## 2 Baseline Algorithm, SAR-SIFT: A SIFT-like Algorithm for SAR Images
### Scale Space Construction  
The differential gradient algorithm may lead to variations in false alarm rates on SAR images, with higher false alarm rates in regions with high reflectivity compared to those in low reflectivity regions. Therefore, the classical differential gradient calculation method is not a constant false alarm rate algorithm, and the use of the ratio of exponentially weighted averages (ROEWA) is more suitable for multiplicative noise than differentiation. To achieve a constant false alarm rate on SAR images, the ratio of exponentially weighted averages [24], an edge detector, can be employed. The ratio of exponential weighted average ROEWA is an improvement on the ratio of average ROA [25] (the ratio of average), which calculates the exponential weighted local average. For instance, the average value of a given point \((c,d)\) with horizontal direction \(i=1\) and vertical direction \(i=3\) are defined as:  
\[ \begin{cases}M_{1,\alpha}(i=1)=\int_{x=R}\int_{y=R^{+}}I(c+x,d+y)\times e^{- \frac{|x|+|y|}{\alpha}}\\ M_{2,\alpha}(i=1)=\int_{x=R}\int_{y=R^{-}}I(c+x,d+y)\times e^{-\frac{|x|+|y|}{ \alpha}}\\ M_{1,\alpha}(i=3)=\int_{x=R}\int_{y=R^{+}}I(a+x,b+y)\times e^{-\frac{|x|+|y|}{ \alpha}}\\ M_{2,\alpha}(i=3)=\int_{x=R}\int_{y=R^{-}}I(a+x,b+y)\times e^{-\frac{|x|+|y|}{ \alpha}}\\ \end{cases}  \tag{1}\]  
Among \(\alpha\) is an exponential weight parameter. The ratio in direction \(i\) and its normalization are defined as:  
\[Ratio_{i,\alpha}=\frac{M_{1,\alpha}(i)}{M_{2,\alpha}(i)} \tag{2}\]  
\[T_{i,\alpha}=max(Ratio_{i,\alpha},\frac{1}{Ratio_{i,\alpha}}) \tag{3}\]  
These ratios \(T_{i,\alpha}\) in \(\alpha\) is calculated along the horizontal \((i=1)\) and vertical \((i=3)\) directions, and the edge image is obtained as follows:  
\[D^{2}_{n,\alpha}=\sqrt{\left(T_{1,\alpha}\right)^{2}+\left(T_{3,\alpha} \right)^{2}} \tag{4}\]  
ROEWA is more accurate and robust to noise in multi-scale edge contexts because the weighted adaptive smoothing parameters \(\alpha\) can be applied to images. The horizontal and vertical gradients of SAR-SIFT are defined as:  
\[ \begin{cases}G_{x,\alpha}=\log\left(R_{1,\alpha}\right)\\ G_{y,\alpha}=\log\left(R_{3,\alpha}\right)\end{cases}  \tag{5}\]Calculate the magnitude and direction of the gradient using the following formula  
\[G_{n,\alpha}=\sqrt{\left(G_{x,\alpha}\right)^{2}+\left(G_{y,\alpha}\right)^{2}} \tag{6}\]  
\[G_{t,\alpha}=\arctan(\frac{G_{y,\alpha}}{G_{x,\alpha}}) \tag{7}\]  
Among them, \(\alpha\) is a parameter used to calculate the exponential weight of the local mean. To obtain negative and positive gradient values, no normalization with minimum (or maximum) values has been applied between the ratio and its reciprocal. Furthermore, due to the weighted parameters allowing for smoothing the image at different scales, this gradient calculation method can be compared with the gradient difference applied to images with Gaussian blur. The gradient calculation method of SAR-SIFT is the ratio gradient GR (gradient by ratio) [26]. The differential gradient can to some extent mitigate speckle noise, but the gradient values (magnetic component, vertical component, and horizontal component) in high-reflectivity areas are higher than those in low-reflectivity areas. For the ratio gradient method, the gradient responses in different reflection regions are more consistent and robust against speckle noise. This novel gradient calculation method will facilitate the application of the SIFT algorithm to SAR images. LoG (the Laplacian of Gaussian) and Hessian matrices rely on second-order derivative calculations, making them less suitable for multiplicative noise. In contrast, the multi-scale Harris function is based on first-order derivative calculations. According to the ratio gradient (GR) calculation method proposed by SAR-SIFT for SAR images, SAR-SIFT adopts a new approach based on this detector. The multi-scale Harris matrix and function are defined as follows:  
\[C(x,y,\sigma)=\sigma^{2}\cdot\mathcal{G}_{\sqrt{2},\sigma}*\left[