# A Novel Iterative Thresholding Algorithm Based on Plug-and-Play Priors for Compressive Sampling
## 2 Compressive Sampling and Fast Iterative Shrinkage-Thresholding Algorithm
For step (c), the wavelet denoisers are available; but we can solve this sub-problem easily, since it has a close form solution, which can be computed with soft thresholding (see Equation (2)). For step (d), we use the BM3D denoiser [15]. The key to success in solving Equation (3) is to determine the denoising threshold of the BM3D denoiser. The original threshold of BM3D is proportional to the noise variance of image; however, it is hard to estimate the noise variance of the current image in the CS reconstruction. Apparently, the denoising threshold \(\delta\) should be reduced along with the iteration as the recovered image becomes more and more clear. Therefore, a natural idea is that it should be proportional to the observation residual in CS reconstruction. \[\delta=s\sqrt{\|y-Ax_{g}\|_{2}^{2}/m}, \tag{6}\]

where \(s\) is a scale factor and can be set empirically. ## 4 Experiments

To evaluate reconstruction performances of the proposed algorithm in items of average PSNR (peak signal to noise ratio), runtime, and visual quality, we compare our algorithm with four excellent algorithms. There are two algorithms based on iterative shrinkage-thresholding method FCSA [1] (using wavelet sparsity and gradient sparsity), WaTMRI [2] (using wavelet sparsity, gradient sparsity and tree sparsity) and two Bayesian CS methods Turbo-AMP [3] (using tree sparsity), D-AMP [6] (using nonlocal sparsity via BM3D denoiser). Three experiments are carried on both natural images and MR (Magnetic Resonance) images with size 256 \(\times\) 256 at four sampling ratios (18, 20, 22, and 25%). The eight natural images and four MR images used in our experiments are shown in Figure 1. The undersampled Fourier matrix is generated by following the previous works [2; 3], which randomly choose more Fourier coefficients from low frequency and less on high frequency. All experiments are on a desktop with 3.80 GHz AMD A10-5800K CPU. Matlab version is 7.11 (R2010b). We set the maximum iterations \(K=10\) for Turbo-AMP and set \(K=50\) for the rest algorithms. For our algorithm PPP-CS, we use the setting \(\alpha=0.001,\beta=0.023\), \(c=1\). The scale factor in Equation (6) is set to 30 in the first 20 iterations, and is fixed at 5 after 20 iterations; since the estimated threshold \(\delta\) is basically steady at that moment. The 3D wavelet transform \(\Gamma_{\text{3D}}\) is composed of 2-D bior1.5 and 1-D Haar. To construct 3D groups by stacking similar patches, we need to set the following parameters: the size of each patch is \(8\times 8\) pixels the size of window for searching matched patches is \(25\times 25\) pixels; the number of best matched patches is 16; and the sliding step to process every next reference patch is set to 6. After the first five iterations, step (d) is only updated every three iterations in order to speed up the algorithm. For the rest algorithms, the default settings in their codes are used. ### Quantative Evaluation

We carry all algorithms on the 12 test images with various sampling ratios, the average PSNR results by running each image five times are shown in Figure 2, from which we can see that the nonlocal sparsity-based methods PPP-CS and D-AMP are better than others. Particularly, the proposed algorithm PPP-CS achieves the highest PSNR results. By jointly using of the global and nonlocal sparsity, the proposed PPP-CS method performs much better than the D-AMP method on all sensing rates. The PSNR results of reconstructed images from the CS measurements with 20% sampling are included in Figure 3. One can see that the proposed PPP-CS method produces the highest PSNRs on almost all test images (except for three cases where PPP-CS slightly falls behind D-AMP). The average PSNR improvement over D-AMP is about 10.54 dB, which validates the superiority of the proposed algorithm in objective quality. Figure 4 gives the average CPU time of different algorithms carried on the 12 test images by running each 5 times. FCSA spends the least CPU time; however, its PSNR results in Figure 1 are relatively poor.