## 3 Proposed Algorithm
### SAR-SIFT Image Registration Algorithm  
The Scale-Invariant Feature Transform (SIFT) algorithm is a classic method used for feature extraction and matching in images. It exhibits features such as scale invariance,rotation invariance, and illumination invariance, which enables it to extract stable feature points under different scales, angles, and lighting conditions. However, due to the unique characteristics of SAR images, such as strong noise, low contrast, and irregular reflections, traditional SIFT algorithms cannot be directly applied to SAR images for feature extraction. The SAR-SIFT algorithm improves the SIFT algorithm by introducing the characteristics of SAR images to adapt to the special properties of SAR images. Specifically, the SAR-SIFT algorithm introduces techniques such as adaptive Gaussian filtering and adaptive local binarization to enhance the contrast and stability of SAR images, thereby achieving more accurate feature point extraction and matching. The following is the specific operation process of the SAR-SIFT algorithm for registration:  
**Establish the SAR-Harris scale space**  
Calculate the gradient of SAR image using the ROEWA. For orientation \(\varphi\), the weighted averages of bilateral windows are as follows:  
\[\left\{ \begin{array}{l}r_{1}(x,y\ |\ \varphi)=\sum_{(x^{\prime},y^{\prime})}g_{1 }(x^{\prime},y^{\prime})I(x+x^{\prime},y+y^{\prime})\\ r_{2}(x,y\ |\ \varphi)=\sum_{(x^{\prime},y^{\prime})}g_{2}(x^{\prime},y^{ \prime})I(x+x^{\prime},y+y^{\prime})\end{array} \right. \tag{6}\]  
In the formula, \(r_{1}\) and \(r_{2}\) are local exponential weighted averages, \(I\) is the SAR image, \(x\) and \(y\) are image pixel coordinates, \(g_{1}\) and \(g_{2}\) are exponential weighted filters, respectively:  
\[\left\{ \begin{array}{l}g_{1}(x,y\ |\ \varphi)=\exp\left(-\frac{x+y}{\alpha} \right)\ \ \ x\cos\varphi+y\sin\varphi>0\\ g_{2}(x,y\ |\ \varphi)=\exp\left(-\frac{x+y}{\alpha}\right)\ \ \ x\cos\varphi+y\sin\varphi<0\end{array} \right. \tag{7}\]  
\(\varphi=0^{\circ}\) and \(\varphi=90^{\circ}\) are horizontal and vertical direction exponential weighted filters. The gradient of the SAR image is calculated based on the ratio of local exponential weighted averages between horizontal and vertical directions. The gradient of an SAR image is as follows:  
\[\left\{ \begin{array}{l}G_{x,a}(x,y)=\log\left(\frac{r_{1}(x,y|\varphi=90^{ \circ})}{r_{2}(x,y|\varphi=90^{\circ})}\right)\\ G_{y,a}(x,y)=\log\left(\frac{r_{1}(x,y|\varphi=0^{\circ})}{r_{2}(x,y|\varphi= 0^{\circ})}\right)\\ G_{n,a}(x,y)=\sqrt{G_{x,a}(x,y)^{2}+G_{y,a}(x,y)^{2}}\\ G_{t,a}(x,y)=\arctan\left(\frac{G_{y,a}(x,y)}{G_{x,a}(x,y)}\right)\end{array} \right. \tag{8}\]  
Then, we used the obtained gradient to establish the SAR-Harris scale space. The expression of multi-scale SAR-Harris function is as follows:  
\[C_{SH}(x,y,a)=g_{\sqrt{2a}}\left[ \begin{array}{cc}(G_{x,a})^{2}&(G_{x,a}) \big{(}G_{y,a}\big{)}\\ \big{(}G_{y,a}\big{)}(G_{x,a})&\big{(}G_{y,a}\big{)}^{2}\end{array} \right] \tag{9}\]  
In the formula, \(g_{\sqrt{2a}}\) represents a Gaussian kernel with a standard deviation of \(\sqrt{2}\). \(C_{SH}(x,y,a)\) represents the value of SAR-Harris scale space at \((x,y,a)\). **Feature points detection and precise localization. **  
For each point in the multi-scale space, detect feature points using the DoG operator. The DoG operator is a scale space transformation that can detect local extreme points at different scales. To obtain more robust extreme points, we adopted the following formula to filter out low-contrast points in the initial extreme points:  
\[R_{SH}(x,y,a)=\det(C_{SH}(x,y,a))-d\times\text{tr}(C_{SH}(x,y,a))^{2} \tag{10}\]  
In the formula, _det_ represents calculating the determinant of the matrix, _tr_ represents calculating the trace of the matrix, and \(d\) is the adjustment parameter set to 0.04. **Assign the main direction. **Next, we used the ROEWA gradient to calculate the gradient histogram in the key point neighborhood.