Solving the Monge–Ampère equations for the inverse reflector problem

Abstract

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-Ampère equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge-Ampère 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-Ampère equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.

Design of Freeform Illumination Optics

“…This is an inverse problem, the number of freeform surfaces which should be used depends on the design requirements. The design methods of freeform illumination optics can be broken into two groups according to the influence of the spatial or angular extent of an actual light source on the design: zero‐étendue algorithms based on ideal source assumption (point light sources or parallel light beams) and algorithms for extended light sources . When the influence of the spatial or angular extent of a light source can be ignored, the light source can be considered as an ideal source (a point source or a parallel beam) and the inverse problem can be converted into a well‐defined mathematical problem.…”

“…Finding such an integrable ray mapping may not be a simple task. A way to circumvent such integrable mapping calculation is to derive a single equation on the optical surface which leads to the advent of the Monge–Ampère (MA) equation method . The elliptic MA equation is a highly nonlinear partial differential equation of second order, it reveals the mathematical essence of illumination design based on ideal source assumption.…”

“…The MA method can satisfy the integrability condition automatically and can be implemented efficiently . Its effectiveness has been proven in a wide variety of applications . One elegant method to find a rigorous solution of that MA equation is usually named the supporting quadric method (SQM) .…”

“…A way to circumvent such integrable mapping calculation is to derive a single equation on the optical surface which leads to the advent of the Monge-Ampère (MA) equation method. [46,47,52,56,[62][63][64][65][66][67]75] The elliptic MA equation is a highly nonlinear partial differential equation of second order, it reveals the mathematical essence of illumination design based on ideal source assumption. The MA method can satisfy the integrability condition automatically and can be implemented efficiently.…”

“…[62] Its effectiveness has been proven in a wide variety of applications. [47,56,[62][63][64][65][66][67]75] One elegant method to find a rigorous solution of that MA equation is usually named the supporting quadric method (SQM). [48,49,53,61,74,80,85,86] The SQM method is a process of calculating a set of quadric surfaces which are used to build a freeform surface, producing a discrete illumination that is an approximation to the prescribed illumination.…”