## 4 Dictionary Pair Learning Algorithm
### Sparse coding phase
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\sigma\) & **Image** & **Barbara** & **Boat** & **Couple** & **[PERSON]** & **[PERSON]** & **Man** & **Average** \\ \hline \multirow{4}{*}{20} & KSVD [9] & 31.01 & 30.50 & 30.15 & 30.27 & 32.51 & 30.26 & 30.78 \\ \cline{2-9}  & [21] & 31.03 & 30.45 & 30.16 & 30.30 & 32.52 & 30.30 & 30.79 \\ \cline{2-9}  & [1] & **31.07** & 30.35 & 30.20 & 30.31 & 32.56 & 30.07 & 30.76 \\ \cline{2-9}  & SSM-NTF & 31.06 & **30.56** & **30.32** & **30.42** & **32.60** & **30.33** & **30.88** \\ \hline \multirow{4}{*}{30} & KSVD [9] & 28.75 & 28.60 & 28.07 & 28.51 & 30.59 & 28.43 & 28.83 \\ \cline{2-9}  & [21] & 28.78 & 28.63 & 28.06 & 28.53 & 30.60 & 28.47 & 28.85 \\ \cline{2-9}  & [1] & **29.07** & 28.48 & 28.22 & 28.64 & 30.60 & 28.26 & 28.88 \\ \cline{2-9}  & SSM-NTF & 29.03 & **28.71** & **28.32** & **28.74** & **30.82** & **28.59** & **29.04** \\ \hline \multirow{4}{*}{40} & KSVD [9] & 27.03 & 27.23 & 26.54 & 27.23 & 29.13 & 27.17 & 27.39 \\ \cline{2-9}  & [21] & 27.05 & 27.18 & 26.59 & 27.21 & 29.10 & 27.12 & 27.38 \\ \cline{2-9}  & [1] & **27.58** & 27.20 & 26.87 & 27.49 & 29.25 & 26.99 & 27.56 \\ \cline{2-9}  & SSM-NTF & 27.52 & **27.49** & **27.07** & **27.65** & **29.43** & **27.41** & **27.76** \\ \hline \multirow{4}{*}{50} & KSVD [9] & 25.71 & 26.05 & 25.42 & 26.29 & 27.92 & 26.18 & 26.26 \\ \cline{2-9}  & [21] & 25.77 & 26.08 & 25.40 & 26.31 & 27.87 & 26.23 & 26.28 \\ \cline{2-9}  & [1] & **26.45** & 26.15 & 25.84 & 26.63 & 28.15 & 26.09 & 26.55 \\ \cline{2-9}  & SSM-NTF & 26.40 & **26.41** & **26.04** & **26.88** & **28.49** & **26.49** & **26.79** \\ \hline \multirow{4}{*}{60} & KSVD [9] & 24.45 & 25.18 & 24.57 & 25.69 & 27.01 & 25.40 & 25.38 \\ \cline{2-9}  & [21] & 24.45 & 25.20 & 24.50 & 25.65 & 27.03 & 25.42 & 25.38 \\ \cline{2-9}  & [1] & **25.64** & 25.33 & 25.04 & 25.91 & 27.22 & 25.38 & 25.75 \\ \cline{2-9}  & SSM-NTF & 25.52 & **25.55** & **25.22** & **26.11** & **27.41** & **25.74** & **25.93** \\ \hline \multirow{4}{*}{70} & KSVD [9] & 23.40 & 24.46 & 23.90 & 25.10 & 26.18 & 24.77 & 24.63 \\ \cline{2-9}  & [21] & 23.46 & 24.46 & 23.93 & 25.15 & 26.19 & 24.79 & 24.66 \\ \cline{2-9}  & [1] & **24.88** & 24.67 & 24.36 & 25.30 & 26.44 & 24.78 & 25.07 \\ \cline{2-9}  & SSM-NTF & 24.74 & **25.05** & **24.60** & **25.50** & **26.61** & **25.06** & **25.26** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PSNR (dB) for nature image denoising results. Figure 3: Visual comparison of reconstruction results by different methods on ’Man’ (\(\sigma=50\)) and ’Couple’ (\(\sigma=40\)). From left to right: original image, noise image, KSVD [9], method of [1], method of [21] and our proposed method. #### 7.2.2 Piecewise Constant Image Denoising

In this subsection, we demonstrate the analytical property of our SSM-NTF model using a synthetic image. The denoising problem which has been widely studied in sparse representation is used as the target application. We start with a piecewise constant of size \(256\times 256\) contaminated by Gaussian white noise with noise level \(\sigma=5\) and extract all possible \(5\times 5\) image pathes. For the denoising we apply the dictionary pair learning algorithm with the parameters \(\gamma_{1}=1.5\), \(\gamma_{3}=1.2\ L/M\), \(A=0.8\) and \(B=1.8\) in parallel with patch denosing with the synthesis KSVD [9] and the analysis KSVD [14]. We apply 100 iterations of the our dictionary learning method on this training set, and learning dictionary pair of size \(25\times 50\). The experimental set of the synthesis KSVD [9] and the analysis KSVD [14] are at their optimal state according to the previous work. The learned dictionary pair \(\mathbf{\Phi}\) which exhibits much like the synthesis dictionary and \(\mathbf{\
abla}\) which exhibits a high resemblance to the analysis dictionary are illustrated in Figure 4. The resulting PSNRs of the denoised images are 45.32 dB for Analysis KSVD, 43.60 dB for Synthesis KSVD, and 45.17 dB for our proposed algorithm.