# Recovering Texture with a Denoising-Process-Aware LMMSE Filter
## 4 Experimental Results
### Accuracy of Texture Covariance Matrix Estimation  
In this section, we compare the accuracy of two texture covariance matrices. One is our estimated texture covariance matrix, and the other is a simply sampled estimate that is calculated as  
\[\mathbf{R}_{\mathbf{t}\mathbf{t}_{i}}^{\mathrm{smp}}=h((\mathbf{T}_{i}+\mathbf{ N}_{i})(\mathbf{T}_{i}+\mathbf{N}_{i})^{\top}-\sigma^{2}\mathbf{I}), \tag{19}\]  
where \(h(\cdot)\) is a function that replaces any negative eigenvalues of the input matrix with zero; this is equivalent to metric projection to a positive-semidefinite matrix.  
We calculate the Frechet distance to evaluate the accuracy of the texture covariance matrix. [PERSON] and [PERSON] introduced the calculation of the Frechet distance between multivariate normal (MVN) distributions [PERSON] and [PERSON] (1958). The Frechet distance is also used to measure the similarity between generated and real images [PERSON] and [PERSON] (1958). We assume that the texture component \(\mathbf{t}\) follows an MVN distribution. Consider a random MVN-distributed variable \(\mathbf{a}\in\mathbb{R}^{d}\) whose mean vector is \(\boldsymbol{\mu}_{\mathbf{a}}=\mathbb{E}[\mathbf{a}]\in\mathbb{R}^{d}\) and whose covariance matrix is \(\mathbf{R}_{\mathbf{a}a}\) (\(\mathbf{a}\sim\mathcal{N}_{d}(\boldsymbol{\mu}_{\mathbf{a}},\mathbf{R}_{ \mathbf{a}a})\)); its probability density is denoted by  
\[p(\mathbf{a})=\frac{1}{(\sqrt{(2\pi)^{d}\det(\mathbf{R}_{\mathbf{a}a})}}\exp( -\frac{1}{2}(\mathbf{a}-\boldsymbol{\mu}_{\mathbf{a}})^{\top}\mathbf{R}_{ \mathbf{a}a}^{-1}(\mathbf{a}-\boldsymbol{\mu}_{\mathbf{a}})). \tag{20}\]  
Additionally, the Frechet distance between the MVN distributions \(\mathcal{N}_{d}(\boldsymbol{\mu}_{\mathbf{a}},\mathbf{R}_{\mathbf{a}a})\) and \(\mathcal{N}_{d}(\boldsymbol{\mu}_{\mathbf{b}},\mathbf{R}_{\mathbf{b}b})\) is calculated as  
\[\left\|\boldsymbol{\mu}_{\mathbf{a}}-\boldsymbol{\mu}_{\mathbf{b}}\right\|_ {2}^{2}+\mathrm{tr}(\mathbf{R}_{\mathbf{a}a}+\mathbf{R}_{\mathbf{b}b}-2( \mathbf{R}_{\mathbf{a}a}\mathbf{R}_{\mathbf{b}b})^{\frac{1}{2}}). \tag{21}\]  
We assume that the mean vector of the texture is \(\mathbf{0}\). Thus, we can calculate the Frechet distance between the estimated texture covariance matrix \(\hat{\mathbf{R}}_{\mathbf{t}\mathbf{t}_{i}}\) and the true texture covariance matrix \(\mathbf{R}_{\mathbf{t}\mathbf{t}_{i}}\) from the ground truth as follows:  
\[\mathrm{Frechet}(\mathbf{R}_{\mathbf{t}\mathbf{t}_{i}},\hat{\mathbf{R}}_{ \mathbf{t}\mathbf{t}_{i}})=\mathrm{tr}(\mathbf{R}_{\mathbf{t}\mathbf{t}_{i}} +\hat{\mathbf{R}}_{\mathbf{t}\mathbf{t}_{i}}-2(\mathbf{R}_{\mathbf{t}\mathbf{ t}_{i}}\hat{\mathbf{R}}_{\mathbf{t}\mathbf{t}_{i}})^{\frac{1}{2}}). \tag{22}\]  
We can similarly calculate the Frechet distance between the simply sampled estimate \(\hat{\mathbf{R}}_{\mathbf{t}\mathbf{t}_{i}}^{\mathrm{smp}}\) and \(\mathbf{R}_{\mathbf{t}\mathbf{t}_{i}}\) by replacing \(\hat{\mathbf{R}}_{\mathbf{t}\mathbf{t}}\) with \(\hat{\mathbf{R}}_{\mathbf{t}\mathbf{t}}^{\mathrm{smp}}\) in Equation (22).  
Figure 8 shows histograms of these Frechet distances. Note that we calculated the Frechet distance for each patch matrix from dataset I. However, we ignored the very few outliers caused by numerical errors. This figure shows that \(\hat{\mathbf{R}}_{\mathbf{t}\mathbf{t}_{i}}\) is more similar to \(\mathbf{R}_{\mathbf{t}\mathbf{t}_{i}}\) than \(\hat{\mathbf{R}}_{\mathbf{t}\mathbf{t}_{i}}^{\mathrm{smp}}\) is at each noise level.  
Figure 7: Histogram of the \(\alpha\) values that maximize the PSNR for each output image.