A fast, large-scale optimal transport algorithm for holographic beam shaping

Optimal transport methods have recently established state of the art accuracy and efficiency for holographic laser beam shaping. However, use of such methods is hindered by severe $\mathcal{O}(N^2)$ memory and $\mathcal{O}(N^2)$ time requirements for large scale input or output images with $N$ total pixels. Here we leverage the dual formulation of the optimal transport problem and the separable structure of the cost to implement algorithms with greatly reduced $\mathcal{O}(N)$ memory and $\mathcal{O}(N\log N)$ to $\mathcal{O}(N^{3\mathrm{/}2})$ time complexity. These algorithms are parallelizable and can solve megapixel-scale beam shaping problems in tens of seconds on a CPU or seconds on a GPU.