# Recovering Texture with a Denoising-Process-Aware LMMSE Filter
## 3 Recovering Texture via Statistical Analysis
### Recovering the Texture of a Denoised Image  
Based on the above discussion, we can calculate the LMMSE filter and estimate \(\mathbf{t}_{i}\). However, \(\mathbf{R}_{(t_{i}+\mathbf{n}_{i})(t_{i}+\mathbf{n}_{i})}\) is not invertible in practice because the number of patches \(M\) is smaller than \(L\). Although an equivalent estimate of \(\mathbf{R}_{(t_{i}+\mathbf{n}_{i})(t_{i}+\mathbf{n}_{i})}\) appears to be \(\mathbf{\hat{R}}_{t_{i}\mathbf{t}_{i}}+\mathbf{\hat{R}}_{t_{i}\mathbf{n}_{i}} +\mathbf{\hat{R}}_{\mathbf{n}_{i}\mathbf{t}}+\sigma^{2}\mathbf{I}\) (where \(\mathbf{I}\) is the identity matrix), this choice does not provide good recovery performance. Instead, substituting \(\mathbf{R}_{t_{i}\mathbf{t}_{i}}+\sigma^{2}\mathbf{I}\) into the sample covariance matrix \(\mathbf{R}_{(t_{i}+\mathbf{n}_{i})(t_{i}+\mathbf{n}_{i})}\) was found to yield the best results in our preliminary experiments. Thus, we calculate the LMMSE filter as  
\[\mathbf{W}_{i}^{\top}=(\mathbf{R}_{t_{i}\mathbf{t}_{i}}+\mathbf{R}_{t_{i} \mathbf{n}_{i}})(\mathbf{R}_{t_{i}\mathbf{t}_{i}}+\sigma^{2}\mathbf{I})^{-1}. \tag{15}\]  
Substituting Equation (13) and (14) into (15), we can estimate \(\mathbf{T}_{i}\) as  
\[\mathbf{\hat{T}}_{i}=(\sigma^{2}\mathbf{F}_{i})(2\sigma^{2}\mathbf{F}_{i}+ \sigma^{2}\mathbf{I})^{-1}(\mathbf{\hat{Y}}_{i}-\mathbf{\hat{S}}_{i}). \tag{16}\]  
Then, we can obtain the desired estimated texture image \(\mathbf{\hat{t}}\) that is obtained by combining each \(\mathbf{\hat{T}}_{i}\) in the same manner used to reconstruct an image from the patches obtained in WNNM. Finally, we can obtain the final estimated image \(\mathbf{\hat{x}}\) with enhanced texture as  
\[\mathbf{\hat{x}}=\mathbf{s}+\mathbf{\hat{t}}. \tag{17}\]  
We use the LMMSE filter to obtain \(\mathbf{\hat{t}}\); however, minimizing the MSE often causes \(\mathbf{\hat{t}}\) to lose clarity. We can obtain a clearer texture-enhanced image as follows:  
\[\mathbf{\hat{x}}=\mathbf{s}+\sigma\mathbf{\hat{t}}, \tag{18}\]where \(\alpha\) is an arbitrary parameter used to control the magnitude of the texture that is added. Note that \(\alpha=1\) means the proposed method applies the LMMSE filter and we can expect to obtain the denoised images with best PSNR if all of our assumptions hold. The entire proposed process is presented in Algorithm 2.  
```
0: Noisy observation \(\mathbf{y}\)  Initialize \(\mathbf{s}^{(0)}=\mathbf{y}\) and \(\mathbf{y}^{(0)}=\mathbf{y}\) for\(k=1:K\)do  Iterative regularization: \(\mathbf{y}^{(k)}=\mathbf{s}^{(k-1)}+\delta(\mathbf{y}-\mathbf{s}^{(k-1)})\) for\(i=1:I\)do  Find the similar patch matrix \(\mathbf{Y}_{i}^{(k)}\)  Estimate the corresponding original patch matrix \(\mathbf{X}_{i}\) via Equation (2); the result is denoted by \(\mathbf{S}_{i}^{(k)}\) ifk=Kthen  Estimate \(\mathbf{R}_{t_{i}\mathbf{t}_{i}}\) and \(\mathbf{R}_{t_{i}\mathbf{n}_{i}}\) using [PERSON]'s lemma via Equation (13) and (14)  Calculate \(\mathbf{W}_{i}^{\prime}\) via Equation (9)  Estimate \(\mathbf{T}_{i}\) via Equation (16) to obtain \(\hat{\mathbf{T}}_{i}\) endif endfor  Use the \(\mathbf{S}_{i}^{(k)}\) to reconstruct the estimated image \(\mathbf{s}^{(k)}\) endfor  Use the \(\hat{\mathbf{T}}_{i}\) to reconstruct the estimated texture component \(\hat{\mathbf{t}}\) Output: Obtain the final estimated image as \(\hat{\mathbf{x}}=\mathbf{s}^{(K)}+\alpha\hat{\mathbf{t}}\)
```  
**Algorithm 2** Texture recovery using the proposed method.