# Image Recovery with Data Missing in the Presence of Salt-and-Pepper Noise
## 5 Numerical Studies
### Salt-and-Pepper Noise and Data Missing  
Finally, in this section, the experiments are conducted in the presence of both salt-and-pepper noise and data missing, which presents a very challenging case to recover the original image. In Figure 10b, the 40% salt-and pepper noise and the 40% data missing are utilized to corrupt the image of _lena_, and the restoration results obtained by the proposed methods are provided in Figure 10c-f. One observes from these figures that the framelet+APG produces a clear image with PSNR of 29.3 dB that is 22.27 dB greater than the corrupted image and 2.9 dB greater than the framelet+ADMM approach. The results of image _house_ in Figure 11 also confirm the conclusion from the image of _lena_. The PSNRs, MAEs, and SSIMs under the different percentages of the noise and data missing are also provided in Tables 1-3. Observing Table 1 reveals that, even when the corruption level is high, say 40%+40%, the highest PSNR of the reconstructed image is 29.06 dB, which is 3.92 dB greater than the lowest PSNR produced by wavelet+ ADMM. From Table 2, one observes that the MAEs obtained by the framelet+APG are the lowest under all cases, suggesting the better recovery performance, and the SSIMs obtained by the framelet+APG are always close to the one, presented in Table 3.  
Finally, the proposed methods are applied to the scenario, where the data missing comes from the text imbedding into a salt-and-pepper noise corrupted image, demonstrated in Figure 12a. The reconstructed images are depicted in Figure 12b-e, where the images obtained by framelet+ADMM, wavelet+APG, and wavelet+ADMM still show the text residuals. On the other hand, the image reconstructed by framelet+APG is satisfactory and preserves great details.  
Figure 10: Restoration results of different algorithms under both salt-and-pepper noise and data missing for image _Cameraman_. (**a**) original image; (**b**) corrupted image with 40% salt-and pepper noise and 40% missing (7.03 dB); (**c**) reconstructed image obtained by framelet+APG (29.3 dB); (**d**) reconstructed image obtained by framelet+ADMM (26.4 dB); (**e**) reconstructed image obtained by wavelet+APG (26.1 dB); (**f**) reconstructed image obtained by wavelet+ADMM (24.7 dB).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Salt-and-Pepper Noise+Missing** & **20\% + 20\%** & **30\% + 30\%** & **40\% + 40\%** \\ \hline framelet+APG & 34.42 & 31.57 & 29.26 \\ framelet+ADMM & 33.17 & 29.56 & 27.35 \\ wavelet+APG & 32.29 & 29.13 & 27.04 \\ wavelet+ADMM & 30.57 & 27.46 & 25.34 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Peak signal-to-noise ratios (PSNRs) for the lena image with different levels of salt-and-pepper noise and data missing.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Salt-and-Pepper Noise+Missing** & **20\% + 20\%** & **30\% + 30\%** & **40\% + 40\%** \\ \hline framelet+APG & 1.54 & 2.57 & 3.82 \\ framelet+ADMM & 2.52 & 3.96 & 5.79 \\ wavelet+APG & 4.76 & 7.06 & 10.42 \\ wavelet+ADMM & 6.23 & 9.87 & 15.43 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mean absolute errors (MAEs) for the lena image with different levels of salt-and-pepper noise and data missing.  
Figure 11: Restoration results of different algorithms under both salt-and-pepper noise and data missing for image _house._ (**a**) original image; (**b**) corrupted image with 40% salt-and pepper noise and 40% missing (7.51 dB); (**c**) reconstructed image obtained by framelet+APG (31.4 dB); (**d**) reconstructed image obtained by framelet+ADMM (28.6 dB); (**e**) reconstructed image obtained by wavelet+APG (29.1 dB); (**f**) reconstructed image obtained by wavelet+ADMM (26.9 dB).