# Constrained Backtracking Matching Pursuit Algorithm for Image Reconstruction in Compressed Sensing
## 4 Experimental Results
From Figure 5, it is noticed that if \(a\leq\frac{1}{4\sqrt{2}}\), all the PSNR values of CBMP with different step sizes are greater than 31. Therefore, the introduction of the threshold operation is necessary for the improvement of greedy pursuit algorithms. At the same time, threshold parameters do not distinctively affect the reconstruction performance if they are satisfied with the constrained condition in CBMP. Meanwhile, the reconstruction performance of CBMP with \(a=\frac{1}{16\sqrt{2}}\) is better than the others; thus, this \(a\) value is regarded as the optimal value in the CBMP. Tables 1 and 2 compare CBMP, SAMP, and IGSAMP in terms of the reconstruction performance (PSNR) on the Lena image with different sampling ratios and initial step sizes. Tables 3 and 4 compare CBMP, SAMP, and IGSAMP in terms of the reconstruction performance (PSNR) on the Peppers image with different sampling ratios and initial step sizes. In Table 1, when the sampling ratio is 0.3, each PSNR value of the CBMP algorithm is greater than that of SAMP and IGSAMP. For example, with the initial step size of 64, the PSNR value of SAMP and IGSAMP is 25.45 dB and 26.23 dB, respectively, but the PSNR value of CBMP is 32.13 dB. Table 2 shows the PSNR values of SAMP, IGSAMP, and CBMP with the same sampling ratio of 0.4. Different step sizes are used. The PSNR values of SAMP with different initial step sizes range from 26.99 dB to 31.60 dB. The PSNR values of IGSAMP are increased from 27.32 dB to 32.46 dB. However, the PSNR values of CBMP are greater than those of SAMP and IGSAMP, achieving 33.9675 dB as the average value. Similarly, Table 3 shows the PSNR values of the Peppers image by SAMP, IGSAMP, and CBMP when the sampling ratio is 0.3. Each PSNR value of the CBMP algorithm is greater than that of SAMP and IGSAMP. Table 4 shows the PSNR values of the Peppers image by SAMP, IGSAMP, and CBMP when the sampling ratio is 0.4. Different step sizes are used. For example, as the initial step size is 512, the PSNR value of SAMP and IGSAMP  
Finally, the reconstructed results of the Lena image using SAMP, IGSAMP, and CBMP are shown in Figures 6 and 7. The sampling rate is 0.3; different step sizes are used. The reconstructed results of the Peppers image using SAMP, IGSAMP, and CBMP are shown in Figures 8 and 9.  
different stages and accurately estimate the true support set of the image to be reconstructed. On the one hand, the energy criterion between the estimated signal and the measurement is used as a constraint. On the other hand, the four-to-one practical rule is considered and improved as another control. Due to the introduction of these composite mechanisms, the reconstruction performance of the proposed algorithm outperforms the greedy pursuit algorithms, including SAMP and IGSAMP. In particular, CBMP offers a stable reconstruction performance, which is insensitive to the initial step size. In our future works, the CBMP algorithm will be applied to neural network framework-based signal reconstruction, including medical image reconstruction. Conceptualization, X.B. and L.L. ; Formal analysis, L.L. ; Funding acquisition, L.L., X.B. and Y.D. ; Investigation, X.B. and L.L. ; Methodology, X.B. ; Supervision, C.K., Y.D. and F.L. ; Writing original draft, X.B. ; Writing review--editing, X.B., L.L., C.K., X.L. and F.L. All authors have read and agreed to the published version of the manuscript. The work is supported by the National Natural Science Foundation of China (Nos. 61866028, 61872298), the Chunhui Project of the Ministry of Education Project Foundation of China (No. Z2017075), the Sichuan Provincial Department of Education Foundation (No. 17 ZB0416) and the Key Project Foundation of Xihua University (No. Z1520908). Not applicable. The authors declare no conflict of interest.