# Image Deblurring Based on Convex Non-Convex Sparse Regularization and Plug-and-Play Algorithm
###### Abstract  
Image deblurring based on sparse regularization has garnered significant attention, but there are still certain limitations that need to be addressed. For instance, convex sparse regularization tends to exhibit biased estimation, which can adversely impact the deblurring performance, while non-convex sparse regularization poses challenges in terms of solving techniques. Furthermore, the performance of the traditional iterative algorithm also needs to be improved. In this paper, we propose an image deblurring method based on convex non-convex (CNC) sparse regularization and a plug-and-play (PnP) algorithm. The utilization of CNC sparse regularization not only mitigates estimation bias but also guarantees the overall convexity of the image deblurring model. The PnP algorithm is an advanced learning-based optimization algorithm that surpasses traditional optimization algorithms in terms of efficiency and performance by utilizing the state-of-the-art denoiser to replace the proximal operator. Numerical experiments verify the performance of our proposed algorithm in image deblurring. image deblurring; plug-and-play algorithm; convex non-convex strategy; sparse regularization +
Footnote †: journal: _Algorithms_  
definition  
these two terms. Image deblurring model (3) has garnered significant attention, with the selection of sparse regularizations and efficient algorithms being crucial factors in ensuring the efficacy of the model. Appropriate regularization accurately quantifies the prior information of an image, thereby enhancing the performance of image deblurring. The \(\ell_{0}\)-norm is commonly utilized as a regularization term for deblurring due to its ability to induce sparsity [3; 12; 15; 16]. However, the corresponding sparse regularization problem is computationally intractable because of the discontinuous and non-convex nature of \(\ell_{0}\)-norm [10]. The \(\ell_{1}\)-norm regularization, serving as a convex approximation of the \(\ell_{0}\)-norm, is an appropriate alternative to replace the \(\ell_{0}\)-norm. Nevertheless, the estimates based on the \(\ell_{1}\)-norm exhibit bias and tend to underestimate components of larger magnitude. In addition, as noted in [17], blurring decreases the peak height. Therefore, unlike other image restoration tasks such as denoising, utilizing the \(\ell_{1}\)-norm regularization not only fails to contribute to image deblurring but also impedes image deblurring, consequently yielding the opposite effect [10; 17]. The non-convex regularization has been proposed for image deblurring, where [18; 19] uses the \(\ell_{p}\)-norm (\(0<p<1\)) as the regularization term and the \(\ell_{1}/\ell_{2}\) regularization is used in [10]. However, their tendency is to generate a multitude of local sub-optimal solutions, thereby posing a challenge to the problem-solving process. To address this shortcoming, CNC sparse regularization is introduced, which overcomes biased estimation through its non-convexity and enables convex optimization by adjusting the non-convex control parameter [20; 21; 22; 23]. A fast and efficient algorithm can enhance the speed and accuracy of deblurring, which is crucial for ensuring the practical application of an image deblurring model. Image deblurring algorithms can be broadly categorized into two groups. The first group comprises traditional iterative algorithms such as the iterative shrinkage-thresholding algorithm (ISTA) [10], alternating direction method of multipliers (ADMM) [18; 24], and half-quadratic splitting (HQS) [2; 3; 15; 16] algorithms. These algorithms work by iteratively refining a solution until a stopping criterion is met, typically accompanied by theoretical proof of algorithmic convergence. However, the traditional iterative algorithm often requires numerous iterations to achieve a satisfactory outcome, resulting in significant computational costs [25; 26].