## 5 Conclusions  
The weighted nuclear norm minimization (WNNM) is a significant extension of the nuclear norm minimization (NNM) model. It utilizes the physical significance of the matrix singular value. Each singular value stands for the component information in images. A larger eigenvalue means more principal component and it needs to shrink less in the optimization process. WNNM treats all eigenvalues unevenly to achieve a better recovery of details. WNNM remains convex and has the analytical optimal solution. When the weights are in descending order, we present an iterative algorithm to solve it using a similar group searching method.  
1. The adaptive patch selection fusion in WNNM guarantees a better sparsity of the original matrix. It has strengthened the low-rank character, which is helpful in recovering the denoising image.
2. Considering all the analyses, the improved denoising algorithm with the WNNM model based on adaptive GSR could improve PSNR and SSIM, especially under a high noise background. It has achieved better noise suppression and attained the best adaptability among all the algorithms in regard to IR image denoising.
3. The algorithm is suitable for infrared image denoising as well as ordinary image denoising. The denoising process constructs a solid foundation in following IR target detection for remote sensing under a high-noise environment.  
However, the fusion computation is complex, and it could not be realized in the high real-time process. In the future, we could pursue a faster computation strategy and seek a more universal parameter setting strategy to achieve a better optimal solution in solving the denoising problem.  
The following supporting information can be downloaded at [[https://www.mdpi.com/article/10.3390/app13095749/s1](https://www.mdpi.com/article/10.3390/app13095749/s1)]([https://www.mdpi.com/article/10.3390/app13095749/s1](https://www.mdpi.com/article/10.3390/app13095749/s1)).  
Conceptualization, Z.Z. ([PERSON]) and J.C.; methodology, J.C.; software, J.C. and L.Q.; validation, J.C., L.Q. and L.D.; formal analysis, Z.Z. ([PERSON]); investigation, J.C.; resources, L.Q.; data curation, L.D.; writing--original draft preparation, J.C.; writing--review and editing, J.C.; visualization, J.C.; supervision, [PERSON] ([PERSON]); project administration, [PERSON] ([PERSON]); funding acquisition, H.H. All authors have read and agreed to the published version of the manuscript.  
The paper is supported by the project of the Military Commission Science and Technology Commission. The program is \"real-time detection technology based on little and dim space group targets\". It is an important project which is devoted to searching the dim and small targets in remote sensing. The project contract is NO.2020-JCJQ-ZD-052-00. It is also supported by the Youth Innovation Promotion Association of the Chinese Academy of Sciences under Grant 2022296.  
Not applicable.  
The data sets were received from common image data sets. They are representative.  
We obtained test data from common data sets and obtained many images with our IR detector.  
We declare that we have no financial or personal relationships with people or organizations that could inappropriately influence our work. There is no professional or other personal interest of any nature or kind in regard to any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled 'A Novel Adaptive Group Sparse Representation Model based on Infrared Image Denoising for Remote Sensing Application'.