Optimal transport problem has gained much attention in image processing field, such as computer vision, image interpolation and medical image registration. In this paper, we incorporate optimal transport into linear inverse problems as a regularization technique. We establish a new variational model based on Benamou-Brenier energy to regularize the evolution path from a template to latent image dynamically. The initial state of the continuity equation can be regarded as a template, which can provide priors for the reconstructed images. 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 special 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：最优传输问题在图像处理领域，如计算机视觉、图像插值和医学图像配准中受到了广泛关注。本文以最优传输作为线性逆问题的正则化技术，基于Benamou-Brenier能量建立了一个新的变分模型，用以动态正则化从模板到潜在图像的演化路径。连续性方程的初始状态可视为模板，为重建图像提供先验信息。同时，我们分析了该变分问题在Radon测度空间中解的存在性。此外，针对一般成像问题，在特殊网格策略下构造了一阶原始-对偶算法进行求解。最后，针对欠采样MRI重建的数值实验表明，所提模型能够以高质量和结构保持性恢复图像。