A Wasserstein distance and total variation regularized model for image reconstruction problems

IF 1.2 4区数学 Q2 MATHEMATICS, APPLIED

Inverse Problems and Imaging Pub Date : 2023-01-01 DOI:10.3934/ipi.2023045

Yiming Gao

引用次数: 0

Abstract

Optimal transport has gained much attention in image processing fields, 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 problems 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.

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图像重建问题的Wasserstein距离和总变分正则化模型

最优传输在计算机视觉、图像插值和医学图像配准等图像处理领域受到广泛关注。最近，Bredies等人(ESAIM: M2AN 54:2351- 2382,2020)和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。本文将Wasserstein距离和总变分作为一种先验正则化方法引入到静态逆问题中。由Benamou-Brenier能量表示的Wasserstein距离度量给定模板与重建图像之间的相似性。同时，我们还分析了这类变分问题在Radon测量空间中解的存在性。在此基础上，构造了一阶原始对偶算法来解决特定网格策略下的一般成像问题。最后，对欠采样的MRI重建进行了数值实验，实验结果表明，该模型能较好地恢复图像，并能保持图像的结构。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Inverse Problems and Imaging 数学-物理：数学物理

CiteScore

2.50

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing. This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.