Optimal Transportation for the Far-field Reflector Problem

TL;DR

This paper tackles the inverse far-field reflector problem by recasting it as an optimal transport problem on the sphere with a generalized Monge–Ampère equation. It develops a Sobolev gradient descent on the dual OT functional, expressed through $\mathcal{J}'(u)=f-g(T_u)|\det(D^{2}u+A)|$, and implements it via a mixed finite element scheme to solve a Poisson-type update for $u$ while computing the sphere Jacobian accurately. The authors prove convergence of the Sobolev descent under suitable regularity and small step-size conditions, and demonstrate robustness and efficiency on several numerical tests, including smooth, off-axis, and discontinuous targets. The approach achieves fast convergence without Jacobian assembly and is readily adaptable within standard FEM frameworks, with potential extensions to near-field problems and further stabilization via $c$-transform techniques.

Abstract

The inverse reflector problem aims to design a freeform reflecting surface that can direct the light from a specified source to produce the desired illumination in the target area, which is significant in the field of geometrical non-imaging optics. Mathematically, it can be formulated as an optimization problem, which is exactly the optimal transportation problem (OT) when the target is in the far field. The gradient of OT is governed by the generalized Monge-Amp`ere equation that models the far-field reflector system. Based on the gradient, this work presents a Sobolev gradient descent method implemented within a finite element framework to solve the corresponding OT. Convergence of the method is established and numerical examples are provided to demonstrate the effectiveness of the method.

Figures (10)

• Figure 1: Far-field reflector system. The reflecting surface $\Gamma$ receives the source intensity $f(x)$ and produces the target intensity $g(y)$, where $x,y \in \mathbb{S}^{2}$.
• Figure 2: Results of Example 6.1. (a) Numerical solution $u_{h}$ on $\mathbb{S}_{-\pi/4}^{2}$, where the mesh size $h=9.82\times 10^{-3}$. (b) Radial distance function $\rho_h := e^{-u_h}$. (c) Magnitude $|\nabla \rho_{h}|$. (d) Vector field $\nabla \rho_{h}$, where the colors correspond to the values of $|\nabla \rho_{h}|$.
• Figure 3: Ray-traced image of Example 6.1. (a) Far-field target intensity $g$ on $\mathbb{S}_{+\pi /4}^{2}$. (b) Far-field ray-traced intensity $g_{h}$ using the numerical solution $u_{h}$, where $h=9.82\times 10^{-3}$. (c) Absolute error $|g-g_{h}|$.
• Figure 4: Convergence curves of Example 6.1 at different discretization levels, residual $\|r\|_{2}$ versus iterations.
• Figure 5: Results of Example 6.2. (a) The source intensity $f$ (orange) and the far-field target intensity $g$ (blue) on $\mathbb{S}^{2}$. (b) Numerical solution $u_{h}$ on $\mathbb{S}_{-\pi /4}^{2}$, where the mesh size $h=9.82\times 10^{-3}$. (c) Numerical solution $\varphi_{h}$ with $h=9.82\times 10^{-3}$.

Theorems & Definitions (18)

• Theorem 2.1: glimm2003opticalwang2004design
• Theorem 2.2: wang2004design
• Corollary 2.3: glimm2003opticalwang2004design
• Definition 3.1
• Lemma 3.2: kim2012parabolic
• Proposition 3.3
• proof : Proof of Proposition \ref{['dual_grad']}
• Corollary 3.4
• Theorem 4.1
• Lemma 4.2