## 4 Dictionary Pair Learning Algorithm
### Sparse coding phase
The proposed SSM-NTF has RIP and the closed-form expression of the sparse coefficients that ensure the stable recovery especially for seriously noise images. The proposed SSM-NTF contains both a synthesis sparse and an analysis system which share the common sparse coefficients without taking into account the thresholding. We also propose an efficient dictionary pair learning algorithm via developing an explicit analytical expression of the inherent relation between the dictionary pair. The proposed algorithm is capable of approximating structures of signals via a pair of adaptive dictionaries. The effectiveness of our proposed SSM-NTF and its corresponding algorithms are demonstrated in image denoising, image super-resolution and image inpainting. The results of numerical experiments show that the proposed SSM-NTF achieves superior to the compared methods in objective and subjective quality on most of the cases. On the other hand, our proposed SSM-NTF is actually a 1D sparse model. The 1D sparse model suffers from high memory as well as high computational costs especially when handling high dimensional data. MD frame can be expressed as the kronecker product of a series of 1D frames. Benefitting from this good characteristic, in future work, we will extend our stable sparse model to propose an MD stable sparse model. Moreover, the proposed SSM-NTF is not effective enough to remove other kinds of noise (e.g., salt and pepper noise) as the loss function of SSM-NTF is gaussian. We would like to improve the performance of our model by changing the loss function. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Image** & **[PERSON]** & **Lena** & **Hill** & **Couple** & **Average** \\ \hline \(DCT\)[9] & 27.11 & 31.30 & 28.53 & 29.21 & 29.04 \\ \hline \(KSVD\)[9] & 28.01 & 31.69 & 28.90 & 29.50 & 29.53 \\ \hline \(SSM-NTF\) & **28.95** & **32.02** & **29.31** & **29.87** & **30.04** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: PSNR (dB) for image inpainting results. Figure 8: Visual quality comparison of text image inpainting results. From left to right: Original image, text image, result of DCT (\(PSNR=27.11\)), KSVD (\(PSNR=28.01\)) and our SSM-NTF method (\(PSNR=28.95\)), respectively. Figure 9: Visual quality comparison of scratch image inpainting results. From left to right: Original image, scratch image, result of DCT (\(PSNR=30.81\)), KSVD (\(PSNR=31.69\)), and our SSM-NTF method (\(PSNR=32.02\)), respectively. **Author Contributions:** [PERSON] derived the theory, analyzed the data, performed the performance and wrote the original draft; [PERSON] and [PERSON] researched the relevant theory, participated in discussions of the work and revised the manuscript; [PERSON] supervised the project. All authors have read and agreed to the published version of the manuscript. **Funding:** This work was supported by NSFC (No.61672066,61976011, U1811463, U19B2039, 61906008, 61906009). Beijing municipal science and technology commission (No.Z171100004417023) and the Scientific Research Common Program of Beijing Municipal Commission of Education (KM202010005018). **Acknowledgments:** This work was supported by Beijing Advanced Innovation Center for Future Internet Technology, Beijing Key Laboratory of Multimedia and Intelligent Software Technology. We deeply appreciate the organizations mentioned above. **Conflicts of Interest:** The authors declare no conflict of interest. ## References

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* [PERSON] et al. (2015) [PERSON]; [PERSON]; [PERSON]; [PERSON] Joint super resolution and denoising from a single depth image. _IEEE Trans. Multimed._**2015**, _17_, 1525-1537,. [CrossRef]
* [PERSON] et al. (2008) [PERSON]; [PERSON]; [PERSON] _Efficient Implementation of the KSVD Algorithm and the Batch-OMP Method_; Tech. Rep.; Department of Computer Science, Technion: Haifa, Israel, 2008. * [PERSON] et al.