## 4 Conclusions  
This study proposed an image-deblurring model based on the LOGS model to mine the high-order prior information of an IR image containing salt-and-pepper noise. The LOGS regularization term was investigated, combining the advantage of the Lp-pseudo-norm in describing salt-and-pepper noise with replacing the low-order term with a high-order term. The proposed IR image-deblurring model (Lp_HOGS) successfully deblurred an IR image under salt-and-pepper noise. Lp_HOGS achieved average PSNR values that were 0.304, 0.784, and 1.287 dB higher for salt-and-pepper noise at 30%, 40% and 50%, respectively, than those of the Lp_LOGS model for a Gaussian blur. Similarly, Lp_HOGS was 0.362, 0.805, and 1.356 dB higher for the box blur and 1.387, 1.774, and 2.372 dB higher for the motion blur. The findings of this study resulted in the following conclusions:  
1. Upon adding the Gaussian blur and different levels of salt-and-pepper noise to a given test image, the Lp_HOGS model exhibited a better deblurring effect than the existing models. This result implies that the Lp-pseudo-norm had a stronger sparse representation ability and the overlapping group sparsity regularization term increased the difference between the smooth and the edge areas of an image.
2. Upon adding the different types of blur and levels of salt-and-pepper noise to a given test image, Lp_HOGS yielded stronger indicators than Lp_LOGS. This advantage became greater as the noise level increased. Therefore, the high-order prior information of the image improved the quality of the deblurred IR images and the stability of salt-and-pepper noise removal.
3. The advantage of Lp_HOGS over Lp_LOGS was most obvious in terms of the motion blur, indicating that adding the prior constraints of the high-order gradient to the model could significantly improve the IR image deblurring effect amid the motion blur.  
A limitation of this approach is that the application of the Lp_HOGS model is time-consuming. In our future work, we will accelerate the process by introducing accelerated ADMM to improve the performance and efficiency of the proposed method. Nevertheless, the proposed Lp_HOGS model provides a new approach to reduce the salt-and-pepper noise impact on IR images.  
Conceptualization, Z.Y. and Y.C.; Methodology, Z.Y.; Software, Z.Y. and J.H.; Validation, Z.Y. and X.O.; Formal analysis, Z.Y.; Investigation, Z.Y. and J.H.; Resources, Y.C.; Data curation, Z.Y. and X.O.; Writing--original draft preparation, Z.Y. and X.O.; Writing--review and editing, Z.Y. and Y.C.; Visualization, Z.Y.; Supervision, Y.C.; Project administration, Y.C.; Funding acquisition, Y.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Natural Science Foundation Project of Fujian Province, grant numbers 2020J05169 and 2020J01816; Principal Foundation of Minnan Normal University, grant number KJ19019; and High-Level Science Research Project of Minnan Normal University, grant number GJ19019.  
The data presented in this study are openly available at [[http://adas.cvc.uab.es/elektra/datasets/far-infra-red/](http://adas.cvc.uab.es/elektra/datasets/far-infra-red/)]([http://adas.cvc.uab.es/elektra/datasets/far-infra-red/](http://adas.cvc.uab.es/elektra/datasets/far-infra-red/)) (accessed on 5 January 2022), [[http://www.dgp.toronto.edu/~~](http://www.dgp.toronto.edu/~~)]([http://www.dgp.toronto.edu/~~](http://www.dgp.toronto.edu/~~)){}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{{}}{}{}{}{}{}{}{}{}{}{}}{{}}{{}}{}{}{}{}{}{}{}{