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Neural network methods for two-dimensional finite-source reflector design

Roel Hacking, Lisa Kusch, Koondanibha Mitra, Martijn Anthonissen, Wilbert IJzerman

Abstract

We address the inverse problem of designing two-dimensional reflectors that transform light from a finite, extended source into a prescribed far-field distribution. We propose a neural network parameterization of the reflector height and develop two differentiable objective functions: (i) a direct change-of-variables loss that pushes the source distribution through the learned inverse mapping, and (ii) a mesh-based loss that maps a target-space grid back to the source, integrates over intersections, and remains continuous even when the source is discontinuous. Gradients are obtained via automatic differentiation and optimized with a robust quasi-Newton method. As a comparison, we formulate a deconvolution baseline built on a simplified finite-source approximation: a 1D monotone mapping is recovered from flux balance, yielding an ordinary differential equation solved in integrating-factor form; this solver is embedded in a modified Van Cittert iteration with nonnegativity clipping and a ray-traced forward operator. Across four benchmarks -- continuous and discontinuous sources, and with/without minimum-height constraints -- we evaluate accuracy by ray-traced normalized mean absolute error (NMAE). Our neural network approach converges faster and achieves consistently lower NMAE than the deconvolution method, and handles height constraints naturally. We discuss how the method may be extended to rotationally symmetric and full three-dimensional settings via iterative correction schemes.

Neural network methods for two-dimensional finite-source reflector design

Abstract

We address the inverse problem of designing two-dimensional reflectors that transform light from a finite, extended source into a prescribed far-field distribution. We propose a neural network parameterization of the reflector height and develop two differentiable objective functions: (i) a direct change-of-variables loss that pushes the source distribution through the learned inverse mapping, and (ii) a mesh-based loss that maps a target-space grid back to the source, integrates over intersections, and remains continuous even when the source is discontinuous. Gradients are obtained via automatic differentiation and optimized with a robust quasi-Newton method. As a comparison, we formulate a deconvolution baseline built on a simplified finite-source approximation: a 1D monotone mapping is recovered from flux balance, yielding an ordinary differential equation solved in integrating-factor form; this solver is embedded in a modified Van Cittert iteration with nonnegativity clipping and a ray-traced forward operator. Across four benchmarks -- continuous and discontinuous sources, and with/without minimum-height constraints -- we evaluate accuracy by ray-traced normalized mean absolute error (NMAE). Our neural network approach converges faster and achieves consistently lower NMAE than the deconvolution method, and handles height constraints naturally. We discuss how the method may be extended to rotationally symmetric and full three-dimensional settings via iterative correction schemes.

Paper Structure

This paper contains 21 sections, 2 theorems, 56 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Finite-source problem setup
  4. Approximate finite-source problem
  5. Methods
  6. Neural network-based solver
  7. Direct method
  8. Mesh-based method
  9. Iterative deconvolution
  10. Numerical examples
  11. Discussion
  12. Conclusions
  13. Funding.
  14. Acknowledgment.
  15. Disclosures.
  16. ...and 6 more sections

Key Result

Proposition 1

Let $\sigma_\lambda(s,\alpha)$ denote the stereographic far-field coordinate produced by a ray emitted from $(s,0)$ at angle $\alpha$ and reflected by $\mathbf{r}_\lambda$. Assume that the limiting ray map $\sigma_\infty:A\to\Sigma$ takes values in a finite interval $\Sigma\subset\mathbb{R}$ and is $\blacktriangleleft$$\blacktriangleleft$

Figures (10)

  • Figure 1: Finite-source-to-far-field reflector system setup.
  • Figure 2: Finite-source-to-far-field reflector system approximation setup.
  • Figure 3: Mesh-based procedure for computing a continuous loss for inverse reflector design. (a) The target space is discretized into quadrilateral cells. (b) Each vertex is mapped to the source space. (c) The source distribution is evaluated. (d) Integration is performed over valid intersections with the source domain. (e) The integrals are assigned back to the original mesh cells. (f) A histogram of the far-field output is obtained.
  • Figure 4: Reflectors for the mapping $m(s) = s^3$ for different values of $h$ computed using Eq. \ref{['eq:approx-sol-expr']}. The left plot shows the mapping, the center plot shows the obtained functions $u$ for different values of $h$, and the right plot shows the corresponding reflectors in the $xz$-plane.
  • Figure 5: Ground-truth reflector, source distribution, corresponding target distribution, and far-field target distribution for Example \ref{['A']}.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Corollary 1: Point-source design problem