A c-transform optimal transport method for high-contrast freeform reflector design
Abstract
Design of high-contrast freeform reflectors is challenging, as the presence of zero-intensity regions leads to degeneracy in the associated Monge-Ampere-type equation. A common remedy is to add a positive artificial background to the target intensity, which improves the regularity of the equation. However, this regularization inevitably reduces the achievable illumination contrast and introduces nonzero intensity into regions that are intended to remain dark. We propose a fast c-transform method based on optimal transport duality for high-contrast freeform reflector design in the far field without artificial background regularization. The method integrates repeated discrete c-transforms into a measure-based dual optimization scheme to maintain the c-concavity of the reflector potential throughout the iteration. This preserves the admissibility of the iterates and enables stable convergence for targets with zero-intensity regions. To efficiently compute the discrete c-transforms, we develop a localized search algorithm that exploits their contact structure and prove its exactness and linear complexity. Numerical experiments demonstrate that the proposed method accurately realizes high-contrast illumination targets with zero background.
Related
- Optimal distributional estimates of the multiple Hilbert transform
- Convergence of Coordinate Ascent Variational Inference for log-concave measures via optimal transport
- Hausdorff dimension of the singular set in optimal transport
- Optimal regularity of optimal transport and degenerations of convex cones
- A Uniform Bound on Optimal Strategy Length in Water Transport Problem
BibTeX
@misc{bao2026ctransform,
title = {{A c-transform optimal transport method for high-contrast freeform reflector design}},
author = {Gang Bao and Yixuan Zhang and Jiaqi Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19663},
eprint = {2609.19663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19663},
}
RIS
TY - UNPB
TI - A c-transform optimal transport method for high-contrast freeform reflector design
AU - Gang Bao
AU - Yixuan Zhang
AU - Jiaqi Yu
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19663
UR - https://arxiv.org/abs/2609.19663
AB - Design of high-contrast freeform reflectors is challenging, as the presence of zero-intensity regions leads to degeneracy in the associated Monge-Ampere-type equation. A common remedy is to add a positive artificial background to the target intensity, which improves the regularity of the equation. However, this regularization inevitably reduces the achievable illumination contrast and introduces nonzero intensity into regions that are intended to remain dark. We propose a fast c-transform method based on optimal transport duality for high-contrast freeform reflector design in the far field without artificial background regularization. The method integrates repeated discrete c-transforms into a measure-based dual optimization scheme to maintain the c-concavity of the reflector potential throughout the iteration. This preserves the admissibility of the iterates and enables stable convergence for targets with zero-intensity regions. To efficiently compute the discrete c-transforms, we develop a localized search algorithm that exploits their contact structure and prove its exactness and linear complexity. Numerical experiments demonstrate that the proposed method accurately realizes high-contrast illumination targets with zero background.
ER -