# A Novel Iterative Thresholding Algorithm Based on Plug-and-Play Priors for Compressive Sampling
## 1 Introduction  
In compressive sampling [1; 2], we aim to estimate an image \(x_{0}\in\mathbb{C}^{n}\) from \(m<n\) linear observations \(y\in\mathbb{C}^{m},y=\mathbf{A}x_{0}\), where \(\mathbf{A}\in\mathbb{C}^{m\times n}\) is the measurement matrix. Since \(m<n\), this problem is ill-posed. 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. On the other hand, in Bayesian CS [8; 12], graphical models are employed to describe the probabilistic relationship between measurements and the original signal. In [8], the structure of the wavelet coefficients is modeled by a hidden Markov tree (HMT).  
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 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). Closely relating to D-AMP, BM3D-CS [15] iteratively adds noise to the missing part of the spectra and then appliesBM3D to the result. However, the above signal models cannot simultaneously utilize the global and nonlocal sparsity. Although the nonlocal sparse regularization can preserve image fine details by clustering similar components, it is based on block matching, and thus introduces visible artifacts in homogeneous regions, manifesting as low-frequency noise [17]. On the other hand, TV and wavelet sparse regularization can maintain the consistency of the entire image.  
In this paper, we propose an iterative thresholding method that is competitive in quality with D-AMP, but is much simpler to implement. We combine three popular denoisers through the use of the plug-and-play prior approach [18; 19], which is able to turn a denoiser into reconstruction solver. For charactering the nonlocal sparsity, we use the nonlocal wavelet denoiser. For representing the global sparsity, we use the wavelet and gradient denoiser. This multiple denoisers problem is then solved based on fast composite splitting technique [5] and the fast proximal method FISTA (fast iterative shrinkage-thresholding algorithm) [20; 21]. In the frame of FISTA, the original composite regularization problem is firstly decomposed into three simpler regularization sub-problems via fast composite splitting technique; then each of them is separately solved by thresholding methods. Experimental results show that our method impressively outperforms previous methods in terms of reconstruction accuracy, and also achieves competitive advantage in computation complexity.