# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 2 Methods
### Sparse Radon Transform with the Elastic Half-Norm Constraint
However, the \(l_{1/2}\) norm is nonconvex and may cause the solution unstable. To address the issue, [PERSON] et al. [37] proposed an elastic half (EH) norm regularization to improve the \(l_{1/2}\) norm constraint. Containing both Tikhonov and sparse regularization in the equation, the elastic half norm effectively mitigates numerical oscillation when dealing with ill-conditioned inverse problems. This leads to an optimal solution that exhibits improved sparsity and stability. Based on this, we propose a novel sparse Radon transform with the EH norm constraint and denote the method as EH-SRT. The method formulates the optimization problem as follows:  
\[\mathbf{m}^{*}=\underset{\mathbf{m}}{\operatorname{argmin}}\frac{1}{2}\| \mathbf{d}-f^{-1}(\mathbf{L}f(\mathbf{m}))\|_{2}^{2}+\lambda\|\mathbf{m}\|_{ 1/2}^{1/2}+\sigma\|\mathbf{m}\|_{2}^{2}, \tag{7}\]  
where \(\|\cdot\|_{1/2}^{1/2}\) represents the \(l_{1/2}\) norm, \(\lambda\) denotes the regularization parameter, and \(\sigma\) is the scaling parameter. Comparing Equation (7) with the normal \(l_{1/2}\) norm, one can find that there is an additional \(\sigma\|\mathbf{m}\|_{2}^{2}\) term, which is used to guarantee a stable and unique solution, because it solves the problem of numerical oscillations induced by the non-conductivity of the conventional \(l_{1/2}\) norm at zero. If \(\sigma=0\), the EH norm regularization will convert to a typical \(l_{1/2}\) regularization. Equation (7) is a nonconvex optimization problem as it containsboth the \(l_{2}\) and \(l_{1/2}\) norm. To solve the optimization problem, an auxiliary variable \(\mathbf{T}\) needs to be introduced and Equation (7) is expressed by  
\[(\mathbf{m}^{*},\mathbf{T}^{*})=\underset{\mathbf{m},\mathbf{T}}{\text{argmin}} \frac{1}{2}\|\mathbf{d}-f^{-1}(\mathbf{L}f(\mathbf{m}))\|_{2}^{2}+\sigma\| \mathbf{m}\|_{2}^{2}+\lambda\|\mathbf{T}\|_{1/2}^{1/2}\ \ \ \text{s.t}\mathbf{T}=\mathbf{m}, \tag{8}\]  
The minimization problem can be solved by converting it as an augmented Lagrangian function, which is represented by  
\[L(\mathbf{m},\mathbf{T},\mathbf{c})=\underset{\mathbf{m},\mathbf{T},\mathbf{c }}{\text{argmin}}\frac{1}{2}\|\mathbf{d}-f^{-1}(\mathbf{L}f(\mathbf{m}))\|_{2 }^{2}+\sigma\|\mathbf{m}\|_{2}^{2}+\lambda\|\mathbf{T}\|_{1/2}^{1/2}+\langle \mathbf{m}-\mathbf{T},\mathbf{c}\rangle+\frac{\xi}{2}\|\mathbf{m}-\mathbf{T} \|_{2}^{2}, \tag{9}\]  
where is a balanced term, which is used to dominate the convergent rate, \(\mathbf{c}\) denotes the Lagrangian multiplier, and \(\mathbf{c}^{\mathbf{T}}\) represents its transpose. Following the alternating direction method of multipliers (ADMM) algorithm [22], Equation (9) can be solved by updating one variable while fixing the other two variables, and the iteration equation is given by  
\[\mathbf{m}^{k+1}=\underset{\mathbf{m}}{\text{argmin}}L(\mathbf{m},\mathbf{T}^ {k},\mathbf{z}^{k})=\underset{\mathbf{m}}{\text{argmin}}\frac{1}{2}\|\mathbf{ d}-f^{-1}(\mathbf{L}f(\mathbf{m}))\|_{2}^{2}+\sigma\|\mathbf{m}\|_{2}^{2}+ \frac{\xi}{2}\|\mathbf{m}-\mathbf{T}^{k}+\mathbf{z}^{k}\|_{2}^{2}, \tag{10}\]  
\[\mathbf{T}^{k+1}=\underset{\mathbf{T}}{\text{argmin}}L(\mathbf{m}^{k+1}, \mathbf{T},\mathbf{z}^{k})=\underset{\mathbf{T}}{\text{argmin}}\gamma\| \mathbf{T}\|_{1/2}^{1/2}+\frac{\xi}{2}\|\mathbf{m}^{k+1}-\mathbf{T}+\mathbf{ z}^{k}\|_{2}^{2} \tag{11}\]  
\[\mathbf{z}^{k+1}=\mathbf{z}^{k}+\mathbf{m}^{k+1}-\mathbf{T}^{k+1} \tag{12}\]  
where \(\mathbf{z}=\mathbf{c}/\xi\) denotes the scaling Lagrangian multiplier and \(k\) represents the \(k\)-th iteration.