# Modulus Stretch-Based Circular SAR Imaging with Contour Thinning
## 2 Analysis of Imaging Algorithms
### Contour Thinning Algorithm  
The process of contour thinning algorithm is as follows:  
1. The _Nff_-point IFFT is performed on the echo data \(S(f_{k},\tau_{n})\) received at each slow time \(\tau_{n}\) to obtain an \(M\)-point inverse transform sequence, and then cyclically shifts with the FFTshift(\(\bullet\)) function [11]. \[s_{0}(m,\tau_{n})=\text{FFTShift}[\text{IFFT}[S(f_{k},\tau_{n})]]\] (12)
2. The real coordinates corresponding to each pixel on the image are calculated, and the \(\Delta R(\mathbf{r},\tau_{n})\) corresponding to each pixel is calculated [11]. \[\left\{\begin{array}{ll}\Delta R(\mathbf{r},\tau_{n})=d_{a0}(\mathbf{r}, \tau_{n})-d_{a}(\mathbf{r},\tau_{n})\\ d_{a0}(\mathbf{r},\tau_{n})=\sqrt{\left(x_{a}(\tau_{n})-x\right)^{2}+\left(y_{ a}(\tau_{n})-y\right)^{2}+\left(z_{a}(\tau_{n})-z\right)^{2}}\\ d_{a}(\mathbf{r},\tau_{n})=\sqrt{x_{a}^{2}(\tau_{n})+y_{a}^{2}(\tau_{n})+z_{a} ^{2}(\tau_{n})}\end{array}\right.,\] (13) where \(d_{a}(\mathbf{r},\tau_{n})\) denotes the distance of the antenna from the origin of the ground at coordinate \(\mathbf{r}\) and the time \(\tau_{n}\). \(d_{a0}(\mathbf{r},\tau_{n})\) denotes the distance of the antenna from the target in the ground.
3. Linear interpolation is performed on \(s_{0}(m,\tau_{n})\) to obtain the estimated values of the corresponding IFFT transformation at all \(\Delta R(\mathbf{r},\tau_{n})\) points, and the sub-imaging data at each time \(\tau_{n}\) is obtained by multiplying by the data the compensation term [11]. \[s_{int}(\mathbf{r},\tau_{n})=\text{Interp}[s_{0}(m,\tau_{n})]\left(\frac{+j4 \pi f_{0}\Delta R(\mathbf{r},\tau_{n})}{c}\right),\] (14) \[S_{int}(\tau_{n})=\left\{s_{int}(\mathbf{r}_{x,y},\tau_{n})\right\}|x,y\in[ 1,L],\] (15) where \(L\) denotes the real side length of the imaged scene.
4. Let \(\theta\) denote the value of the synthetic aperture. The sub-images of all \(\tau_{n}\) in the range of the synthetic aperture are superimposed, and then the modulus is stretched for each sub-aperture image. Finally, all the sub-aperture images are superimposed to obtain the final imaging results. \[I_{a}^{\prime}=\sum_{\theta_{i}}\psi\left(\sum_{\tau_{n}\in\theta}S_{int}( \tau_{n})\right)\] (16)  
The usual contrast stretch transformation includes gamma transformation, piecewise linear transformation, histogram equalization, and so on. Among them, histogram transformation will improve the brightness of dark areas when the low-brightness area is in the majority, which is not in line with our requirement to significantly reduce the low-modulus part. Gamma transformation and piecewise linear transformation, on the other hand, can adjust parameters so that highlights remain and dark areas are significantly reduced. Therefore, this paper mainly studies the effect of modulus stretch on image contour thinning by using gamma transformation and piecewise linear transformation.  
Gamma transformation in image processing is based on real number transformation, while radar data is complex data, so it is slightly modified to achieve modulus stretch. Gamma transformation based on modulus stretch is shown as follows:  
\[\psi_{1}(x)=|x|^{y}x \tag{17}\]  
Similarly, the piecewise linear transformation of complex numbers is given in Equation (18). Where \(T\) is the threshold value, \(k_{1}\) is the enhancement coefficient, and \(k_{2}\) is the inhibition coefficient.  
\[\psi_{2}(x)=\left\{\begin{array}{l}k_{1}x,\,|x|\geq T\\ k_{2}x,\,|x|<T\end{array}\right. \tag{18}\]