as utilizing some new features of Minkowski sums.

The research of the first and third authors was supported in part by NSF under Grants CCR-9623585 and CCR-9988468. The research of the second author was supported in part by NSF under Grants MIP-9701416 and CCR-9988468, and by HP Labs, Bristol, England, under an external research program grant. The majority of the research of the third author was done when the author was a graduate student at the CSE Dept., Univ. of Notre Dame, and supported in part by a fellowship from the Center for Applied Mathematics, and by a summer graduate research fellowship from the Graduate School, Univ. of Notre Dame, Notre Dame, Indiana, USA. D.T. Lee and S.-H. Teng (Eds.): ISAAC 2000, LNCS 1969, pp. 491-502, 2000.

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Notre Dame, 46556, Notre Dame, IN, USA

    Danny Z. Chen & Xiaobo Hu

  2. Department of Computer Science and Engineering, State University of New York at Buffalo, 226 Bell Hall, Box 602000, 14260, Buffalo, NY, USA

    Jinhui Xu

Authors
  1. Danny Z. Chen
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  2. Xiaobo Hu
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  3. Jinhui Xu
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Editor information

Editors and Affiliations

  1. Karlsruhe University, Germany

    Gerhard Goos

  2. Cornell University, NY, USA

    Juris Hartmanis

  3. Utrecht University, The Netherlands

    Jan van Leeuwen

  4. Academia Sinica, Institute of Information Science, 128 Academia Road, Section 2, Nankang, 115, Taipei, Taiwan, R.O.C.

    D. T. Lee

  5. Department of Computer Science and Akamai Technologies, University of Illinois at Urbana Champaign, 500 Technology Square, 02139, Cambridge, MA, USA

    Shang-Hua Teng

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© 2000 Springer-Verlag Berlin Heidelberg

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Chen, D.Z., Hu, X., Xu, J. (2000). Optimal Beam Penetrations in Two and Three Dimensions. In: Goos, G., Hartmanis, J., van Leeuwen, J., Lee, D.T., Teng, SH. (eds) Algorithms and Computation. ISAAC 2000. Lecture Notes in Computer Science, vol 1969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-40996-3_42

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  • DOI: https://doi.org/10.1007/3-540-40996-3_42

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41255-7

  • Online ISBN: 978-3-540-40996-0

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