Optimal transportation plays a fundamental role in many fields in engineering and medicine, including surface parameterization in graphics, registration in computer vision, and generative models in deep learning. For quadratic distance cost, optimal transportation map is the gradient of the Brenier potential, which can be obtained by solving the Monge-Ampère equation. Furthermore, it is induced to a geometric convex optimization problem. The Monge-Ampère equation is highly non-linear, and during the solving process, the intermediate solutions have to be strictly convex. Specifically, the accuracy of the discrete solution heavily depends on the sampling pattern of the target measure. In this work, we propose a self-adaptive sampling algorithm which greatly reduces the sampling bias and improves the accuracy and robustness of the discrete solutions. Experimental results demonstrate the efficiency and efficacy of our method.

最佳运输在工程和医学的许多领域都发挥着基础性作用，包括图形中的表面参数化、计算机视觉中的配准以及深度学习中的生成模型。对于二次距离成本，最优运输图是 Brenier 势的梯度，可以通过求解 Monge-Ampère 方程得到。此外，它被引入几何凸优化问题。Monge-Ampère 方程是高度非线性的，在求解过程中，中间解必须是严格凸的。具体来说，离散解的准确性在很大程度上取决于目标度量的采样模式。在这项工作中，我们提出了一种自适应采样算法，该算法大大降低了采样偏差，提高了离散解的准确性和鲁棒性。