## 4 Denoising Methods
### BP-ADMM Methods  
In this section we formulate our problem and derive an efficient distributed optimization algorithm via ADMM. In the case of noise, according to Equations (7) and (9) of the paper, it can be concluded that:  
\[\widetilde{x}=\underset{x}{\text{argmin}}\|x\|_{1},\ \ \ s.t.\|y-Mx\|_{2}^{2}\leq\varepsilon \tag{15}\]  
where \(\varepsilon\) represents a very small normal number error. In addition, according to the Lagrange multiplier theorem, a constant \(\lambda\) can be introduced to make the above optimization problem with constraints become an unconstrained minimization problem. Firstly, we set up the objective function:  
\[\widetilde{x}=L(x,\lambda)=\underset{x}{\text{argmin}}\frac{1}{2}\|y-Mx\|_{2}^ {2}+\lambda\|x\|_{1} \tag{16}\]  
In order to apply the ADMM to the BP method, we first transform the ADMM formula to an equivalent constrained problem. Then according to \(yy=Hx\) of Section 4.2, it can be described as the following formula:  
\[\underset{x\in X}{\text{min}}F(x)=\frac{1}{2}\|y-Mx\|_{2}^{2}+\lambda\left\|H ^{-}y\right\|_{1} \tag{17}\]  
We introduce new variables \(z=Ax\). Then the ECG denoising and reconstruction problem can be transformed into an unconstrained form, and the corresponding augmented Lagrange function is:  
\[L_{\rho}(x,y,z,\lambda)=\sum\limits_{i=1}^{r}\left(\|z\|+\lambda^{T}(Mx-y)+ \frac{\rho}{2}\|Mx-y\|_{2}^{2}+u^{T}(Ax-z)+\frac{\mu}{2}\|Ax-z\|_{2}^{2}\right) \tag{18}\]  
where \(\lambda\) is a dual variable, and \(\rho\) is the penalty operator. Use the alternate direction multiplier of the ADMM algorithm to solve the problem. Therefore, the variables \(x\) and \(z\) are updated in sequence, and then the dual variables are updated. In the second step of ADMM, substituting the augmented Lagrange function in Equation (18) into the ADMM algorithm framework shown in Equation (15), the new ADMM algorithm process can be drawn as:  
\[x^{(k+1)}=\underset{x}{\text{argmin}}\Big{(}\lambda^{T}(Mx-y)+\frac{\rho}{2} \|Mx-y\|_{2}^{2}+v^{T}(Ax-z)+\frac{\mu}{2}\|Ax-z\|_{2}^{2}\Big{)} \tag{19}\]\[z^{(k+1)}=\underset{z}{\text{argmin}}\Big{(}\|z\|+\lambda^{T}(Ax-z)+\frac{\rho}{2} \|Ax-z\|_{2}^{2}\Big{)} \tag{20}\]  
\[\lambda^{(k+1)}=\lambda^{k}+\rho(Mx^{k}-y^{k}) \tag{21}\]  
\[v^{(k+1)}=v^{k}+\mu(Ax^{k}-z) \tag{22}\]  
Let us define:  
\[G_{k}(x)=\lambda^{T}(Mx-y)+\frac{\rho}{2}\|Mx-y\|_{2}^{2}+v^{T}(Ax-z)+\frac{ \mu}{2}\|Ax-z\|_{2}^{2} \tag{23}\]  
Obviously, to derive \(G_{k}(x)\) in Equation (18) with respect to \(x\), the derivation process is as follows:  
\[\frac{dG_{k}(x)}{dx}=M^{T}\lambda+\rho M^{T}(Mx-y)+A^{T}v+\mu A^{T}(Ax-z) \tag{24}\]  
Finally, we update the dual variables. Let Equation (24) be 0, and then we get:  
\[\widetilde{x}=L(x,\lambda)=\underset{x}{\text{argmin}}\frac{1}{2}\|y-Mx\|_{2}^ {2}+\lambda\|x\|_{1} \tag{16}\]0  
\[\widetilde{x}=L(x,\lambda)=\underset{x}{\text{argmin}}\frac{1}{2}\|y-Mx\|_{2}^ {2}+\lambda\|x\|_{1} \tag{16}\]1  
The multiplier term and the penalty term are completely squared, and after discarding some constant terms, the optimization sub-problem can be transformed into:  
\[\widetilde{x}=L(x,\lambda)=\underset{x}{\text{argmin}}\frac{1}{2}\|y-Mx\|_{2}^ {2}+\lambda\|x\|_{1} \tag{16}\]2  
In addition, the soft threshold function, the soft threshold operator \(S\) is defined as  
\[S_{soft}(x,T)=\left\{ \begin{array}{ll}x+T&x\leq-T\\ 0&|x|\leq T\\ x-T&x\geq T\end{array} \right. \tag{28}\]  
Its iterative process turns into a soft threshold for point-by-point operation:  
\[z^{(k+1)}=S_{\frac{1}{\rho}}\Big{(}Ax+\frac{\lambda}{\rho}\Big{)} \tag{29}\]  
But different from the general iterative algorithm, the ADMM algorithm is a dual-condition stop threshold judgment on the stop criterion of iterative convergence, that is, both the original problem residual and the dual residual must reach the convergence threshold. In the ADMM algorithm, the selection of parameters \(\lambda\) and \(\rho\) have a relatively important impact on the accuracy of the image reconstruction algorithm [46].