Optimal transport has gained much attention in image processing field, such as computer vision, image interpolation and medical image registration. Recently, Bredies et al. (ESAIM:M2AN 54:2351-2382, 2020) and Schmitzer et al. (IEEE T MED IMAGING 39:1626-1635, 2019) established the framework of optimal transport regularization for dynamic inverse problems. In this paper, we incorporate Wasserstein distance, together with total variation, into static inverse problems as a prior regularization. The Wasserstein distance formulated by Benamou-Brenier energy measures the similarity between the given template and the reconstructed image. Also, we analyze the existence of solutions of such variational problem in Radon measure space. Moreover, the first-order primal-dual algorithm is constructed for solving this general imaging problem in a specific grid strategy. Finally, numerical experiments for undersampled MRI reconstruction are presented which show that our proposed model can recover images well with high quality and structure preservation.

翻译：最优传输在图像处理领域（如计算机视觉、图像插值与医学图像配准）中受到广泛关注。近期，Bredies等人（ESAIM:M2AN 54:2351-2382, 2020）与Schmitzer等人（IEEE T MED IMAGING 39:1626-1635, 2019）建立了动态逆问题的最优传输正则化框架。本文提出将Wasserstein距离与全变分相结合，作为静态逆问题中的先验正则项。其中，基于Benamou-Bourier能量公式的Wasserstein距离用于度量给定模板与重建图像之间的相似性。同时，我们分析了该变分问题在Radon测度空间中的解存在性。进一步地，构建了一种基于特定网格策略的一阶原始-对偶算法以求解该通用成像问题。最后，针对欠采样MRI重建的数值实验表明，所提模型能够高质量地恢复图像并有效保持结构信息。