## 4 Dictionary Pair Learning Algorithm
### Sparse coding phase
Thus, the dictionary size of [37] is larger than the sum of our dictionaries. Table 3 shows the objective evaluation results of our proposed SSM-NTF compared with bicubic interpolation and [37]. On average, our SSM-NTF presents best in PSNR. Figure 7 presents the corresponding visual comparison of the illumination SR results of Image 12. We can observe that the result of bicubic interpolation is too smooth and the result of [37] suffers from obvious ringing artifact and noises. The HR reconstruction of our SSM-NTF method provides more clear details. Figure 5: Visual quality comparison of denoising results for piecewise constant image. Images of the absolute errors are displayed in the dynamic range [0,20] (from left to right): Original image, noise image, analysis KSVD [14], synthesis KSVD [9], our proposed method. Figure 6: Test images for image super-resolution performance evaluation [18]. ### Image Inpainting

To illustrate the potential applicability of our proposed SSM-NTF model on image inpainting, we apply it to the applications of text removal. 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. We operate on the image 'Adar','Lena', 'Couple', 'Hill' with super-imposed text of various fonts. In this experiment, we applied our SSM-NTF model to image inpainting in a way similar to the non-blind KSVD inpainting algorithm [9], which requires the knowledge of which pixels are corrupted and required inpainting. Actually, only the non-corrupted pixels are used to training the dictionary pair and inpainting the images. We operate our method on pathes of size \(10\times 10\) that extract from the images with overlap of 1 pixel between adjacent. The trained dictionary pair are of size \(100\times 200\). The KSVD algorithm in this experiment is dealing with patches of size \(8\times 8\) that extract from the images with overlap of 1 pixel between adjacent. The dictionary size is \(64\times 256\) at its best performance according to [9]. The patch inpainting stage is followed solving Problem (41). Table 4 shows the objective evaluation results of our proposed SSM-NTF compared with DCT and KSVD [9]. The visual comparisons are shown in Figures 8 and 9. We find that the proposed SSM-NTF method is able to eliminate text of fonts completely while the KSVD is dull. Our SSM-NTF method achieves better performance in terms of both subjective and objective quality. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline _Image_ & \(1\) & \(2\) & \(3\) & \(4\) & \(5\) & \(6\) & \(7\) & \(8\) \\ \hline _Bicubic_[37] & 32.69 & 26.32 & 26.04 & 28.39 & 28.70 & 25.81 & 37.21 & 24.40 \\ \hline \(SC\)[37] & 32.89 & 26.53 & 26.16 & 28.51 & 29.09 & 26.25 & 37.59 & 24.61 \\ \hline \(SSM-NTF\) & **33.10** & **26.75** & **26.41** & **28.82** & **29.31** & **26.40** & **37.49** & **24.77** \\ \hline _Image_ & \(9\) & \(10\) & \(11\) & \(12\) & \(13\) & \(14\) & \(15\) & _Average_ \\ \hline _Bicubic_[37] & 31.40 & 28.72 & 33.24 & 30.28 & 26.49 & 28.11 & 31.90 & 29.31 \\ \hline \(SC\)[37] & 31.73 & **29.35** & 33.50 & 30.62 & 26.75 & 28.74 & 32.13 & 29.63 \\ \hline \(SSM-NTF\) & **31.90** & **29.49** & **33.66** & **30.99** & **26.95** & **29.02** & **32.19** & **29.82** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: PSNR (dB) for 3\(\times\) SR reconstructions results. Figure 7: Visual quality comparison of SR results for Image 12 corresponding to Table 3. From left to right: Original image, result of bicubic interpolation (\(PSNR=30.28\)), [32] (\(PSNR=30.62\)) and our SSM-NTF method (\(PSNR=30.99\)), respectively. ## 8 Conclusions

In this paper, we propose a stable sparse model with non-tight frame (SSM-NTF) and further formulate a dictionary pair learning model to stably recover the signals. We theoretically analyze the rationality of the approximation for RIP with the non-tight frame condition.