最优传输理论及其在图像处理中的应用

A review of optimal transport in image processing

最优传输问题是寻找概率测度间的最优传输变换的一类特殊的优化问题,近年来在众多领域得到了广泛的关注.针对传统最优传输问题存在的计算量过大、正则性缺失等问题,学者们提出了多种改进的最优传输模型和算法,用于处理实际中的各种问题.简述最优传输问题的基本理论和方法,介绍Wasserstein距离的概念及其衍生出的Wasserstein重心,探讨离散化最优传输模型及其在正则化等方面的改进,讨论求解最优传输问题的算法进展,综述Wasserstein距离在图像处理领域的简单应用,并展望有待进一步研究的工作.

The optimal transport problem which has attracted wide attentions in many fields in recent years,is a special kind of optimization problem discussed in the probabilistic measure space.In order to overcome the disadvantages of traditional optimal transport models,such as complex computation and lack of regularity,many different kinds of improved optimal transport models and algorithms are proposed to deal with various practical problems.Firstly,this paper briefly describes the basic theory and methods of optimal transport,and further introduces the concept of Wasserstein distance and Wasserstein barycenters.And then,the discrete optimal transport model and the improved regularization models are discussed.Besides,a short summary of the algorithms to solve optimal transport problem is given.Then,from Wasserstein distance aspect,a review of applications in several areas of image processing is briefly discussed.At last,the further research work is prospected.