# A Novel Iterative Thresholding Algorithm Based on Plug-and-Play Priors for Compressive Sampling
## 2 Compressive Sampling and Fast Iterative Shrinkage-Thresholding Algorithm
Note that PPP-CS spends 62.05% less CPU time to achieve higher PSNR than D-AMP, for the result that the Bayesian algorithm AMP is slower than our fast iterative shrinkage-thresholding algorithm. Figure 1: Twelve experimental test images. Figure 4: Average CPU time (s) comparisons on the 12 test images with various sampling ratios. Figure 3: Average PSNR (dB) of different images with 20% sampling. Figure 2: Average PSNR (peak signal to noise ratio) (dB) comparisons on the 12 test images with various sampling ratios. Figure 5 gives the performance comparisons between different methods in terms of the CPU time over the PSNR with 20% sampling. The PPP-CS obtains the best reconstruction results by achieving the highest PSNR in less CPU time after five seconds. Since PPP-CS is based on FISTA, it can converge fast. The D-AMP is inferior to the PPP-CS for most of the time, because of its higher cost in each iteration. These results show the effectiveness of acceleration strategies in the PPP-CS inherited from the FISTA for the image reconstruction. ### Visual Quality Evaluation

Figure 6 provides the CS recovery results for image Cameraman in the case of sample rate = 20% using the test algorithms. For each algorithm, we compute the PSNR as well as the structural similarity index (SSIM) which better reflects the visual quality of the images. From Figure 6, it is clear to see that the nonlocal sparsity-based methods PPP-CS and D-AMP are still better than others. The proposed algorithm enjoys great advantages over other competing algorithms in producing more clean screens, e.g., on the area of sky. Due to the joint use of the global and nonlocal sparsity and adaptive thresholding strategy, the elimination of artifacts is more effective by applying PPP-CS. \[x^{k}=soft(x_{g},\lambda c)=\text{sgn}(x_{g})\cdot\text{max}(abs(x_{g})- \lambda c,0). \tag{2}\]0 Performance comparison (CPU time vs. PSNR) with 20% sampling. \[x^{k}=soft(x_{g},\lambda c)=\text{sgn}(x_{g})\cdot\text{max}(abs(x_{g})- \lambda c,0). \tag{2}\]1 _Cont._

The corresponding the iterations vs. PSNR curves and iterations vs. SSIM curves are given in Figures 7 and 8. Note that our algorithm achieves significant improvement in comparison with other algorithms both visually and quantitatively (PSNR and SSIM). \[x^{k}=soft(x_{g},\lambda c)=\text{sgn}(x_{g})\cdot\text{max}(abs(x_{g})- \lambda c,0). \tag{2}\]2 Visual quality comparison of image CS recovery using different algorithms for image “Cameraman” in the case of sample rate = 20%. (**a**) The original; (**b**) FCSA (PSNR = 25.92 dB, SSIM = 0.6784); (**c**) WaTMRI (PSNR = 27.07 dB, SSIM = 0.7451); (**d**) Turbo-AMP (PSNR = 27.48 dB, SSIM = 0.8523); (**e**) D-AMP (PSNR = 31.01 dB, SSIM = 0.9329); (**f**) PPP-CS (PSNR = 32.73 dB, SSIM = 0.9448). \[x^{k}=soft(x_{g},\lambda c)=\text{sgn}(x_{g})\cdot\text{max}(abs(x_{g})- \lambda c,0). \tag{2}\]3 Performance comparison (iterations vs. PSNR) on “Cameraman” with 20% sampling. ## 5 Conclusions

A new image compressive sampling recovery algorithm based on the FISTA algorithm and plug-and-play priors is proposed in this paper. Our contributions are as follows: (1) jointly using of the global and nonlocal sparsity to maintain entire consistency and preserve details in image CS recovery, which can be achieved through the use of the plug-and-play prior approach; (2) applying the FISTA algorithm and fast composite splitting technique to turn the complicated composite constraint problem into three independent shrinkage-thresholding problems, and obtain a simple implementation and a fast convergence rate. Finally, we combine the FISTA algorithm with plug-and-play priors for the first time, making it easy to use the powerful denoisers in CS reconstruction. In the experiments, our algorithm is shown to outperform four excellent algorithms in reconstructed quality, and also achieve competitive advantage in runtime. This work is partially supported by the National Natural Science Foundation of China (No. 61302120), the Science and Technology Planning Project of Guangdong Province (No.