# Recovering Texture with a Denoising-Process-Aware LMMSE Filter
## 4 Experimental Results
### Image Denoising
\begin{table}
\begin{tabular}{c c c c} \hline  & **BM3D** & **GHP** & **WNNM** & **Proposed** \\ \hline Image01 & 28.51/0.817 & 28.57/0.818 & 28.75/0.822 & **28.83**/**0.824** \\ Image02 & 28.03/0.688 & 28.03/**0.700** & 28.07/0.684 & **28.13**/0.694 \\ Image03 & 26.08/0.719 & 26.21/0.738 & 26.29/0.727 & **26.39**/**0.740** \\ Image04 & 24.57/0.703 & 24.68/0.718 & 24.81/0.714 & **24.92**/**0.729** \\ Image05 & 26.04/0.813 & 26.28/0.819 & 26.38/0.823 & **26.51**/**0.831** \\ Image06 & 28.22/0.762 & 28.43/**0.778** & 28.50/0.770 & **28.59**/0.776 \\ Image07 & 29.37/0.756 & 29.24/0.757 & 29.43/0.755 & **29.45**/**0.759** \\ Image08 & 25.13/0.724 & 25.34/0.744 & 25.46/0.738 & **25.60**/**0.751** \\ Image09 & 29.48/0.854 & 29.47/0.855 & 29.65/**0.856** & **29.70**/0.855 \\ Image10 & 26.00/0.718 & 26.09/0.733 & 26.14/0.725 & **26.25**/**0.738** \\ \hline Average & 27.14/0.755 & 27.24/0.766 & 27.35/0.761 & **27.44**/**0.770** \\ \hline \end{tabular}
\end{table} 
Table 3: The PSNR [dB] and SSIM results obtained after denoising on dataset I (with a noise standard deviation of \(\sigma=30\)). \begin{table}
\begin{tabular}{c c c} \hline  & **WNNM** & **Proposed** \\  & **(WNNM + Texture Recovery)** \\ \hline \(\sigma=10\) & 149 & 169 \\ \(\sigma=20\) & 148 & 167 \\ \(\sigma=30\) & 311 & 334 \\ \hline \end{tabular}
\end{table} 
Table 5: The averages of computational time (in seconds) of WNNM and the proposed method. A histogram of the \(\alpha_{\text{SSIM}}\) values obtained on BSD100 is shown in Figure 6. We note that this histogram shows the frequency under three noise levels (\(\sigma=10\), \(20\), \(30\)). This figure indicates that there is no single optimal value of \(\alpha_{\text{SSIM}}\) that is common to all images. Nevertheless, the figure also shows that choosing \(\alpha=2\) will almost always increase the expected value of the SSIM. Moreover, as with \(\alpha_{\text{SSIM}}\), we searched for the value of \(\alpha\) that would maximize the PSNR of the output image, and denoted it by \(\alpha_{\text{PSNR}}\). We obtained \(\alpha_{\text{PSNR}}\) for each image via a line search in the range from \(0\) to \(8\) in increments of \(0.01\). Figure 7 shows a histogram of the values of \(\alpha_{\text{PSNR}}\). We confirmed that \(\alpha=1\) produces the denoised images with the best PSNR in most cases. Figure 4: Performance comparison of GHP, WNNM, and the proposed method. Figure 5: True texture \(\mathbf{t}\) and estimated texture \(\hat{\mathbf{t}}\) of the proposed method. The pixel value of each image is multiplied by a factor of 3. Figure 6: Histogram of the \(\alpha\) values that maximize the SSIM for each output image. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **WNNM** & **Proposed** & **DnCNN** \\ \hline Image01 & 32.38/0.902 & **32.45**/**0.905** & 32.70/0.911 \\ Image02 & 31.00/0.824 & **31.13**/**0.834** & 31.38/0.845 \\ Image03 & 29.53/0.853 & **29.67**/**0.864** & 29.89/0.872 \\ Image04 & 28.37/0.847 & **28.45**/**0.856** & 28.59/0.860 \\ Image05 & 30.30/0.916 & **30.37**/**0.919** & 30.53/0.922 \\ Image06 & 31.69/0.876 & **31.81**/**0.882** & 31.82/0.886 \\ Image07 & 32.37/0.852 & **32.45**/**0.858** & 32.68/0.867 \\ Image08 & 29.20/0.862 & **29.32**/**0.871** & 29.58/0.877 \\ Image09 & 33.00/0.919 & **33.08**/**0.921** & 33.35/0.927 \\ Image10 & 29.47/0.858 & **29.60**/**0.867** & 29.79/0.874 \\ \hline Average & 30.73/0.871 & **30.83**/**0.878** & 31.03/0.884 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of non-learning-based methods (WNNM and the proposed method) and a learning-based method (DnCNN). The PSNR [dB] and SSIM results obtained after denoising on dataset I (with a noise standard deviation of \(\sigma=15\)). Note that the proposed method does not require any training and is fully explainable. The bold value means the highest PSNR or SSIM in non-learning-based methods.