Solving the Monge-Amp\`ere Equations for the Inverse Reflector Problem

The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the target, e.g., a projected image on a screen. This optical problem can mathematically be understood as a problem of optimal transport and equivalently be expressed by a secondary boundary value problem of the Monge-Amp\`ere equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge-Amp\`ere equation is numerically solved using a collocation method based on tensor-product B-splines, in which nested iteration techniques are applied to ensure the convergence of the nonlinear solver and to speed up the calculation. In the numerical method special care has to be taken for the constraint: It enters the discrete problem formulation via a Picard-type iteration. Numerical results are presented as well for benchmark problems for the standard Monge-Amp\`ere equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.Comment: 28 pages, 8 figures, 2 tables; Keywords: Inverse reflector problem, elliptic Monge-Amp\`ere equation, B-spline collocation method, Picard-type iteration; Minor revision: reference [59] to a recent preprint has been added and a few typos have been correcte

Fast inverse reflector design (FIRD)

This paper presents a new inverse reflector design method using a GPU-based computation of outgoing light distribution from reflectors. We propose a fast method to obtain the outgoing light distribution of a parametrized reflector, and then compare it with the desired illumination. The new method works completely in the GPU. We trace millions of rays using a hierarchical height-field representation of the reflector. Multiple reflections are taken into account. The parameters that define the reflector shape are optimized in an iterative procedure in order for the resulting light distribution to be as close as possible to the desired, user-provided one. We show that our method can calculate reflector lighting at least one order of magnitude faster than previous methods, even with millions of rays, complex geometries and light sources. © 2009 The Eurographics Association and Blackwell Publishing Ltd.This work was done under grant TIN2007-67120 from the Spanish Government, and under grant 7th Framework Programme-Capacities (grant No 222550, project EcoStreet-Light) from the UE.Peer Reviewe

Fast Inverse Reflector Design (FIRD)

This paper presents a new inverse reflector design method using a GPU-based computation of outgoing light distribution from reflectors. We propose a fast method to obtain the outgoing light distribution of a parametrized reflector, and then compare it with the desired illumination. The new method works completely in the GPU. We trace millions of rays using a hierarchical height-field representation of the reflector. Multiple reflections are taken into account. The parameters that define the reflector shape are optimized in an iterative procedure in order for the resulting light distribution to be as close as possible to the desired, user-provided one. We show that our method can calculate reflector lighting at least one order of magnitude faster than previous methods, even with millions of rays, complex geometries and light sourcesThis work was done under grant TIN2007-67120 from the Spanish Government, and under grant 7th Framework Programme-Capacities (grant No 222550, project EcoStreetLight) from the U