# A Hierarchical Fusion SAR Image Change-Detection Method Based on HF-CRF Model
## 2 Materials and Methods
### Boundary Entropy Dynamic Fusion CRF Model  
CRF [59] is a well-known discriminative model, widely utilized in various domains such as image segmentation [60]. The CRF model for change detection [61] comprises two components. Modeled as a probability distribution map, the unary potential function represents the intrinsic energy of each pixel, which is generated by processing the clustering algorithm with either the temporal phase map [62] or the semantic segmentation network [53]. The pair-wise potential function models the second-order neighborhood potential energy, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **FP** & **FN** & **OE** & **PCC** & **Kappa** \\ \hline U-Net & 5076 & 3451 & 8527 & 91.60\% & 0.6969 \\ D-DRUNet-S & 3908 & 274 & 4182 & 95.88\% & 0.8583 \\ D-DRUNet & 1206 & 894 & 2100 & 97.93\% & 0.9229 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance metrics of different methods on Ottawa dataset. Figure 5: Visualization of guide mask region division (the Berne data). Figure 6: Experimental test result graph of Ottawa data section. The red dashed boxes mark the false alarms detected by the D-DRUNet-S method. incorporating both the positional and color information as feature functions, which can be expressed using Equation (1) [59], where \(Z(X|Y)\) is the regularization constant, \(E(X|Y)\) is the energy function, and \(\psi_{u}(x_{i})\), \(\psi_{p}(x_{i},x_{j})\) are the expressions for the one-dimensional potential function and the two-dimensional potential function, respectively. \[ \begin{array}{c}P(X|Y)=\frac{1}{Z(Y)}e^{-E(X|Y)}\\ E(X|Y)=\sum\limits_{i}\psi_{u}(x_{i})+\sum\limits_{i<j}\psi_{p}(x_{i},x_{j}) \end{array}  \tag{1}\]  
The fully connected CRF model calculates the pair-wise potential function by considering all pixels in the image as neighborhood relations. It employs the mean-field inference model [45] to achieve the model solution through downsampling convolution. Herein, we introduced a hybrid pair-wise potential function using the fully connected CRF model. The segmentation result obtained from D-DRUNet serves as a unary potential function, while the multimodal difference map is incorporated as Equations (2)-(6). \(\alpha_{i}\) and \(\beta_{i}\) are the corresponding weights of the first pixel of the two modal difference images; \(P(x_{i}=l_{k})\) is the probability that the predicted label \(x_{i}\) of the \(i\)-th pixel point in the network segmentation result is \(l_{k}\); \  
nique is utilized to enhance the boundary information in the difference maps, whereby its significance in the dynamic weight calculation is increased. We choose three different sharpening operators for performance comparison. Both the Sobel operator and the Prewitt operator use a pixel window to compute the first-order derivatives of the image in the x and y directions, respectively, with the edge operator for the convolution summation operation, thus processing each pixel point of the image to extract the edges of the image. The Laplacian operators are based on the second-order derivatives, specifically, the Laplacian operator with a four-neighborhood template is used in this paper. The parameters of operators are shown in the Table 2. Figure 8a,d show the sharpening results of the sharpening operators [64; 65] on the Berne dataset, and (e) to (h) show the figures of the sharpening results on the Ottawa dataset. The comparison shows that the Sobel operator [66] has the optimal boundary strengthening effect on both datasets, as shown in the blue box. Therefore, we use the Sobel operator as the final sharpening operator. In summary, the proposed local boundary entropy pair-wise potential function is constructed as follows. * First, sharpening image by using the Sobel operator to obtain sharpened log-difference images \(X_{L}\) and sharpened mean-difference images \(X_{M}\), respectively.