## 2 The Tensor Modeling of Structured Sparse SAR Imaging  
### Received Signal Model  
For convenience of statement hereafter, several notions of this paper are explained as follows. The scalar is indicated by italic letter, e.g., \(s\); the vector and matrix are indicated by bold italic letters, e.g., \(\mathbf{v},\mathbf{M}\); the tensor is indicated by calligraphic capital letter, e.g., \(\mathcal{T}\); the index of a set is indicated by script letter, e.g., \(\mathcal{I}\). Here we consider the point-scattering models [3]. The spotlight SAR geometry is illustrated in Figure 1. The target scene is represented by \(O(x,y,z)\) coordinate system. The azimuth and pitch angles of antenna relating to the center of the scene are \(\phi\) and \(\theta\). The distance between target and antenna is \(R\left(x,y,z\right)=R_{c}-x\cos\left(\theta\right)\cos\left(\phi\right)+y\cos \left(\theta\right)\sin\left(\phi\right)+z\sin\left(\theta\right)\), where \(R_{c}\) denotes the distance between the antenna and the center of target area. The time delay for the target point \(\left(x,y,z\right)\) in SAR echo signal is \(\tau\left(x,y,z\right)=2R\left(x,y,z\right)/c\), where \(c\) is the speed of light. After matched filtering, the reflection signal is  
\[y\left(t,\theta,\phi\right)=\sum_{x,y,z}g\left(x,y,z\right)s\left(t-\tau\left( x,y,z\right)\right)+e\left(t\right), \tag{1}\]  
where \(g\left(x,y,z\right)\) is the scattering function of scatterer placed at \(\left(x,y,z\right)\); \(s\left(t\right)\) is the autocorrelation function of transmitted signal; \(e\left(t\right)\) is the additive noise. In the following discussion the noise is neglected for notational simplicity. Then the phase history can be obtained through Fourier transform:  
\[Y\left(f,\theta,\phi\right)=C\left(f\right)\sum_{x,y,z}g\left(x,y,z\right)\exp \left(H_{x,y,z}\left(f,\theta,\phi\right)\right), \tag{2}\]  
Figure 1: The geometry of spotlight SAR for point scatterers. where \(C\left(f\right)=S\left(f\right)\exp\left(-j2\pi f\frac{2R_{c}}{c}\right)\) can be ignored as it is a constant for all scatterers, and the phase factor is \(H_{x,y,z}\left(f,\theta,\phi\right)=j\frac{4\pi}{c}\left(xf\cos\theta\cos\phi+ yf\cos\theta\sin\phi+zf\sin\theta\right)\). For decoupling between the parameters in the phase factor, an interpolation approach is applied and the polarization coordinates is transformed to a Cartesian coordinates [33]. Assume there are \(M_{1},M_{2},M_{3}\) grid points along the directions of the axis of x, y, z, respectively; and the number of grid points for \(f,\theta,\phi\) are \(P_{1},P_{2},P_{3}\), respectively. Then for the scatterer at \(\left(x_{j},y_{k},z_{l}\right)\), the steering vector is \(\mathbf{a}_{j,k,l}=\left[b_{j,k,l}\left(1,1,1\right),\ldots,b_{j,k,l}\left(P_{1}, P_{2},1\right),\ldots,b_{j,k,l}\left(1,1,P_{3}\right),\ldots,b_{j,k,l} \left(P_{1},P_{2},P_{3}\right)\right]^{T}\), where \(b_{j,k,l}\left(p_{1},p_{2},p_{3}\right)=\exp\left(j\frac{4\pi}{c}\left(x_{j}u _{p_{1}}+y_{k}v_{p_{2}}+z_{l}w_{p_{3}}\right)\right)\). Let \(M=M_{1}\times M_{2}\times M_{3}\), \(P=P_{1}\times P_{2}\times P_{3}\). Define the steering matrix \(\mathbf{A}\in\mathbb{C}^{P\times M}\) as  
\[\mathbf{A}=\left[\mathbf{a}_{1,1,1},\ldots,\mathbf{a}_{M_{1},M_{2},1},\ldots,\mathbf{a}_{1,1,M _{3}},\ldots,\mathbf{a}_{M_{1},M_{2},M_{3}}\right], \tag{3}\]  
then the received signal model (2) can be rewritten as  
\[\mathbf{y}=\mathbf{A}\mathbf{s}, \tag{4}\]  
where \(\mathbf{s}=\left[g\left(x_{1},y_{1},z_{1}\right),\ldots,g\left(x_{M_{1}},y_{M_{2} },z_{1}\right),\ldots,g\left(x_{1},y_{1},z_{M_{3}}\right),\ldots,g\left(x_{M_{ 1}},y_{M_{2}},z_{M_{3}}\right)\right]^{T}\). Note that each element in the steering matrix  
\[a_{p_{1}p_{2}p_{3},m_{1}m_{2}m_{3}}=\exp\left(j\frac{4\pi}{c}\left(x_{m_{1}}u _{p_{1}}+y_{m_{2}}v_{p_{2}}+z_{m_{3}}w_{p_{3}}\right)\right),\]  
then (3) can be rewritten as  
\[\mathbf{A}=\mathbf{A}_{3}\otimes\mathbf{A}_{2}\otimes\mathbf{A}_{1}, \tag{5}\]  
where \(\otimes\) denotes the Kronecker product.