Far-field reflector problem and intersection of paraboloids
- 157 Downloads
In this article, we propose a numerical approach to the far field reflector problem which is an inverse problem arising in geometric optics. Caffarelli et al. (Contemp Math 226:13–32, 1999) proposed an algorithm that involves the computation of the intersection of the convex hull of confocal paraboloids. We show that computing this intersection amounts to computing the intersection of a power diagram (a generalization of the Voronoi diagram) with the unit sphere. This allows us to provide an algorithm that computes efficiently the intersection of confocal paraboloids using the exact geometric computation paradigm. Furthermore, using an optimal transport formulation, we cast the far field reflector problem into a concave maximization problem. This allows us to numerically solve the far field reflector problem with up to 15k paraboloids. We also investigate other geometric optic problems that involve union of confocal paraboloids and also intersection and union of confocal ellipsoids. In all these cases, we show that the computation of these surfaces is equivalent to the computation of the intersection of a power diagram with the unit sphere.
Mathematics Subject Classification68U05 Computer graphics; computational geometry 65K15 Mathematical programming, optimization and variational techniques 52A41 Convex functions and convex programs 35Q99 Equations of mathematical physics and other areas of application
The authors would like to thank Dominique Attali, Olivier Devillers and Francis Lazarus for interesting discussions. Olivier Devillers suggested the approach used in the proof of the lower complexity bound for intersection of ellipsoids. P. M. M. de Castro is supported by Grant FACEPE/INRIA, APQ-0055-1.03/12. Q. Mérigot and B. Thibert would like to acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-11-BS01-014-01 (TOMMI) and ANR-13-BS01-0008-03 (TOPDATA) respectively.
- 4.Boissonnat, J.-D., Karavelas, M.I.: On the combinatorial complexity of Euclidean Voronoi cells and convex hulls of d-dimensional spheres. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Philadelphia, PA, USA), SODA ’03. Society for Industrial and Applied Mathematics, pp. 305–312 (2003)Google Scholar
- 9.Cgal, Computational Geometry Algorithms Library. http://www.cgal.org
- 10.Delage, C.: CGAL-based first prototype implementation of Möbius diagram in 2D. Technical Report ECG-TR-241208-01, INRIA Sophia-Antipolis, 2003Google Scholar
- 14.Kitagawa, J.: An iterative scheme for solving the optimal transportation problem (2012) (preprint). arXiv:1208.5172
- 16.Mérigot, Q.: A multiscale approach to optimal transport. In: Computer Graphics Forum, vol. 30, pp. 1583–1592. Wiley Online Library, New York (2011)Google Scholar
- 18.Oliker, V.I.: Mathematical aspects of design of beam shaping surfaces in geometrical optics. In: Trends in Nonlinear Analysis, pp. 193–224 (2003)Google Scholar
- 19.Oliker, V.I.: A rigorous method for synthesis of offset shaped reflector antennas. Comput. Lett. 2(1–2), 1–2 (2006)Google Scholar