## 2 Materials and Methods
### Improved Feature Extraction Algorithm
#### 2.2.2 Removing Non-Scale Key Points and Key Points Detected  
Although the SAR-SIFT algorithm mentions the concept of multiple scales, it does not take the positional information of the extracted feature points into account. It only extracts feature points at each layer in the multi-scale space, and such feature points do not have high-scale invariance. To advance the scale invariance of feature points, this paper uses the spatial maximum value to obtain key points with scale invariance. As shown in Equation (5), in this algorithm process, the feature points that do not have scale invariance are eliminated by comparing whether the R value of the current layer feature point is greater than that of the adjacent layer feature point. When the feature point is at the topmost layer, it is compared with the second topmost layer value, and when the feature point is at the bottommost layer, it is compared with the second bottommost layer.  
\[\left\{\begin{array}{c}l(x,y,k)>l(x,y,k+1),l(x,y,k)>0\,,k=1\\ l(x,y,k)>l(x,y,k+1),l(x,y,k)>l(x,y,k-1),l(x,y,k)>0\,,1<k<n\\ l(x,y,k)>l(x,y,k-1),l(x,y,k)>0\,,k=n\end{array}\right. \tag{5}\]  
Figure 2: STSU–SAR–SIFT flowchart.  
In this equation, \(k\) represents the current feature scale level, and \(n\) represents the entirety of feature scales; \(x\), \(y\) represents the position of the pixel.  
Some of the key points detected by the aforementioned method cannot be accurately located; therefore, the Laplacian of Gaussian (LoG)operator is used to refine their feature scale and position. The LoG function is defined as follows:  
\[\left|\text{LoG}(x,y,\sigma)\right|=\sigma^{2}\big{|}\mathbf{G}_{xx}(x,y,\sigma)+ \mathbf{G}_{yy}(x,y,\sigma)\big{|} \tag{6}\]  
where represents the current scale layer, \(G_{xx}\) is obtained by taking the second-order derivative of the signal in the horizontal direction using the Laplacian operator, and \(G_{yy}\) is obtained by taking the second-order derivative of the signal in the vertical direction using the Laplacian operator.  
The improved response function derived from Equation (4) can strengthen the robustness of feature point extraction. Although the Gaussian blur applied in creating the scale space reduces the impact of multiplicative noise, the SAR-SIFT algorithm still suffers from the presence of false feature points caused by gradient calculation. To address this issue, we apply the SUSAN operator to suppress noise in feature points, as it is more robust to multiplicative noise due to its independence from differential operations. The noise reduction process is conducted after the detection of feature point candidates based on maximum response. By doing so, we obtain a set of feature points that are more accurate and less affected by noise. The detailed algorithm is described in Equation (7).  
\[\left\{\begin{aligned} & c((x,y),(x_{0},y_{0}))=\begin{cases}1& \left|I(x,y)-I(x_{0},y_{0})\right|\leqslant j\\ 0&\left|I(x,y)-I(x_{0},y_{0})\right|>j\end{cases}\\ & n(x_{0},y_{0})=\Sigma_{r}c((x,y),(x_{0},y_{0}))\\ & R(x_{0},y_{0})=\begin{cases}1&g>n(x_{0},y_{0})\\ 0&g\leq n(x_{0},y_{0})\end{cases}\end{aligned}\right. \tag{7}\]  
Here, \((x_{0},y_{0})\) denotes the position of the current feature point, and \(I(x,y)\) represents the pixel intensity at \(I(x,y)\). When the difference between the pixel values around the feature point and that of the feature point itself is below a threshold j, they are considered homogeneous. \(n(x_{0},y_{0})\) represents the number of homogeneous pixels in a given region; \(r\) refers to the circular region centered at the feature point \((x_{0},y_{0})\); and \(R(x_{0},y_{0})\) determines whether the current feature point should be retained. If the number of homogeneous pixels is below the threshold \(g\), the point is retained; otherwise, it is suppressed.