# Modulus Stretch-Based Circular SAR Imaging with Contour Thinning
## 2 Analysis of Imaging Algorithms
### Backprojection Algorithm
\right]\] \[=\left[\begin{array}{c|c|c}\left|\sum_{m=1}^{N_{p}}S_{int}( \mathbf{r}_{1,1},\tau_{n})\right|&\left|\sum_{n=1}^{N_{p}}S_{int}(\mathbf{r}_ {1,2},\tau_{n})\right|&\cdots&\left|\sum_{n=1}^{N_{p}}S_{int}(\mathbf{r}_{1,L },\tau_{n})\right|\\ \left|\sum_{n=1}^{N_{p}}S_{int}(\mathbf{r}_{2,1},\tau_{n})&\left|\sum_{n=1}^{N _{p}}S_{int}(\mathbf{r}_{2,2},\tau_{n})\right|&\cdots&\left|\sum_{n=1}^{N_{p} }S_{int}(\mathbf{r}_{2,L},\tau_{n})\right|\\ \vdots&\vdots&\vdots&\vdots\\ \left|\sum_{n=1}^{N_{p}}S_{int}(\mathbf{The angle of the stripes changes with different \(\tau_{n}\). The scattering properties of the target also exhibit different echo data as the angle changes, as shown in Figure 1.  
The final imaging result of the scene is the superposition of this series of striped images. It can be intuitively concluded that the wider the sub-image stripes at each \(\tau_{n}\) time, the larger the area of the final imaging target after the superposition. If the imaging result is not for the purpose of restoring the true scattering properties of the target, but for the purpose of clearer and thinner contours of the target, then another idea can be considered. Consider introducing a transformation function \(\psi(\cdot)\) into Equation (6). The purpose of the transformation function is to make the contour in the image clearer and thinner, but to avoid changing the overall shape of the original contour in the image. In order to satisfy this requirement, \(\psi(\cdot)\) should be a monotone function. The transformed image is denoted as  
\[I^{\prime}_{n}=\sum_{n=1}^{N_{p}}\psi(S_{int}(\tau_{n})) \tag{8}\]  
In order to evaluate the thinning degree of contour after transformation, appropriate evaluation indexes should be defined. Considering that the clearer and thinner the contour, the smaller the target area, the target area can be used as one of the evaluation factors. However, if the area decreases and the perimeter also decreases, it becomes the reduction of the overall area, rather than the thinning of the contour width, which fails to achieve the purpose of contour thinning. Therefore, the perimeter of the area should also be taken into account. Based on the above two points, a simple and intuitive evaluation index \(D(I)\) of contour thinning degree is defined. \(D(I)\) is defined as the perimeter of the pixels of all the target areas in a binary image divided by the area of the pixels of all the target areas. \[D(I)=\frac{\text{pixel perimeter of target}}{\text{pixel area of target}} \tag{9}\]  
After introducing the evaluation index of contour thinning degree, the optimization problem can be described as follows:  
\[\underset{\psi(\cdot)}{\text{argmax}}\;D(I^{\prime}_{m}),\;s.t.\;\psi(\cdot) \in\textit{monotone function} \tag{10}\]  
Equation (10) is a compound function optimization problem, which cannot be solved directly. It probably does not have an optimal solution, but a suboptimal solution can be constructed based on given conditions. The problem description is modified to  
\[\text{Find the appropriate }\psi(\cdot),s.t.\Bigg{\{} \begin{array}{c}D(I^{ \prime}_{m})>D(I_{m})\\ \psi(\cdot)\in\textit{monotone function}\end{array}  \tag{11}\]  
Figure 1: Single azimuth stripe image: (**a**) azimuth = 0°; (**b**) azimuth = 45°; (**c**) azimuth = 100°. Intuitively, there are two ways to improve the contour thinning degree, one is to increase the perimeter, the other is to reduce the area. Firstly, the perimeter can be increased by adding new area, or by corroding the existing area to make the edges more complicated. However, neither method obviously makes the contour thinner. Therefore, it can only be achieved by reducing the area. If the stripes of the sub-aperture image can be narrowed, it is possible to reduce the target area of the final superimposed image. Reducing the area can be achieved by stretching the modulus of the sub-aperture image, that is, enhancing the region with high modulus and greatly reducing the region with low modulus.