# Recovering Texture with a Denoising-Process-Aware LMMSE Filter
## 4 Experimental Results
### Image Denoising
\begin{table}
\begin{tabular}{c c c c c} \hline  & **BM3D** & **GHP** & **WNNM** & **Proposed** \\ \hline Image01 & 30.47/0.868 & 30.58/0.869 & 30.81/0.872 & **30.91**/**0.875** \\ Image02 & 29.55/0.765 & 29.67/**0.778** & 29.64/0.765 & **29.77**/**0.777** \\ Image03 & 27.82/0.796 & 28.10/**0.816** & 28.06/0.802 & **28.21**/**0.815** \\ Image04 & 26.61/0.793 & 26.74/0.803 & 26.80/0.798 & **26.91**/**0.809** \\ Image05 & 28.30/0.881 & 28.48/0.884 & 28.58/0.883 & **28.70**/**0.888** \\ Image06 & 29.96/0.828 & 30.21/**0.841** & 30.25/0.833 & **30.38**/**0.841** \\ Image07 & 30.97/0.812 & 31.00/0.816 & 31.03/0.811 & **31.11**/**0.817** \\ Image08 & 27.22/0.806 & 27.41/0.818 & 27.54/0.815 & **27.69**/**0.827** \\ Image09 & 31.35/0.895 & 31.39/**0.896** & 31.52/0.895 & **31.60**/**0.896** \\ Image10 & 27.82/0.803 & 27.98/0.813 & 27.96/0.807 & **28.11**/**0.819** \\ \hline Average & 29.01/0.825 & 29.16/0.833 & 29.22/0.828 & **29.34**/**0.837** \\ \hline \end{tabular}
\end{table} 
Table 2: The PSNR [dB] and SSIM results obtained after denoising on dataset I (with a noise standard deviation of \(\sigma=20\)). We also compare the denoising performance of the proposed method with that of the DnCNN [9] on dataset I. The noise level \(\sigma\) was set to 15 and the texture scaling parameter \(\alpha\) was set to 1. We used the PyTorch implementation of DnCNN and the pretrained model parameters for noise level \(\sigma=15\), which is published by the authors of [9]. The experimental results are shown in Table 6. The improvement of DnCNN over WNNM is 0.30 dB in the average of PSNR while that of the proposed method is 0.10 dB. Note that the proposed method requires no training process and is fully explainable while the denoising mechanism of DnCNN is black box. Next, to observe the effect of the parameter \(\alpha\), we searched for the value of this parameter that would maximize the SSIM of the output image, denoted by \(\alpha_{\text{SSIM}}\). We obtained \(\alpha_{\text{SSIM}}\) via a line search in the range from 0 to 8 in increments of 0.01. For several examples, the original image, the noisy observation, the output image obtained via GHP, the output image obtained via WNNM, and an output image obtained with our method are shown in Figure 4. From this figure, we can observe that the subjective quality and SSIM can be drastically improved by properly choosing the parameter \(\alpha\). For example, as shown in Figure 4p-t for the image named 196073, the SSIM value obtained with our method is 0.073 higher than that achieved via WNNM. Moreover, we show \(\mathbf{t}\) and \(\alpha\hat{\mathbf{t}}\) of each image of Figure 4 in Figure 5. Please note that the pixel value of each image is multiplied by a factor of 3. This figure shows that \(\hat{\mathbf{t}}\) is similar to \(\mathbf{t}\) in each textured region such as surface of a stone, a coral, the skin of a snake, and the surface of the sea. \begin{table}
\begin{tabular}{c c c c c} \hline  & **BM3D** & **GHP** & **WNNM** & **Proposed** & \(p\)**-Value** \\ \hline \(\sigma=10\) & 33.13/0.913 & 33.10/0.911 & 33.36/0.916 & **33.42**/**0.920** & \(5.3\times 10^{-16}/3.8\times 10^{-20}\) \\ \(\sigma=20\) & 29.42/0.824 & 29.49/0.832 & 29.64/0.829 & **29.74**/**0.837** & \(4.6\times 10^{-26}/3.3\times 10^{-19}\) \\ \(\sigma=30\) & 27.56/0.759 & 27.58/0.769 & 27.78/0.766 & **27.84**/**0.773** & \(1.7\times 10^{-16}/6.8\times 10^{-13}\) \\ \hline Average & 30.03/0.832 & 30.06/0.837 & 30.26/0.837 & **30.33**/**0.843** & – \\ \hline \end{tabular}
\end{table} 
Table 4: The PSNR [dB] and SSIM results obtained after denoising on BSD100. The \(p\)-values were calculated by one-tailed paired \(t\)-tests on the difference of each PSNR and SSIM between the proposed method and WNNM.