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Computing Optimal Beams in Two and Three Dimensions

Author

Listed:
  • Danny Z. Chen

    (University of Notre Dame)

  • Xiaobo (Sharon) Hu

    (University of Notre Dame)

  • Jinhui Xu

    (State University of New York at Buffalo)

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 dimensional spaces. 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 pierce the 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.

Suggested Citation

  • Danny Z. Chen & Xiaobo (Sharon) Hu & Jinhui Xu, 2003. "Computing Optimal Beams in Two and Three Dimensions," Journal of Combinatorial Optimization, Springer, vol. 7(2), pages 111-136, June.
    • Handle: RePEc:spr:jcomop:v:7:y:2003:i:2:d:10.1023_a:1024484412699
    DOI: 10.1023/A:1024484412699
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    Cited by:

    1. Yam Ki Cheung & Ovidiu Daescu, 2011. "Line facility location in weighted regions," Journal of Combinatorial Optimization, Springer, vol. 22(1), pages 52-70, July.
    2. Azadeh Tabatabaei & Mohammad Ghodsi, 2015. "Walking in streets with minimal sensing," Journal of Combinatorial Optimization, Springer, vol. 30(2), pages 387-401, August.

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