# Recovering Texture with a Denoising-Process-Aware LMMSE Filter
## Appendix B Experimental Results to Confirm the Adopted Assumptions  
### The Texture Follows an MVN Distribution  
We assume that \(\mathbf{t}\) follows an MVN distribution. Based on this assumption, we calculate the Frechet distance between the true texture covariance matrix \(\mathbf{R}_{\mathbf{t}\mathbf{t}_{i}}\) and its estimation. We confirm this assumption by considering the true texture patch matrix \(\mathbf{T}_{i}\). Unfortunately, the number of patches contained in \(\mathbf{T}_{i}\) is fewer than the number of pixels in \(\mathbf{t}_{i}\). Simply testing whether the texture vector \(\mathbf{t}_{i}\) follows an MVN distribution in such a situation would be overly optimistic. Therefore, in this experiment, we employed the Kolmogorov-Smirnov test to confirm that each pixel of \(\mathbf{t}_{i}\) follows a (univariate) normal distribution. To determine each index of \(\mathbf{t}_{i}\) corresponding to \(\mathbf{T}_{i}\), 20 images from BSD100 and 40 patch indices from each image were randomly selected. The noise level \(\sigma\) was set to 20. The results of our Kolmogorov-Smirnov test at a significance level of 0.05 show that, for 97% of the pixel values of the texture component, there is no clear evidence that they do not follow a normal distribution. The reader might suppose that the fact that each pixel of \(\mathbf{t}_{i}\) follows a normal distribution does not necessarily imply that \(\mathbf{t}_{i}\) follows an MVN distribution. However, the former strongly implies the latter in practice. An Observed Patch Matrix and the Corresponding WNNM Output Patch Matrix Have Similar Singular Matrices  
In this subsection, we confirm that \(\mathbf{Y}_{i}\) and \(\mathbf{S}_{i}\) have similar right and left singular matrices. We implicitly apply this assumption when introducing the simple linear approximation of the WNNM denoising process. To confirm this, we evaluate how close the two matrices \(\hat{\Sigma}_{\mathbf{Y}_{i}}\) and \(\hat{\Sigma}_{\mathbf{S}_{i}}\), which are defined as  
\[\hat{\Sigma}_{\mathbf{Y}_{i}}=\mathbf{U}_{\mathbf{S}_{i}}^{-}\mathbf{Y}_{i} \mathbf{V}_{\mathbf{S}_{i}}\text{ and }\hat{\Sigma}_{\mathbf{S}_{i}}=\mathbf{U}_{ \mathbf{Y}_{i}}^{-}\mathbf{S}_{i}\mathbf{V}_{\mathbf{Y}_{i}}, \tag{10}\]  
are to diagonal matrices. Note that if \(\mathbf{Y}_{i}\) and \(\mathbf{S}_{i}\) have identical right and left singular matrices, then \(\hat{\Sigma}_{\mathbf{Y}_{i}}\) and \(\hat{\Sigma}_{\mathbf{S}_{i}}\) must be diagonal matrices. To evaluate the _diagonality_ of a matrix, we introduce an evaluation function \(\mathrm{dm}(\cdot)\) that is defined as  
\[\mathrm{dm}(\mathbf{A})=\frac{\frac{1}{|P|}\sum_{(i,j)\in P}(\mathbf{A}_{i,j} )^{2}}{\frac{1}{|P|}\sum_{(i,j)\in P}(\mathbf{A}_{i,j})^{2}+\frac{1}{|D|}\sum_ {(i,j)\in D}(\mathbf{A}_{i,j})^{2}}, \tag{11}\]  
where \(P\) and \(D\) are the sets of the indices of the nondiagonal and diagonal elements of \(\mathbf{A}\in\mathbb{R}^{m\times l}\), respectively. Note that \(\mathrm{dm}(\mathbf{A})=0\) means that \(\mathbf{A}\) is a diagonal matrix and that \(\mathrm{dm}(\mathbf{A})=1\) means that \(\mathbf{A}\) is a matrix whose diagonal elements are all zero. For this experiment, we used the same images and patches and the same noise level described in Section B.1. The experimental results show that the average values of \(\mathrm{dm}(\hat{\Sigma}_{\mathbf{Y}_{i}})\) and \(\mathrm{dm}(\hat{\Sigma}_{\mathbf{S}_{i}})\) are 0.0768 and 0.0480, respectively. Note that when the maximum singular value of \(\mathbf{S}_{i}\) is small, no elements of \(\hat{\Sigma}_{\mathbf{Y}_{i}}\) affect the estimation accuracy of \(\mathbf{T}_{i}\) because the LMMSE filter \(\mathbf{W}_{i}^{*}\) becomes an almost zero matrix. Thus, when the maximum singular value of \(\mathbf{S}_{i}\) is 0.1 or less, we exclude \(\mathrm{dm}(\hat{\Sigma}_{\mathbf{Y}_{i}})\) from the calculation of the average value as given above.