# Fusion Network for Change Detection of High-Resolution Panchromatic Imagery
## 3 Proposed Fusion Network for Change Detection with Panchromatic Imagery
### Fusion Network for Change Detection
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(I^{1}\) and \(I^{2}\) & Cropped temporal input image in time 1 and 2, respectively \\ \(N^{1}\) and \(N^{2}\) & Patch network 1 and 2 correspond to the back-end network \\ \(N^{3}\) & Patch network 3 correspond to the front-end network \\ \(F^{i}_{l,d_{r}}\) & Feature maps of the \(l\)-th layer at the \(r\)-th dense block and the \(i\)-th network \\ \(P^{1}\) and \(P^{2}\) & Outputs of \(N^{1}\) and \(N^{2}\), respectively \\ \(D\) & Dissimilarity distance \\ \(O\) & Change detection probability output of \(N^{3}\) \\  & Incorporation process of a batch normalization (BN), a \(3\times 3\) \\ \(H^{i}_{l-1,d_{r}}\) & convolution, and ReLU of the \((l-1)\)-th layer at the \(r\)-th dense block and the \(i\)-th network \\  & A concatenation of the feature-maps of all of previous layers, layer 0, \\ \([F^{i}_{0,d_{r}}\ F^{i}_{1,d_{r}}\,\dots,\ F^{i}_{l-1,d_{r}}]\) & \(\dots,\) and layer \(l-1\) \\ \(L\) & Proposed loss function \\ \(E_{c}\) & Contrastive loss function \\ \(E_{B}\) & Weighted binary cross entropy loss function \\ Y & Ground truth \\ \(Ls\) and \(L_{D}\) & Partial loss function for a pair of similar and dissimilar pixels, \\ \(m\) & Margin value \\ \(a\) & Weighted loss \\ \(W\) & Proposed weighted function \\ \(C\) and \(U\) & Changed and unchanged numbers of pixels, respectively \\ \(N\) & The number of full dataset \\ \(\beta_{c}\) and \(\beta_{u}\) & Penalization weights for false-negative and false-positive errors, \\  & respectively \\ \(M_{1}\) & Change map for first prediction \\ \(M_{2}\) & Change map for second prediction \\ \(Nb\) & Local information of \(M_{1}\) \\ T & Tested temporal images \\ \(m\) and \(n\) & Size of T \\ \(s\) & Size of \(l\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols used in the proposed fusion network for change detection. where \([F_{0,d,t}^{i},\ F_{1,d,t}^{i},\ \dots,\ F_{l-1,d,t}^{i}]\) indicates a concatenation of the feature-maps of all of previous layers, layer 0,..., and layer (\(l-1\)). In addition, \(H(\cdot)\) incorporates a batch normalization (BN), a \(3\times 3\) convolution, and ReLU. A pair of temporal images were cropped into two patches (\(40\times 40\)) (\(I^{1}\) and \(I^{2}\)) by sliding the window and fed into \(N^{1}\) and \(N^{2}\), respectively. The dissimilarity distance (D) was then computed based on the Euclidean distance, which is defined as follows:  
\[D=\left\|P^{1}-P^{2}\right\|_{2} \tag{2}\]  
where \(P^{1}\) and \(P^{2}\) are the outputs of \(N^{1}\) and \(N^{2}\) activated by sigmoid function, respectively. The proposed method applies a pixel-wise change detection by inspecting the neighboring pixels. The \(40\ \times\ 40\) patch images identify a change corresponding to the center pixel of the patch. Thus, when the value of \(D\) is close to 1, the center of \(I\) is assigned to a changed pixel. In addition, \(I^{1}\) and \(I^{2}\) were concatenated to be fed into \(N^{3}\). The same dense convolution architecture was employed in this branch network to generate the change detection probability (\(O\)). The dual outputs (\(D\) and \(O\)) are a result of this fusion network. In addition, a post-processing step during the inference stage was proposed based on these outputs (\(D\) and \(O\)) to achieve a proper prediction.