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

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

cs.LGApr 2, 2026

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

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

Organizations: Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands · Signify, High Tech Campus 7, 5656 AE, Eindhoven, The Netherlands

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. The reflector height is represented by a neural network and optimized with two objective functions: a direct change-of-variables loss based on the closed-form inverse ray map, and a mesh-based loss that maps target cells back to the source and remains usable for discontinuous sources. Gradients are computed by automatic differentiation and minimized with a robust quasi-Newton method. As a baseline, we adapt a deconvolution pipeline built on a simplified finite-source approximation: a one-dimensional monotone map is recovered from flux balance, converted to a reflector by an integrating-factor ODE solve, and embedded in a modified Van Cittert iteration with nonnegativity clipping and ray-traced feedback. Across four benchmarks, covering continuous and discontinuous sources and minimum-height constraints, accuracy is measured by ray-traced normalized mean absolute error. On the two main benchmarks, the neural method reaches errors of about 2e-5 and 5e-5 within a few seconds on one NVIDIA RTX 4090 GPU, compared with 4e-3 and 5e-2 for the deconvolution baseline after several hundred seconds. The results show that the neural formulation is both more accurate and substantially faster, while still supporting practical height constraints. We also discuss extensions to rotationally symmetric and full three-dimensional reflector design through iterative correction schemes.

Figures & tables

Figure 1: Finite-source-to-far-field reflector system setup.

Figure 2: Finite-source-to-far-field reflector system approximation setup.

Symbol	Meaning
$\Omega$	Spatial support of the linear light source (source base domain)
$A$	Angular source domain (radian angles from positive $x$ -axis)
$\mathcal{S}$	Full source domain, $\mathcal{S}:=\Omega\times A$
$\Sigma$	Stereographic far-field target domain
$\mathcal{T}$	Full target domain, $\mathcal{T}:=\Omega\times\Sigma$
$f(s,\alpha)$	Source distribution function over $\mathcal{S}$

Table 1: Symbols used throughout the paper.

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 $m(s)=(2s+s^{3})/3$ for different values of $h$ computed using Eq. ( 35 ). 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.

Panel of Figure 5 — Figure 5: Ground-truth reflector, source distribution, corresponding target distribution, and far-field target distribution for Example A .

Panel of Figure 6 — Figure 6: Final reflectors, pointwise absolute reflector error, and convergence history for the neural-network and deconvolution methods applied to Example A . In (a) the ground-truth reflector is drawn as a dashed black curve on top of the predictions so that the otherwise-coincident neural-network curve remains visible. Panel (b) gives the pointwise absolute deviation from ground truth.

Panel of Figure 7 — Figure 7: Ground-truth reflector, source distribution, corresponding target distribution, and far-field target distribution for Example B .

Panel of Figure 8 — Figure 8: Final reflectors, pointwise absolute reflector error, and convergence history for the neural-network and deconvolution methods applied to Example B . In (a) the ground-truth reflector is drawn as a dashed black curve on top of the predictions so that the otherwise-coincident neural-network curve remains visible. Panel (b) gives the pointwise absolute deviation from ground truth.

Example	Method	Best RT NMAE	Time-to-best	Iterations	Final loss
A	NN	$2.16\times 10^{-5}$	$0.34$ s	$209$	$1.29\times 10^{-11}$
A	Deconv	$4.44\times 10^{-3}$	$410$ s	$900$	—
B	NN	$4.65\times 10^{-5}$	$2.54$ s	$278$	$6.74\times 10^{-12}$
B	Deconv	$5.02\times 10^{-2}$	$611$ s	$3000$	—

Table 2: Summary of accuracy and computational cost for the neural-network (NN) and deconvolution (Deconv) methods on Example A and Example B . “Best RT NMAE” is the smallest ray-traced NMAE attained over the run, “time-to-best” is the wall-clock time at which that minimum is reached, “Iterations” is the corresponding number of accepted/applied updates, and “Final loss” is the value of

\mathcal{L}(\boldsymbol{\theta})

at the end of training for the neural method (not defined for the deconvolution baseline). All runs are on a single NVIDIA RTX 4090 GPU.

Figure 9: NMAE for the neural-network and deconvolution methods as a function of minimum reflector height

h_{\min}

for Example C .

Figure 10: NMAE for the neural-network and deconvolution methods as a function of minimum reflector height

h_{\min}

for Example D .

Figure 11: Hyperparameter sensitivity for the direct change-of-variables method on Example A : ray-tracing NMAE as a function of optimization time for one-factor sweeps over (a) MLP depth, (b) MLP width, (c) far-field sample count

n_{\sigma}

, and (d)

p

-sample count

n_{p}

(five settings

\times

five seeds each). Solid curves are per-setting means in log space; shaded bands are 95 % confidence intervals. The dashed line is the single-run baseline from Figure 6 .

Figure 12: Hyperparameter sensitivity for the mesh-based method on Example B : ray-tracing NMAE as a function of optimization time for one-factor sweeps over (a) MLP depth, (b) MLP width, (c)

\sigma

-mesh resolution

n_{\sigma}^{\mathrm{mesh}}

, and (d)

p

-mesh resolution

n_{p}^{\mathrm{mesh}}

(five settings

\times

five seeds each). Solid curves are per-setting means in log space; shaded bands are 95 % confidence intervals. The dashed line is the single-run baseline from Figure 8 .

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Direct Optimization of a 3D Finite-Source Reflector via Neural-Network Parameterization

Roel Hacking, Lisa Kusch, Martijn Anthonissen, Wilbert IJzerman

Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands · Signify, High Tech Campus 7, 5656 AE Eindhoven, The Netherlands

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We present a direct optimization method for three-dimensional freeform reflectors that transform the light of a finite-étendue source into a prescribed far-field angular intensity distribution. The reflector profile is represented by a small neural network (a multilayer perceptron), which is trained end-to-end through a differentiable ray-tracing objective. We furthermore parameterize the emission directions in gnomonic coordinates, and show how we use this to ensure that every emitted ray intersects the reflector. At each iteration, the network is converted to a bicubic spline representation for ray-tracing efficiency, and intersections with this smooth surface are solved by a damped Newton solve, with gradients computed via the implicit function theorem. The traced output distribution is compared with the desired target on a 'soft' histogram, under an

H^{-1}

-type spectral weighting that emphasizes long-range transport of flux to improve convergence. Optimization is performed using a BFGS method with self-scaled Broyden updates and a plateau-perturbation rule to prevent stalling. The method converges reliably within seconds on a single GPU for all examples tested.

Abstract

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