# Sparse Parabolic Radon Transform with Nonconvex Mixed Regularization for Multiple Attenuation
## 2 Methodology
### High-Resolution Parabolic Radon Transform with \(L_{q_{1}}-L_{q_{2}}\) Mixed Regularization  
To suppress multiple \(\mathbf{m}_{2}\) from \(\mathbf{d}\), we use \(L_{q_{1}}-L_{q_{2}}\) mixed regularization with \(0<q_{1},q_{2}<1\) for sparsity promotion and use the following formulation:  
\[\min_{\mathbf{m}_{1}\mathbf{m}_{2}}\biggl{\{}\frac{1}{\beta}\|\mathbf{A}_{1}\mathbf{m}_{1}+\bm {A}_{2}\mathbf{m}_{2}-\mathbf{d}\|_{2}^{2}+\mu\|\mathbf{m}_{1}\|_{q_{1}}^{q_{1}}+\|\mathbf{m}_ {2}\|_{q_{2}}^{q_{2}}\biggr{\}} \tag{12}\]  
where \(\mu\) is a positive parameter, \(\beta>0\) is a penalty parameter, \(\mathbf{A}=\mathbf{A}_{1}=\mathbf{A}_{2}\) is the Radon operator, \(\mathbf{m}_{1}\) is primary and \(\mathbf{m}_{2}\) is multiple. To determine \(q_{1}\) and \(q_{2}\), we explore the \(L_{q}\)(\(0<q<1\)) regularization. [PERSON] et al. [25] introduced \(L_{q}\)(\(0<q<1\)) regularization and proved that the \(L_{q}\) regularization can obtain much sparser solutions than \(L_{1}\) regularization. Meanwhile, they proposed \(L_{1/2}\) regularization, and it is the most representative regularization method for \(L_{q}\)(\(0<q<1\)) regularization. The study shows that the \(L_{1/2}\) regularization is the sparsest and the most robust among the \(L_{q}\)(\(1/2\leq q<1\)) regularization, and when \(0<q<1/2\), the \(L_{q}\) regularization has similar properties to the \(L_{1/2}\) regularization. Therefore, we select \(q_{1}=q_{2}=1/2\) in this paper. \(L_{q_{1}}-L_{q_{2}}\) mixed regularization is the nonconvex problem, and it is a very complicated process to solve. In this paper, we use an improved alternative direction method of multipliers algorithm (ADMM) to approximate the solution [34]. The ADMM algorithm can be used to solve many high-dimensional problems and uses a decomposition-coordination procedure to decouple the variables [34; 35; 36; 40]. This algorithm enables the difficult global problem to be properly solved. The Problem (12) can be formulated as  
\[\min_{\mathbf{m}_{1}\mathbf{m}_{2}}\biggl{\{}\|\mathbf{A}_{1}\mathbf{m}_{1}+\mathbf{A}_{2}\mathbf{m} _{2}-\mathbf{d}\|_{2}^{2}+\beta\mu\|\mathbf{z}_{1}\|_{q_{1}}^{q_{1}}+\beta\|\mathbf{z}_{2} \|_{q_{2}}^{q_{2}}\biggr{\}} \tag{13}\]  
where the auxiliary variables are \(\mathbf{z}_{1}=\mathbf{m}_{1}\), \(\mathbf{z}_{2}=\mathbf{m}_{2}\). The augmented Lagrangian function is  
Figure 2: Unit ball pictures for (**a**) \(L_{0}\), (**b**) \(L_{2}\), (**c**) \(L_{1}\) and (**d**) \(L_{1/2}\) regularization. \[L(\mathbf{m}_{1},\mathbf{m}_{2},z_{1},z_{2},\ \mathbf{w}_{1},\mathbf{w}_{2})\] \[=\|\mathbf{A}_{1}\mathbf{m}_{1}+A_{2}\mathbf{m}_{2}-\mathbf{d}\|_{2}^{2}+\beta\mu\| \mathbf{z}_{1}\|_{q_{1}}^{q_{1}}+\beta\|\mathbf{z}_{2}\|_{q_{2}}^{q_{2}}+\langle\mathbf{w}_ {1},\mathbf{m}_{1}-\mathbf{z}_{1}\rangle+\langle\mathbf{w}_{2},\mathbf{m}_{2}-\mathbf{z}_{2}\rangle\] \[+\tfrac{\rho_{1}}{2}\|\mathbf{m}_{1}-\mathbf{z}_{1}\|_{2}^{2}+\tfrac{\rho_ {2}}{2}\|\mathbf{m}_{2}-\mathbf{z}_{2}\|_{2}^{2}\]  
where \(\rho_{1}\) and \(\rho_{2}\) are positive penalty parameters and \(\mathbf{w}_{1}\) and \(\mathbf{w}_{2}\) are the dual variables.