## 3 The Proposed Method  
### Sparse Reconstruction Based on Kronecker Dictionary  
Compressed sensing can reconstruct the source signal from the undersampled measurements. Applying the technique to SAR super-resolution imaging still faces many challenges. The sparse imaging method is mainly to solve an ill-posed linear inverse problem by using the sparsity of radar cross section (RCS). For the large scale data, the classic \(l_{0}\) procedure will be rigorously challenged by many computation problems. For instance, considering a scene with \(1024\times 1024\) observation points, the classic reconstruction approach will deal with data in length up to \(1048576\), which leads to a tremendous linear system, consuming computation resource significantly. From the analysis above, it is known that the target scene has the sparsity and the scatterers usually clustered together to blocky areas. Therefore, considering the structured sparse feature, the SAR imaging can be treated as a sparse reconstruction problem with Kronecker dictionary. A multiway structured sparse matching pursuit algorithm can be constructed to estimate the scatter coefficients with much fewer iterations. Firstly, the SAR echo signal must be decoupled, for which the tensor data \(\mathcal{Y}\in\mathbb{R}^{P_{1}\times P_{2}\times\cdots\times P_{N}}\), and dictionaries along each mode \(\mathbf{A}_{n}\in\mathbb{R}^{P_{n}\times M_{n}}\), \(n=1,2,\ldots,N\) have to be constructed. Our purpose is reconstructing the nonzero scatter coefficients from \(\mathcal{Y}\), i.e., finding a \((K_{1},K_{2},\ldots,K_{N})\)-structured sparse representation corresponding to \(\mathbf{A}_{n}\left(n=1,2,\ldots,N\right)\). Because of the sparsity of target scene, only a fewer rows need to be randomly extracted from the dictionary. Let \(\mathbf{B}_{n}=\mathbf{A}_{n}\left(:,\mathscr{I}_{n}\right),\;n=1,2,\ldots,N\), and then the Tucker decomposition of the original signal can be rewritten as  
\[\hat{\mathbf{y}}=\left(\mathbf{B}_{N}\otimes\mathbf{B}_{N-1}\otimes\cdots\otimes\mathbf{B}_{1} \right)\mathbf{s}_{nz}, \tag{13}\]  
where \(\mathbf{s}_{nz}\in\mathbb{R}^{K}\) is the vectorization of all nonzero elements. Consequently, the solution of the problem can be described as  
\[\mathbf{s}_{nz}=\operatorname*{arg\,min}_{\mathbf{u}}\left\|\left(\mathbf{B}_{N}\otimes \mathbf{B}_{N-1}\otimes\cdots\otimes\mathbf{B}_{1}\right)\mathbf{u}-\mathbf{y}\right\|_{2}^{2}. \tag{14}\]  
Defining \(\mathbf{B}=\mathbf{B}_{N}\otimes\mathbf{B}_{N-1}\otimes\cdots\otimes\mathbf{B}_{1}\), the solution of (14) can be written as \(\mathbf{s}_{nz}=\left(\mathbf{B}^{T}\mathbf{B}\right)^{-1}\mathbf{B}\mathbf{y}\). Efficient calculation steps can be deduced by Cholesky decomposition [40]. ```
0: decoupled echo signal \(\mathcal{Y}\); the number of maximum nonzero entries \(k_{max}\); dictionaries along each mode \(\{\mathbf{A}_{1},\mathbf{A}_{2},...,\mathbf{A}_{N}\}\); the threshold of error \(\epsilon\). 0: scattering coefficients estimate \(\hat{\mathcal{S}}\). 1: Initialization: \(k\gets 1\); \(\mathcal{R}\leftarrow\mathcal{Y}\); \(\hat{\mathcal{S}}\leftarrow\mathbf{0}\); \(\mathscr{I}_{n}\leftarrow\mathcal{O}\), \(n=1,2,\ldots,N\).