## 3 Results  
In the experiments, four true CT images and simulated projections were used to study the performance of our algorithm under degraded conditions. The experimental setting is as follows: The image size is set to be \(512\times 512\). The known projection data are obtained by convolution of the CT system matrix \(\mathbf{A}\) and the true CT image. The size of the CT system matrix is different according to different image sizes. The system matrix of the fan beam scanning mode [25] is used in the experiment. The point spread function of the CT image is approximated as a standard Gauss function process with different sizes, \(20\times 20\), \(27\times 27\), \(18\times 18\), \(15\times 15\), respectively and the same variance, \(\sigma^{2}~{}=~{}1\). The white Gaussian noise is added to the known projection data to obtain the known noisy projection data with signal-to-noise ratio (SNR) 20 dB and 40 dB. All known measurement projection data is acquired through binary matrices to verify the applicability of the method. The sampling ratio varies from 40%-100%. To simulate fewer data and make a comparative analysis, a few data were obtained by random sampling of complete data. The sampling ratio rate represents the amount of sample data. The lower the sampling rate, the fewer the data used in the experiment. In other words, fewer data represent lower radiation doses. One can initialize the point spread function to a Dirac delta function. \(\psi\) is a sparse representation matrix and represents wavelet sparse transform in this paper [14]. Here the Haar basis was chosen as wavelet transformation basis. The stopping criterion of the iteration is accomplished by relative error. That is \(\|\mathbf{h}(l+1)-\mathbf{h}(l)\|/\|\mathbf{h}(l)\|\leq tol\) and \(tol\) is a relatively small constant. In the simulation experiment, \(tol~{}=~{}10^{-6}\). When the relative error is less than \(tol\), the algorithm terminates. Figures 1-4 present CT image blind restoration and reconstruction results using the proposed approach for four different images degraded with different Gaussian PSF. Good results were obtained in terms of clarity contrast, detail retention, and so on. Although the image quality decreases with the decrease in the sampling ratio, it is acceptable in general. From Figures 1-4, it is clear that the proposed algorithm has good reconstruction performance in the case of high sampling ratio. With the decrease of sampling ratio, the image quality also decreases. Moreover, until a lower sampling ratio of 0.4, an intuitive and acceptable image can be reconstructed for medical diagnosis services. In the case of high signal-to-noise ratio, the proposed method can obtain better reconstruction results at a sampling rate of 40%. Other methods cannot achieve good reconstruction at this sampling rate. With the decrease of the signal-to-noise ratio, the reconstruction results can be obtained at the sampling rate of 40%, but the quality will decrease significantly. The reconstruction results of the other three methods can be obtained in the Supplementary Materials. For quantitative comparison, the peak signal-to-noise ratio (PSNR) and structural similarity index metric (SSIM) are used to evaluate the algorithm performance at SNR = 40 dB. About these two metrics, the ideal value of PSNR is +\(\infty\), the ideal values of SSIM are 1. These two metrics can only be used in the simulated experiments because they require a referred image. The above two metrics tabulated for each experiment are the average values over 10 times repetitions. The results are shown in Tables 2 and 3. The reported values are the average and standard deviation of the 10 runs. The less standard deviation judges the robustness of the proposed approach method. Because the compound Gaussian distribution provides better prior information, the reconstruction method based on hierarchical Bayesian structure has better performance in the case of high sampling rate. Figure 1: Experimental images and results of the first CT image.