# Spaceborne Sparse SAR Imaging Mode Design: From Theory to Implementation
## 3 Design and Performance Analysis of Sparse SAR Imaging Mode
### Mixed-Norm Regularization-Based Sparse SAR Imaging  
\(L_{q}\)**-norm Regularization-Based Sparse Imaging. ** Let \(N_{p}\) and \(N_{Q}\) denote the number of points in the azimuth and range directions, and let \(\textbf{X}\in\mathbb{C}^{N_{p}\times N_{Q}}\) and \(\textbf{Y}\in\mathbb{C}^{N_{x}\times N_{t}}\) denote the backscattered coefficient of the surveillance region and echo data, respectively. \(\mathcal{I}(\cdot)\) and  
Figure 8: AASR versus PRF at different azimuth positions in Scan mode. Figure 10: The relationship between RASR and PRF in Scan mode. Figure 9: The relationship between RASR and incidence angle in Scan mode. \(\mathcal{G}(\cdot)\) are the imaging and inverse imaging operators of the MF-based algorithm. The echo data under sparse parameters can be regarded as the echo data received under traditional parameters with azimuth downsampling. Then, the approximated observation-based sparse SAR imaging model can be expressed as  
\[\mathbf{Y}\ =\ \mathbf{H}\circ\mathcal{G}(\mathbf{X})\ +\ \mathbf{N} \tag{16}\]  
where \(\mathbf{H}\) is the downsampling matrix in the azimuth direction, \(\mathbf{N}\) is the noise, and \(\circ\) is the Hadamard product operator. For the model in (16), the considered scene can be reconstructed by solving the \(L_{q}(0<q\leq 1)\)-norm regularization problem [34; 39], i.e.,  
\[\hat{\mathbf{X}}\ =\ \arg\min_{\mathbf{X}}\left\{\|\mathbf{Y}-\mathbf{H}\circ \mathcal{G}(\mathbf{X})\|_{\text{F}}^{2}\ +\ \beta\|\mathbf{X}\|_{q}\right\} \tag{17}\]  
where \(\hat{\mathbf{X}}\) is the reconstructed image of the considered 2-D scene, and \(\beta\) is the regularization parameter. \(L_{2,q}\)-norm Regularization-Based Sparse ImagingThe above \(L_{q}\)-norm regularization-based sparse imaging method can recover the large-scale surveillance area, but it fails to achieve ambiguity suppression. The sparse SAR imaging mode inevitably produces azimuth ambiguity due to the adoption of a downsampled beam position. To achieve high-quality sparse imaging in the designed modes, an \(L_{2,q}\)-norm regularization-based method is proposed and introduced for the data processing of the collected echo data. Because the azimuth ambiguities are considered in the construction of the imaging model, the proposed method can achieve ambiguity suppression and hence obtain the unambiguous sparse image of the considered scene. The unambiguous imaging model can be written as [41]  
\[\mathbf{Y}\ =\ \mathbf{H}\circ\left(\mathcal{G}(\mathbf{X})+\sum_{i}\mathcal{ G}_{i}(\mathbf{X}_{i})\right)\ +\ \mathbf{N},\ i\in\mathbb{Z},i\
eq 0 \tag{18}\]  
where \(\mathcal{G}_{i}(\cdot)\) refers to the inverse imaging operators for ambiguity areas, and \(\mathbf{X}_{i}\in\mathbb{C}^{N_{p}\times N_{\mathbb{Q}}}\) with \(i\in\mathbb{Z}^{-}\) and \(i\in\mathbb{Z}^{+}\) are the left and right ambiguity areas. Then, the considered scene can be recovered by solving the \(L_{2,q}\)-norm regularization problem, i.e.,  
\[\hat{\mathbf{X}}\ =\ \arg\min_{\mathbf{X}}\left\{\left\|\mathbf{Y}-\mathbf{H} \circ\left(\mathcal{G}(\mathbf{X})+\sum_{i}\mathcal{G}_{i}(\mathbf{X}_{i}) \right)\right\|_{\text{F}}^{2}\ +\ \beta_{1}\|\mathbf{X}_{\text{all}}\|_{2,q}^{q}\ +\ \beta_{2}\|\mathbf{X}\|_{1}\right\} \tag{19}\]  
where \(\hat{\mathbf{X}}\) is the reconstructed image of the considered scene, \(\beta_{2}\) is the regularization parameter which controls the sparsity of \(\mathbf{X}\), and \(\beta_{1}\) controls the sparsity of the whole scene \(\mathbf{X}_{\text{all}}\), which can be expressed as  
\[\|\mathbf{X}_{\text{all}}\|_{2,q}^{q}=\left(\sum_{np=1}^{N_{p}}\left(|\mathbf{ X}_{np}|+\sum_{i}|\mathbf{X}_{i,np}|\right)^{q}\right)^{1/q} \tag{20}\]  
where \(\mathbf{X}_{np}\) and \(\mathbf{X}_{i,np}\) are the \(np\)th rows of \(\mathbf{X}\) and \(\mathbf{X}_{i}\), and \(np=1,2,\cdots,N_{P}\). The detailed iterative process of solving the regularization problem in (20) is depicted in [41] when \(q=1\). Then, the focused unambiguous image can be obtained.