Compressive Sensing via Nonlocal Low-Rank Regularization

Abstract

Sparsity has been widely exploited for exact reconstruction of a signal from a small number of random measurements. Recent advances have suggested that structured or group sparsity often leads to more powerful signal reconstruction techniques in various compressed sensing (CS) studies. 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 and justify the benefit of such a strategy using extensive experiments. To further improve the computational efficiency of the proposed algorithm, we have developed a fast implementation using the alternative direction multiplier method technique. Experimental results have shown that the proposed NLR-CS algorithm can significantly outperform existing state-of-the-art CS techniques for image recovery.

A Novel Iterative Thresholding Algorithm Based on Plug-and-Play Priors for Compressive Sampling

“…In order to solve this ill-posed problem, one needs some prior knowledge [3,4]. Therefore, researchers have explored the use of sophisticated structures to reflect the image priors, such as minimal total variation [5], wavelet trees [6,7], Markov mixture models [8], and non-local sparsity [9][10][11][12][13][14][15].…”

“…To establish these complicated models for image CS recovery, penalty functions are frequently used to encourage solutions of a certain form [5,6,11]. In FCSA (Fast Composite Splitting Algorithm) [5], sparsity penalty terms in wavelet and gradient domain are jointly used to constrain the solution space; while WaTMRI (wavelet tree sparsity magnetic resonance imaging) [6] superadds a tree sparse regularization to the objective function of FCSA, further forcing the parent-child wavelet coefficients to be zeros or non-zeros together.…”

“…Among these structures, nonlocal sparsity [11,12,15], which exploits the self-repetitive structure exhibited often in natural images has shown great potential. In NLR-CS (CS via nonlocal low-rank regularization) [11], a patch-based low-rank regularization model is built to enforce the low-rank property over the sets of nonlocal similar patches; while RCoS (CS recovery via collaborative sparsity) [14] uses a 3D sparsity term to maintain image nonlocal consistency.…”

“…In NLR-CS (CS via nonlocal low-rank regularization) [11], a patch-based low-rank regularization model is built to enforce the low-rank property over the sets of nonlocal similar patches; while RCoS (CS recovery via collaborative sparsity) [14] uses a 3D sparsity term to maintain image nonlocal consistency. In contrast, the D-AMP (denoising-based approximate message passing) algorithm [12] exploits the image self-similarity through the use of nonlocal based denoisers, such as the BM3D (block-matching and 3D filtering) denoiser [16], which performs hard or soft thresholding with a 3D orthogonal dictionary (3D filtering) on 3D image blocks built by stacking similar patches together (block-matching).…”

Reference Information Based Remote Sensing Image Reconstruction with Generalized Nonconvex Low-Rank Approximation

“…The principle of low-rank approximation is that similar patches are grouped to share a similar underlying structure and form a low-rank matrix appropriately [5,6], and it can be solved by using the efficient singular value decomposition as the optimization tool. Using the low-rank framework, Dong et al [7] proposed a nonlocal low-rank algorithm, called the spatially-adaptive iterative singular-value thresholding method. Making use of similar information from another relative band, it is easy to extend these nonconvex penalty functions on singular values of a matrix to improve low-rank matrix recovery performance.…”