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On-line Access: 2024-08-27

Received: 2023-10-17

Revision Accepted: 2024-05-08

Crosschecked: 2021-08-19

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 ORCID:

Na Lei

https://orcid.org/0000-0003-3361-0756

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Frontiers of Information Technology & Electronic Engineering  2021 Vol.22 No.9 P.1207-1220

http://doi.org/10.1631/FITEE.2000250


Robust and accurate optimal transportation map by self-adaptive sampling


Author(s):  Yingshi Wang, Xiaopeng Zheng, Wei Chen, Xin Qi, Yuxue Ren, Na Lei, Xianfeng Gu

Affiliation(s):  Department of Computer Science, Inner Mongolia University of Finance and Economics, Hohhot 010010, China; more

Corresponding email(s):   nalei@dlut.edu.cn, gu@cs.stonybrook.edu

Key Words:  Optimal transportation, Monge-Ampère equation, Self-adaptive sampling


Yingshi Wang, Xiaopeng Zheng, Wei Chen, Xin Qi, Yuxue Ren, Na Lei, Xianfeng Gu. Robust and accurate optimal transportation map by self-adaptive sampling[J]. Frontiers of Information Technology & Electronic Engineering, 2021, 22(9): 1207-1220.

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author="Yingshi Wang, Xiaopeng Zheng, Wei Chen, Xin Qi, Yuxue Ren, Na Lei, Xianfeng Gu",
journal="Frontiers of Information Technology & Electronic Engineering",
volume="22",
number="9",
pages="1207-1220",
year="2021",
publisher="Zhejiang University Press & Springer",
doi="10.1631/FITEE.2000250"
}

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%A Yingshi Wang
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%A Wei Chen
%A Xin Qi
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%A Xianfeng Gu
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%I Zhejiang University Press & Springer
%DOI 10.1631/FITEE.2000250

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T1 - Robust and accurate optimal transportation map by self-adaptive sampling
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A1 - Xiaopeng Zheng
A1 - Wei Chen
A1 - Xin Qi
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A1 - Na Lei
A1 - Xianfeng Gu
J0 - Frontiers of Information Technology & Electronic Engineering
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DOI - 10.1631/FITEE.2000250


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.

基于自适应采样的鲁棒精确最优传输映射

王应时1,郑晓朋2,陈伟2,3,齐鑫4,任玉雪3,雷娜2,3,顾险峰4
1内蒙古财经大学计算机系,中国呼和浩特市,010010
2大连理工大学软件学院,中国大连市,116620
3首都师范大学北京成像理论与技术高精尖创新中心,中国北京市,100048
4石溪大学计算机系,美国纽约州石溪镇,11794
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。

关键词:最优传输;Monge-Ampère方程;自适应采样

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

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