Compressive Sensing via Nonlocal Low-Rank Regularization论文总结

Abstract：

  In this paper, we propose a nonlocal low-rank regularization (NLR) approach toward exploiting structured sparsity and explore its application into CS of both photographic and MRI images. 

   We also propose the use of a nonconvex log det(X) as a smooth surrogate function for the rank instead of the convex nuclear norm .

I.Introduction:

    In this paper, we propose a unified （统一的）variational（变化的） framework for nonlocal low-rank regularization of CS recovery. 

   To exploit the nonlocal sparsity of natural or medical images, we propose to regularize the CS recovery by patch grouping and low-rank approximation. 

    Specifically, for each exemplar（标本） image patch we group a set of similar image patches to form a data matrix X. Since each patch contain similar structures, the rank of this data matrix X is low implying a useful image prior. To more efficiently solve the problem of rank minimization, we propose to use the log det(X) as a smooth surrogate function for the rank (instead of using the convex nuclear norm), which lends itself to iterative singular-value thresholding. 

II. BACKGROUND

III. NONLOCAL LOW-RANK REGULARIZATION FOR CS RECOVERY

    The proposed regularization model consists of two components: patch groupingfor characterizing self-similarity of a signal and low-rank approximation for sparsity enforcement.

      所提出方法的基本假设是自相似性在我们的信号中是丰富的。可以发现大量的大小为  的相似块，在位置i，表示为，对于每个样本块，我们可以用K-近邻搜索得到一个局部窗口

                                         

 T是设定的阈值，Gi是这些相似块位置的集合。对于每个样本块可以得到一个数据矩阵=，

的每一列表示一个和相似的块。

     因为存在噪声，所以将数据矩阵建模为=， Li  ：the low-rank matrix ，Wi ：the Gaussian noise matrix 

可以用如下优化问题来解决：