Abstract
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.
摘要
最优传输在工程、 医疗等各领域扮演着重要角色, 包括图形学中的曲面参数化、计算机视觉中的注册、 深度学习中的生成模型等. 对于平方距离传输成本, 最优传输映射是Brenier势的梯度, 可通过求解Monge-Ampère方程得到. 此外, 最优传输映射可归结为几何凸优化问题. Monge-Ampère方程高度非线性, 在求解过程中, 中间解需要始终保持严格凸. 特别地, 离散解的精确性严重依赖于目标测度的采样. 因此, 提出一种自适应采样算法, 极大减少采样偏差, 同时提高离散解的精确性和鲁棒性. 实验结果验证了所提算法的有效性和高效性.
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Yingshi WANG, Na LEI, and Xianfeng GU offered the insights and proposed the algorithms. Xiaopeng ZHENG, Wei CHEN, Xin QI, and Yuxue REN implemented the algorithms and visualized the data. All the authors have collaboratively organized the materials, conducted the experiments, drafted the manuscript, and revised and finalized the paper.
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Yingshi WANG, Xiaopeng ZHENG, Wei CHEN, Xin QI, Yuxue REN, Na LEI, and Xianfeng GU declare that they have no conflict of interest.
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Project supported by the National Numerical Wind Tunnel Project, China (No. NNW2019ZT5-B13) and the National Natural Science Foundation of China (Nos. 61907005, 61772105, 61936002, and 61720106005)
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Wang, Y., Zheng, X., Chen, W. et al. Robust and accurate optimal transportation map by self-adaptive sampling. Front Inform Technol Electron Eng 22, 1207–1220 (2021). https://doi.org/10.1631/FITEE.2000250
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DOI: https://doi.org/10.1631/FITEE.2000250