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.
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References
P.K. Agarwal and M. Sharir, “Pipes, cigar, and kreplach: The union of Minkowski sums in three dimensions,” in Proc. 15th Annual Symposium on Computational Geometry, 1999, pp. 143–153.
N.M. Amato, M.T. Goodrich, and E.A. Ramos, “Computing the arrangement of curve segments: Divide-and-conquer algorithms via sampling. in Proc. 11th Annual ACM-SIAM Symposium on Discrete Algorithms, 2000, pp. 705–706.
T. Asano, L.J. Guibas, and T. Tokuyama, “Walking in an arrangement topologically,” Int. J. of Computational Geometry and Applications, vol. 4, pp. 123–151, 1994.
T. Asano and T. Tokuyama, “Topological walk revisited,” in Proc. 6th Canadian Conf. on Comp. Geometry, 1994, pp. 1–6.
G.K. Bahr, J.G. Kereiakes, H. Horowitz, R. Finney, J. Galvin, and K. Goode, “The method of linear programming applied to radiation treatment planning,” Radiology, vol. 91, pp. 686–693, 1968.
T. Bortfeld, J. Bürkelbach, R. Boesecke, and W. Schlegel, “Methods of image reconstruction from projections applied to conformation radiotherapy,” Phys. Med. Biol., vol. 38, pp. 291–304, 1993.
T. Bortfeld and W. Schlegel, “Optimization of beam orientations radiation therapy: Some theoretical considerations,” Phys. Med. Biol., vol. 35, pp. 1423–1434, 1990.
A.L. Boyer, T.R. Bortfeld, L. Kahler, and T.J. Waldron, “MLC modulation of x-ray beams in discrete steps,” in Proc. 11th Conf. on the Use of Computers in Radiation Therapy, 1994, pp. 178–179.
A.L. Boyer, G.E. Desobry, and N.H Wells, “Potential and limitations of invariant kernel conformal therapy,” Med. Phys., vol. 18, pp. 703–712, 1991.
A. Brahme, “Optimization of stationary and moving beam radiation therapy techniques,” Radiother. Oncol., vol. 12, pp. 129–140, 1988.
A. Brahme, “Inverse radiation therapy planning: Principles and possibilities,” in Proc. 11th Conf. on the Use of Computers in Radiation Therapy, 1994a, pp. 6–7.
A. Brahme, “Optimization of radiation therapy,” Int. J. Radiat. Oncol. Biol. Phys., vol. 28, pp. 785–787, 1994b.
R.D. Bucholz, “Introduction to the journal of image guided surgery,” Journal of Image Guided Surgery, vol. 1, no. 1, pp. 1–11, 1995.
C.W. Burckhardt, P. Flury, and D. Glauser, “Stereotactic brain surgery,” IEEE Engineering in Medicine and biology, vol. 14, no. 3, pp. 314–317, 1995.
Y. Censor, M.D. Altschuler, and W.D. Powlis, “A computational solution of the inverse problem in radiation-therapy treatment planning,” Applied Math. and Computation, vol. 25, pp. 57–87, 1988.
B. Chazelle, “A faster deterministic algorithm for minimum spanning trees,” in Proc. 38th Ann. IEEE Symp. Found. Comp. Sci., 1997, pp. 22–31.
D.Z. Chen, O. Daescu, X.S. Hu, X.Wu, and J. Xu, “Determining an optimal penetration among weighted regions in two and three dimensions,” in Proc. 15th ACM Annual Symposium on Computational Geometry, 1999, pp. 322–331.
L. Craig, J.L. Zhou, and A.L. Tits, User’s Guide for CFSQP Version 2.5. Electrical Eng. Dept. and Institute for Systems Research, University of Maryland, TR-94-16r1, 1994.
F. Durand, G. Drettakis, and C. Puech, “The 3D visibility complex, a new approach to the problems of accurate visibility,” in Proc. 7th Eurographic Workshop on Rendering, 1996, pp. 245–257.
H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag: New York, 1987.
H. Edelsbrunner and L.J. Guibas, “Topologically sweeping an arrangement,” Journal of Computer and System Sciences, vol. 38, pp. 165–194, 1989.
H. Edelsbrunner, L.J. Guibas, J. Pach, R. Pollack, R. Seidel, and M. Sharir, “Arrangements of curves in the plane: Topology, combinatorics, and algorithms,” Theoretical Computer Science, vol. 92, pp. 319–336, 1992.
A. Gustafsson, B.K. Lind, and A. Brahme, “A generalized pencil beam algorithm for optimization of radiation therapy,” Med. Phys., vol. 21, pp. 343–356, 1994.
T. Holmes and T.R. Mackie, “A comparison of three inverse treatment planning algorithms,” Phys. Med. Biol., vol. 39, pp. 91–106, 1994.
J. Legras, B. Legras, J.P. Lambert, and P. Aletti, “The use of a microcomputer for non-linear optimization of doses in external radiotherapy,” Phys. Med. Biol., vol. 31, pp. 1353–1359, 1986.
B.K. Lind, “Properties of an algorithm for solving the inverse problem in radiation therapy,” in Proc. 9th Int. Conf. on the Use of Computers in Radiation Therapy, 1987 pp. 235–239.
B.K. Lind and A. Brahme, “Optimization of radiation therapy dose distributions with scanned photon beams,” Inv. Prob., vol. 16, pp. 415–426, 1990.
S.C. McDonald and P. Rubin, “Optimization of external beam radiation therapy,” Int. J. Radiat. Oncol. Biol. Phys., vol. 2, pp. 307–317, 1977.
W.D. Powlis, M.D. Altschuler, Y. Censor, and E.L. Buhle, “Semi-automated radiotherapy treatment planning with a mathematical model to satisfy treatment goals,” Int. J. Radiat. Oncol. Biol. Phys., vol. 16, pp. 271–276, 1989.
F.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction, Springer-Verlag: New York, 1985.
S. Rivière, Visibility computations in 2D polygonal scenes. Ph.D. thesi, Université Joseph Fourier, Grenoble, France, 1997.
A. Schweikard, J.R. Adler, and J.-C. Latombe, “Motion planning in stereotaxic radiosurgery,” IEEE Trans. on Robotics and Automation, vol. 9, pp. 764–774, 1993.
A. Schweikard, R. Tombropoulos, L. Kavraki, J.R. Adler, and J.-C. Latombe, “Treatment planning for a radiosurgical system with general kinematics,” in Proc. IEEE International Conference on Robotics and Automation, 1994, pp. 1720–1727.
M. Sharir and P.K. Agarwal, Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, 1995.
R.Z. Tombropoulos, J.R. Adler, and J.-C. Latombe, “CARABEAMER: A treatment planner for a robotic radiosurgical system with general kinematics,” Medical Image Analysis, vol. 3, pp. 1–28, 1999.
S. Webb, “Optimization of conformal radiotherapy dose distributions by simulated annealing,” Phys. Med. Biol., vol. 34, pp. 1349–1369, 1989.
S. Webb,“Optimizing the planning of intensity-modulated radiotherapy,” Phys. Med. Biol., vol. 39, pp. 2229–2246, 1994.
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Chen, D.Z., Hu, X.(. & Xu, J. Computing Optimal Beams in Two and Three Dimensions. Journal of Combinatorial Optimization 7, 111–136 (2003). https://doi.org/10.1023/A:1024484412699
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DOI: https://doi.org/10.1023/A:1024484412699