Point Source Regularization of the Finite Source Reflector Problem

Jean-David Benamou (1) , Guillaume Chazareix (1) , Wilbert L Ijzerman (2) , Giorgi Rukhaia (1)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 Signify Research

We address the “freeform optics” inverse problem of designing a reflector surface mapping a prescribed source distribution of light to a prescribed far field distribution, for a finite light source. When the finite source reduces to a point source, the light source distribution has support only on the optics ray directions. In this setting the inverse problem is well posed for arbitrary source and target probability distributions. It can be recast as an Optimal Transportation problem and has been studied both mathematically and nu-merically. We are not aware of any similar mathematical formulation in the finite source case: i.e. the source has an “´etendue” with support both in space and directions. We propose to leverage the well-posed variational formulation of the point source problem to build a smooth parameterization of the reflec-tor and the reflection map. Under this parameterization we can construct a smooth loss/misfit function to optimize for the best solution in this class of reflectors. Both steps, the parameterization and the loss, are related to Optimal Transportation distances. We also take advantage of recent progress in the numerical approximation and resolution of these mathematical objects to perform a numerical study.

Mots clés: Inverse reflector problem; Optimal transportation; Non-linear optimization

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Jean-David Benamou, Guillaume Chazareix, Wilbert L Ijzerman, Giorgi Rukhaia. Point Source Regularization of the Finite Source Reflector Problem. 2021. ⟨hal-03344571v1⟩