# Image Deblurring Based on Convex Non-Convex Sparse Regularization and Plug-and-Play Algorithm
## 3 Image Deblurring Model with CNC Sparse Regularization  
In this section, we write out the corresponding image deblurring model based on CNC sparse regularization and further prove the overall convexity of the objective function.  
Firstly, we introduce the CNC sparse regularization, which can be formulated as  
\[\phi_{b}(x)=\|x\|_{1}-s_{b}(x), \tag{10}\]  
where \(s_{b}(x)=\min_{b}\Bigl{\{}\|v\|_{1}+\frac{b^{2}}{2}\|x-v\|_{2}^{2}\Bigr{\}}\) is the Moreau envelope of \(\|x\|_{1}\), and the parameter \(b\) controls the non-convexity of \(\phi_{b}(x)\).  
By substituting \(\phi_{b}(x)\) for \(\phi(x)\) in problem (3), we obtain the image deblurring model with the CNC sparse regularization.  
\[\min_{x}\quad\frac{1}{2}\|y-F^{-1}\Lambda Fx\|_{2}^{2}+\lambda\phi_{b}(x). \tag{11}\]  
Although the CNC sparse regularization is non-convex, the overall convexity of the objective function can be ensured by adjusting the non-convex parameters \(b\). The following theorem focuses on the convexity condition to determine the circumstances under which the objective function can maintain its convexity.  
**Theorem 1**.: _For \(\lambda>0\), \(b>0\), the objective function of the image deblurring model is defined as_  
\[T(x)=\frac{1}{2}\|y-F^{-1}\Lambda Fx\|_{2}^{2}+\lambda\phi_{b}(x). \tag{12}\]  
_When \(b^{2}I\preceq\frac{1}{\lambda}F^{T}\Lambda^{T}\Lambda F\), the objective function is convex; when \(b^{2}I\prec\frac{1}{\lambda}F^{T}\Lambda^{T}\Lambda F\), the objective function is strictly convex._Proof.: Consider the objective function  
\[T(x)= \frac{1}{2}\|y-F^{-1}\Lambda Fx\|_{2}^{2}+\lambda\phi_{b}(x)\] \[= \frac{1}{2}\|y-F^{-1}\Lambda Fx\|_{2}^{2}+\lambda(\|x\|_{1}-s_{b}( x))\] \[= \frac{1}{2}\|y-F^{-1}\Lambda Fx\|_{2}^{2}+\lambda\bigg{\{}\|x\|_{ 1}-\min_{v}\{\|v\|_{1}+\frac{b^{2}}{2}\|x-v\|_{2}^{2}\}\bigg{\}}\] \[= \frac{1}{2}\|y\|_{2}^{2}-y^{\mathrm{T}}(F^{-1}\Lambda Fx)+\frac{ 1}{2}\|F^{-1}\Lambda Fx\|_{2}^{2}+\lambda\|x\|_{1}\] \[-\lambda\min_{v}\bigg{\{}\|v\|_{1}+\frac{b^{2}}{2}\|x\|_{2}^{2}-b ^{2}v^{\mathrm{T}}x+\frac{b^{2}}{2}\|v\|_{2}^{2}\bigg{\}}\] \[= \frac{1}{2}\|F^{-1}\Lambda Fx\|_{2}^{2}-\frac{\lambda b^{2}}{2} \|x\|_{2}^{2}+\frac{1}{2}\|y\|_{2}^{2}-y^{\mathrm{T}}(F^{-1}\Lambda Fx)\] \[+\lambda\|x\|_{1}-\lambda\min_{v}\bigg{\{}\|v\|_{1}-b^{2}v^{ \mathrm{T}}x+\frac{b^{2}}{2}\|v\|_{2}^{2}\bigg{\}}\] \[= \frac{1}{2}x^{\mathrm{T}}\Big{(}F^{\mathrm{T}}\Lambda^{\mathrm{T }}\Lambda F-\lambda b^{2}I\Big{)}x+\frac{1}{2}\|y\|_{2}^{2}-y^{\mathrm{T}}(F^{ -1}\Lambda Fx)\] \[+\lambda\|x\|_{1}+\lambda\max_{v}\bigg{\{}-\|v\|_{1}+b^{2}v^{ \mathrm{T}}x-\frac{b^{2}}{2}\|v\|_{2}^{2}\bigg{\}}. \tag{13}\]  
Since the three terms \(\frac{1}{2}\|y\|_{2}^{2},-\|v\|_{1},\) and \(-\frac{b^{2}}{2}\|v\|_{2}^{2}\) are independent of \(x\), they can be regarded as constants. The term \(\lambda\|x\|_{1}\) is convex. The term \(-y^{T}(F^{-1}\Lambda Fx)\) is linear, so this term is convex. The term \(b^{2}v^{\mathrm{T}}x\) is affine in \(x\) and the maximum of the convex functions set is convex, and therefore the term \(\lambda\max_{v}\big{\{}-\|v\|_{1}+b^{2}v^{\mathrm{T}}x-\frac{b^{2}}{2}\|v\|_{2 }^{2}\big{\}}\) is convex.  
In summary, the convexity of the objective function (12) depends on the convexity of the following term  
\[\frac{1}{2}x^{\mathrm{T}}\Big{(}F^{\mathrm{T}}\Lambda^{\mathrm{T }}\Lambda F-\lambda b^{2}I\Big{)}x. \tag{14}\]  
The convexity of this term is determined by \(F^{\mathrm{T}}\Lambda^{\mathrm{T}}\Lambda F-\lambda b^{2}I\). If \(F^{\mathrm{T}}\Lambda^{\mathrm{T}}\Lambda F-\lambda b^{2}I\) is positive semi-definite, i.e. \(b^{2}I\preceq\frac{1}{4}F^{\mathrm{T}}\Lambda^{\mathrm{T}}\Lambda F\), then the objective function is convex. If \(F^{\mathrm{T}}\Lambda^{\mathrm{T}}\Lambda F-\lambda b^{2}I\) is positive definite, i.e. \(b^{2}I\prec\frac{1}{4}F^{\mathrm{T}}\Lambda^{\mathrm{T}}\Lambda F\), then the objective function is strictly convex.