# Modulus Stretch-Based Circular SAR Imaging with Contour Thinning
## 2 Analysis of Imaging Algorithms  
### Backprojection Algorithm  
According to the principle of SAR detection, the radar transmits a linear continuous frequency modulation wave to the target and receives the echo by matched filter. Radar echo data are functions of receiving frequency \(f\) and slow time \(\tau\), which is denoted by \(S(f,\tau)\). Let \(\mathbf{r}=(x,y,z)\) denote the target coordinates in the detection scene. Let \(\Delta R(\mathbf{r},\tau)\) denote the difference between the distance from the sensor to the origin and the distance from the sensor to the target with the coordinate \(\mathbf{r}\) at time \(\tau\). The output of the matched filter is the scattering response of the target at coordinate \(\mathbf{r}\) in the scene, which is denoted by \(I(\mathbf{r})\), is given by [15]  
\[I(\mathbf{r})=\iint S(f,\tau)\exp\left(\frac{+j4\pi f\Delta R(\mathbf{r},\tau) }{c}\right)dfd\tau, \tag{1}\]  
where \(c\) denotes the constant speed of light. Since Equation (1) is calculated point-by-point and the speed is too slow, it is rewritten to facilitate the use of the Inverse Fast Fourier Transform (IFFT) to obtain the backprojection algorithm. Rewrite Equation (1) as a discrete form [11]:  
\[I(\mathbf{r})=\frac{1}{N_{p}K}\sum_{n=1}^{N_{p}}\sum_{k=1}^{K}S(f_{k},\tau_{n })\exp\left(\frac{+j4\pi f_{k}\Delta R(\mathbf{r},\tau_{n})}{c}\right), \tag{2}\]  
where \(Np\) denotes the number of sampling points of the sensor on the circular orbit. \(K\) denotes the sampling frequency of the echo of the sensor at each circular orbital position. First consider the inner summation, which can be regarded as the IFFT of \(S(f_{k},\tau_{n})\). After transformation, the inner summation can be rewritten as [11]:  
\[s(m,\tau_{n})=N_{fft}\text{FF}\text{shift}\{\text{IFFT}[S(f_{k},\tau_{n})]\}\exp \biggl{(}\frac{+j2\pi f_{1}\Delta R(m,\tau_{n})}{cN_{fft}}\biggr{)}, \tag{3}\]  
where _Nfft_ denotes the length of the IFFT. To put \(m=1\) at the center of the range profile, the function FFTshift(\(\bullet\)) is applied to the output of the IFFT. Since \(m\) is not the coordinate value in the real scene, it is necessary to interpolate the real coordinate value in order to obtain the IFFT sequence corresponding to the real coordinate approximately. The interpolated expression is denoted by [11]  
\[s_{int}(\mathbf{r},\tau_{n})=\text{Interp}[s(m,\tau_{n})], \tag{4}\]  
where \(\mathbf{r}\) represents the true coordinates in the scene. Interpolating all the coordinates in the scene, the overall imaging result of the scene at time \(\tau_{n}\) is obtained, \(S_{int}(\tau_{n})\), given by  
\[S_{int}(\tau_{n})=\left[ \begin{array}{cccc}s_{int}(\mathbf{r}_{1,1},\tau_{ n})&s_{int}(\mathbf{r}_{1,2},\tau_{n})&\cdots&s_{int}(\mathbf{r}_{1,L},\tau_{n}) \\ s_{int}(\mathbf{r}_{2,1},\tau_{n})&s_{int}(\mathbf{r}_{2,2},\tau_{n})&\cdots&s _{int}(\mathbf{r}_{2,L},\tau_{n})\\ \vdots&\vdots&\vdots&\vdots\\ s_{int}(\mathbf{r}_{L,1},\tau_{n})&s_{int}(\mathbf{r}_{L,2},\tau_{n})&\cdots&s _{int}(\mathbf{r}_{L,L},\tau_{n})\end{array} \right] \tag{5}\]  
where \(L\) represents the pixel length of the final image size. Finally, all the data at the slow time are superimposed together to obtain the coherent imaging result of backprojection, represented by _Ia_, as follows [11]:  
\[I_{a}=\sum_{n=1}^{N_{p}}S_{int}(\tau_{n}) \tag{6}\]  
Since the echo data is complex, the final _Ia_ is also a complex matrix, so the modulus value is used to represent the grayscale in the image. Let \(\text{abs}(\bullet)\) represent the modulus of each element in the matrix, and the grayscale matrix of the image is denoted by  
\[I_{m}=\text{abs}(I_{a}) =\left[ \begin{array}{c|c|c|c}\left|I_{a}(1,1)\right|&\left|I_{ a}(1,2)\right|&\cdots&\left|I_{a}(1,L)\right|\\ \left|I_{a}(2,1)\right|&\left|I_{a}(2,2)\right|&\cdots&\left|I_{a}(2,L)\right| \end{array} \right]\] (7) \[=\left[ \begin{array}{c|c|c}\left|I_{a}(1,1)\right|&\left|I_{a}(1,2)\right|&\cdots&\left|I_{a}(1,L)\right|\\ \left|I_{a}(2,1)\right|&\left|I_{a}(2,2)\right|&\cdots&\left|I_{a}(L,L) \right|\end{array}