Computing Optimal Beams in Two and Three Dimensions

Danny Z. Chen (), Xiaobo (Sharon) Hu () and Jinhui Xu ()
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Danny Z. Chen: University of Notre Dame
Xiaobo (Sharon) Hu: University of Notre Dame
Jinhui Xu: State University of New York at Buffalo

Journal of Combinatorial Optimization, 2003, vol. 7, issue 2, No 1, 136 pages

Abstract: 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.

Keywords: optimal beams; piercing problems; covering problems; curve arrangements; radiosurgery (search for similar items in EconPapers)
Date: 2003
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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DOI: 10.1023/A:1024484412699

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