End-to-end Surface Optimization for Light Control

TL;DR

This work tackles the inverse design of free-form optical surfaces to reproduce a target light pattern. It introduces an end-to-end optimization framework on a triangle mesh that couples a differentiable, flux-based rendering model with a face-based optimal transport initialization and a piecewise smoothness regularization to ensure manufacturability. The method directly minimizes the difference between rendered and target images, enabling high-fidelity caustic reproduction and producing physically fabricable surfaces validated through CNC-milled prototypes. The combination of differentiable rendering, iterative OT-guided initialization, and fabrication-aware regularization yields accurate light control with practical applicability in art, architecture, medical devices, and energy harvesting.

Abstract

Designing a freeform surface to reflect or refract light to achieve a target distribution is a challenging inverse problem. In this paper, we propose an end-to-end optimization strategy for an optical surface mesh. Our formulation leverages a novel differentiable rendering model, and is directly driven by the difference between the resulting light distribution and the target distribution. We also enforce geometric constraints related to fabrication requirements, to facilitate CNC milling and polishing of the designed surface. To address the issue of local minima, we formulate a face-based optimal transport problem between the current mesh and the target distribution, which makes effective large changes to the surface shape. The combination of our optimal transport update and rendering-guided optimization produces an optical surface design with a resulting image closely resembling the target, while the geometric constraints in our optimization help to ensure consistency between the rendering model and the final physical results. The effectiveness of our algorithm is demonstrated on a variety of target images using both simulated rendering and physical prototypes.

Figures (21)

• Figure 1: Pipeline of our algorithm. We iterate between a face-based optimal transport that adjusts correspondence between mesh faces and target image regions, and a rendering guided optimization that improves the details of the resulting image and reduces its difference from the target.
• Figure 2: Our rendering process. The light rays pass through the front surface $U$ of the lens without refraction. At each triangle $t_i$ on the back surfaces, the light rays refract according to the normal of $t_i$, forming a corresponding triangle $t'_i$ on the receptive plane. The total flux of $t'_i$ is determined by the area of the projection $P_U(t_i)$ from $t_i$ to $U$. These refracted triangles collectively form the resulting image $g$ after applying the inverse gamma correction $\gamma^{-1}(\cdot)$ (see Eq. \ref{['eq:image_term']}).
• Figure 3: An illustration of different lens shapes that produce the same distribution of light. We prefer a smoother lens shape as it facilitates fabrication.
• Figure 4: An illustration of our discrete optimal transport process. Given the current image triangles resulting from our rendering model, we associate the total flux of each triangle to its centroid $\mathbf{c}'_i$, and optimize a partition of the target image such that the flux at $\mathbf{c}'_i$ can be minimally transported to its corresponding cell $\Omega_i$ in the partition to match the target image. Then we compute the flux-weighted weighted centroid $\widetilde{\mathbf{c}}_i$ of each cell to guide the update of the lens shape.
• Figure 5: Our iterative optimization process. We show the resulting images derived from the intermediate results of our optimization process. The numbers on top are the resolution of the target image (the leftmost column) or the intermediate mesh (the other columns). The label at the bottom indicates the completed numbers of iterations for OT or rendering guided optimization at the current resolution.