## 4 Conclusions  
In this paper, we proposed and evaluated an image deblurring method based on sparse representation. We determined the image prior based on the nonzero measurement in the gradient  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test Images** & **NRIQA** & \begin{tabular}{c} **Method in** \\ **[**35**]** \\ \end{tabular} & \begin{tabular}{c} **Method in** \\ **[**36**]** \\ \end{tabular} & \begin{tabular}{c} **Method in** \\ **[**37**]** \\ \end{tabular} & \
\begin{tabular}{c} **Method in** \\ **[**38**]** \\ \end{tabular} &
\begin{tabular}{c} **Proposed** \\ **Method** \\ \end{tabular} \\ \hline \multirow{2}{*}{“Tower.bmp”} & BIBQ & 46.347 & 38.829 & 41.496 & 39.855 & 37.293 \\ \cline{2-7}  & SSEQ & 37.831 & 46.839 & 40.268 & 36.637 & 35.693 \\ \hline \multirow{2}{*}{“Chimney.bmp”} & BIBQ & 57.800 & 52.974 & 58.437 & 52.889 & 51.953 \\ \cline{2-7}  & SSEQ & 69.902 & 52.127 & 59.659 & 49.884 & 33.326 \\ \hline \multirow{2}{*}{“Pole.bmp”} & BIBQ & 52.357 & 45.612 & 46.507 & 45.246 & 41.586 \\ \cline{2-7}  & SSEQ & 45.693 & 52.588 & 44.672 & 39.171 & 37.514 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Non-reference image quality assessment data of the comparison results of Figures 10–12.  
Figure 11: Comparison results for the “Chimney.bmp” image: (**a**) blurry image; (**b**) method in [35]; (**c**) method in [36]; (**d**) method in [37]; (**e**) method in [38]; (**f**) proposed method.  
Figure 12: Comparison results for the “Pole.bmp” image: (**a**) blurry image; (**b**) method in [35]; (**c**) method in [36]; (**d**) method in [37]; (**e**) method in [38]; (**f**) proposed method.  
domain of natural images, which is an efficient prior for blur kernel estimation. We optimized the cost function via the split Bergman iteration that turned the non-convex problem into an alternating scheme, and the blur kernel could be estimated correctly in a coarse-to-fine manner for avoiding local minima. Subsequently, we revealed the latent image through a similar technique with the help of the hyper-Laplacian distribution-based prior with the estimated blur kernel. In the interim, to reduce ringing effects, we preprocessed the image boundaries to achieve second-order smoothness. Moreover, the deblurring algorithm converged quickly owing to the analytical solutions of the subproblems. The results of experiments and comparative analysis shows that the proposed method can obtain improved results compared with other similar deblurring methods. In addition, our method is effective when dealing with atmospheric turbulence blurs in real-life applications. Because our deblurring framework is based on a linear system, it works well on spatially invariant blurs in most scenarios. In the future, the deblurring model can be extended to nonlinear blurs for wider applications.  
Conceptualization, H.Y., X.S. and S.C.; methodology, H.Y. and S.C.; software, H.Y. and S.C.; validation, H.Y., X.S. and S.C.; formal analysis, H.Y. and X.S.; investigation, H.Y. and S.C.; resources, H.Y.; data curation, H.Y.; writing--original draft preparation, H.Y. and X.S.; writing--review and editing, H.Y. and S.C.; visualization, H.Y. and X.S.; supervision, H.Y. and S.C.; project administration, H.Y. and X.S.; funding acquisition, X.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by High-tech Innovation Fund of Chinese Academy of Sciences, grant number GQRC_19_19.  
We would like to thank Editage (www.editage.cn) for English language editing.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.