# Image Recovery with Data Missing in the Presence of Salt-and-Pepper Noise
## 5 Numerical Studies
### Data Missing Only  
In this section, the perforce of different algorithms is provided in the case of the data missing only. The original image and degraded image with 70% random missing are presented in Figure 4a,b, and the reconstructed images by proposed methods and other approaches are provided in Figure 4c-h. In this case, the framelet+APG approach is able to recover the original clean image in great details and hence produces the best PSNR of 29.1 dB. The second best PSNR is provided by the FoE approach, but burry effect is still present. In Figure 5, using 100 randomly selected images from SIPI Image Database, the indexes of PSNR, MAE and SSIM are provided in which the framelet+APG approach produces the best and consistent PSNR across the range of low and high missing levels. Especially, when the missing level is low, say 10%, the PSNR is more than 40 dB that is a 2 dB gain greater than the second best one, and when the missing level is high, say 70%, the PSNR obtained is still close to 35 dB. In addition to that, the value of the SSIM obtained by the framelet+APG is close to one even when the missing level is high as 70%. In summary, the recovery performance obtained by framelet+APG is satisfactory in terms of both visual and objective standards. Figure 6 shows the restoration results of different algorithms under a specific data missing pattern. A text message imbedded in the clean image is given in Figure 6a. The reconstructed images are provided in Figure 6b-f. However, all the methods are able to recover the original image, but the image obtained by the framelet+APG is visually superior to other schemes. Judging from the close-up portion of the Figure 6, compared to the other images obtained, the great details of the image obtained by the framelet+APG approach are preserved, demonstrated in Figure 7. The recovered text images in Figure 8 show again that the framelet+APG algorithm produces best result, whereas the text recovered by wavelet+ADMM shows faded effect. In practice, the illegal action of covering licence plate is somehow a common means to avoid penalties. To save manpower, the reconstruction of the original plate from the degraded one is desired, and the proposed methods are applied in attempt to that end. In order to simulate the illegal action, one number is completely missed in the original plate, shown in Figure 9b. In this data missing problem, the proposed approaches are applied to reconstruct the original plate and the results are provided in Figure 9c-f. It is observed that the APG based approaches are able to recover the original number, whereas there are still some missing parts in the image obtained by the ADMM based approaches. The experiment in this case is somehow ideal in the way that, if the intention is to avoid penalty, the entire plate will be covered. If that is the case, we suspect that the technology would not be of much use except for the local enforcements. Nonetheless, the proposed methods still will be useful in recovering the partially covered plates. Figure 3: Results of PSNR, MAE, and SSIM at different noise levels. (**a**) PSNR, (**b**) MAE, (**c**) SSIM. PSNR: peak signal-to-noise ratio, MAE: mean absolute error, SSIM: structural similarity index. Figure 4: Restoration results of different algorithms under random data missing only. (**a**) original image; (**b**) corrupted image with 70% missing (7.23 dB); (**c**) reconstructed image obtained by framelet+APG (29.1 dB); (**d**) reconstructed image obtained by framelet+ADMM (26.7 dB); (**e**) reconstructed image obtained by wavelet+APG (25.9 dB); (**f**) reconstructed image obtained by wavelet+ADMM (23.6 dB); (**g**) reconstructed image obtained by BPFA (27.8 dB); (**h**) reconstructed image obtained by FoE (28.3 dB). FoE: fields of experts. Figure 5: Results of PSNR, MAE, and SSIM at different missing levels. (**a**)PSNR, (**b**)MAE, (**c**) SSIM  
Figure 6: Restoration results of different algorithms under a specific data missing pattern.