## 4 Dictionary Pair Learning Algorithm
### Sparse coding phase
The figure shows that our dictionary pair learning method is able to capture the features of the piecewise constant image. Figure 5 shows the absolute difference images for each of the three methods. Note that these images are displayed in the dynamic range \([0,20]\)to make the differences more pronounced. Our proposed approach leads to a much better denoising result than the synthesis KSVD and is comparable with the analysis KSVD. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & KSVD [9] & 22.80 & 23.85 & 23.44 & 24.69 & 25.34 & 24.19 & 24.05 \\ \cline{2-10} \multirow{2}{*}{80} & [21] & 22.87 & 23.93 & 23.47 & 24.76 & 25.40 & 24.27 & 24.09 \\ \cline{2-10}  & [1] & **24.18** & 24.09 & 23.79 & 24.77 & 25.74 & 24.25 & 24.47 \\ \cline{2-10}  & SSM-NTF & 24.11 & **24.42** & **24.02** & **24.91** & **26.02** & **24.58** & **24.68** \\ \hline \multirow{4}{*}{90} & KSVD [9] & 22.31 & 23.34 & 22.98 & 24.31 & 24.96 & 23.78 & 23.61 \\ \cline{2-10}  & [21] & 22.31 & 23.38 & 23.01 & 24.36 & 24.97 & 23.84 & 23.64 \\ \cline{2-10}  & [1] & **23.50** & 23.56 & 23.27 & 24.30 & 25.14 & 23.79 & 23.93 \\ \cline{2-10}  & SSM-NTF & 23.41 & **23.84** & **23.47** & **24.58** & **25.43** & **24.13** & **24.14** \\ \hline \multirow{4}{*}{100} & KSVD [9] & 21.88 & 22.92 & 22.64 & 23.96 & 24.44 & 23.39 & 23.21 \\ \cline{2-10}  & [21] & 21.89 & 22.95 & 22.67 & 23.99 & 24.47 & 23.43 & 23.23 \\ \cline{2-10}  & [1] & **23.01** & 23.11 & 22.83 & 23.86 & 24.59 & 23.38 & 23.46 \\ \cline{2-10}  & SSM-NTF & 22.90 & **23.46** & **23.05** & **24.16** & **24.90** & **23.68** & **23.69** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._

Figure 4: The exemplified dictionaries training by the piecewise constant. From left to right: Synthesis KSVD (\(25\times 100\)), our proposed dictionary pair (\(25\times 50\)) and analysis KSVD (\(25\times 50\)). ### Super Resolution

We evaluate our SSM-NTF in comparison with two examplar-based scheme [37] for image Super Resolution (SR) Problem (41) with a bicubic filter. Figure 6 shows the 15 test natural images [18] with both rich texture and structure. All the schemes are applied to the illumination channel, where the scale factor is 3, we always use \(3\times 3\) low-resolution patches with overlap of 1 pixel between adjacent patches, corresponding to \(9\times 9\) patches with overlap of 3 pixels for the high-resolution patches. In these experiments, we have used the following parameters: \(A=0.8,B=1.8\), \(\gamma_{1}=1.1\) and \(\gamma_{3}=1.2\)\(L/M\) where \(L\) and \(M\) are the sample and dictionary size, respectively. In our scheme, dictionary learning is performed between HR and middle-level (MR) images which are the first-, and second-order derivatives of the upsampled version of one LR image by a factor of 2. The four \(1D\) filters used to extract the derivatives are:

\[ \begin{split}\mathbf{f}_{1}&=[-1,0,1],\qquad \mathbf{f}_{2}=\mathbf{f}_{1}^{T}\\ \mathbf{f}_{3}&=[1,0,-2,0,1],\quad\mathbf{f}_{4}= \mathbf{f}_{3}^{T}\end{split}  \tag{45}\]

We train two pairs of HR/LR dictionaries \(\{\mathbf{\Phi}_{h},\mathbf{\Psi}_{h}\}\) and \(\{\mathbf{\Phi}_{l},\mathbf{\Psi}_{l}\}\) from 100,000 HR/LR patch pairs \([\mathbf{X}_{h},\mathbf{X}_{l}]\) randomly sampled from the collected natural images which are also used in [37] where \(\mathbf{X}_{h}\) is sampled from the HR images and \(\mathbf{X}_{l}\) is sampled from the four feature images. The feature images are obtained by applying the four filters to the upsampled LR image. Given \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\) and the four MR feature images, the sparse coefficients \(\mathbf{Y}\) and threshold value \(\lambda\) can be calculated by Algorithm 1. With the theory in [37], the HR image can be recovered via Algorithm 3. In the experiment, our HR dictionary pair are of size \(81\times 450\) and MR ones are of size \(144\times 450\). The dictionary size of [37] is \(81\times 1024\) (HR) and \(144\times 1024\) (MR) at its best performance as stated in the paper.