发表机构
School of Mathematical Sciences, Zhejiang University; School of Mathematical Sciences, Peking University（浙江大学数学科学学院; 北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高对比度自由曲面反射器设计中零强度区域导致的方程退化问题，提出一种基于最优传输对偶的快速c-变换方法，无需人工背景正则化，通过局部搜索算法保持c-凹性，实现稳定收敛并精确实现零背景高对比度照明。
AI 中文摘要
高对比度自由曲面反射器的设计具有挑战性，因为零强度区域的存在导致相关的Monge-Ampere型方程出现退化。常见的补救措施是向目标强度添加正的人工背景，这改善了方程的 regularity。然而，这种正则化不可避免地降低了可实现的照明对比度，并将非零强度引入到本应保持黑暗的区域。我们提出了一种基于最优传输对偶的快速c-变换方法，用于远场中无人工背景正则化的高对比度自由曲面反射器设计。该方法将重复的离散c-变换集成到基于测度的对偶优化方案中，以在整个迭代过程中保持反射器势的c-凹性。这保持了迭代的容许性，并使得对于具有零强度区域的目标能够稳定收敛。为了高效计算离散c-变换，我们开发了一种利用其接触结构的局部搜索算法，并证明了其精确性和线性复杂度。数值实验表明，所提出的方法能够准确实现具有零背景的高对比度照明目标。
英文摘要
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.
