## 2 Methodology  
### Network Architecture  
The overall framework of the proposed BCSR-STF is illustrated in Figure 1. It comprises three primary components: feature extraction, feature fusion, and a prediction head. The feature extraction module employs a pyramid structure to facilitate multi-scale and multi-level feature representation. The feature fusion module consists of multi-level PSTFR to iteratively refine the fused features, enhancing detail and ensuring coherence across scales. Lastly, the prediction head reconstructs the refined features into a fused image, producing the desired output. The model operates with three input images: \(C_{l_{0}}\), \(F_{t_{0}}\), and \(C_{t_{1}}\). Here, \(C_{l_{0}}\) and \(F_{t_{0}}\) denote the coarse and fine images in the reference data, respectively, while \(C_{t_{1}}\) refers to the coarse image in the predicted data. \(F_{t_{1}}\) is the network output, i.e., the fine image in the predicted data. In the feature extraction stage, a multi-scale patch embedding strategy [41] is first employed to transform the input images \(C_{t_{0}}\), \(C_{t_{1}}\), and \(F_{t_{0}}\) into serialized tokens. This approach utilizes convolutional kernels of four sizes (2 \(\times\) 2, 4 \(\times\) 4, 8 \(\times\) 8, and 16 \(\times\) 16), all with a fixed stride of 2, to sample and embed features at varying spatial scales. Smaller kernels emphasize fine-grained local details, while larger kernels capture broader contextual information. The resulting features are embedded into a unified token sequence, preserving multi-scale spatial information for subsequent processing. \[L_{0}^{0}=PE_{\omega_{1}}(C_{t_{0}}) \tag{1}\]  
\[L_{1}^{0}=PE_{\omega_{1}}(C_{t_{1}}) \tag{2}\]  
\[H_{0}^{0}=PE_{\omega_{2}}(F_{t_{0}}) \tag{3}\]  
\(PE_{\omega}(\cdot)\) represents the multi-scale patch embedding module parameterized by \(\omega\). The feature extractors for \(C_{t_{0}}\) and \(C_{t_{1}}\) are based on a Siamese Network, which is used to assess the similarities and differences between \(C_{t_{0}}\) and \(C_{t_{1}}\). The low-resolution features extracted from \(C_{t_{0}}\) and \(C_{t_{1}}\) are represented as \(\left\{L_{0}^{(1)},\ldots,L_{0}^{(i)},L_{0}^{(i+1)}\ldots L_{0}^{(N)}\right\}\) and \(\left\{L_{1}^{(1)},\ldots L_{1}^{(i)},L_{1}^{(i+1)}\ldots L_{1}^{(N)}\right\}\), respectively. The resolution of these features progressively decreases from the finest level 1 to the coarsest level \(N\), where the size of the current-level feature \(L_{t}^{(i)}\) is twice that of the feature at the next level \(L_{t}^{(i+1)}\). Similarly, the high-resolution features extracted from \(F_{t_{0}}\) are represented as \(\left\{H_{0}^{(1)},\ldots,H_{0}^{(i)},H_{0}^{(i+1)}\ldots H_{0}^{(N)}\right\}\). Formally, we define  
\[L_{t}^{(i)}=p_{\phi_{1}}^{(i)}\Big{(}E_{\phi_{1}}^{(i)}\Big{(}L_{t}^{(i-1)} \Big{)}\Big{)} \tag{4}\]  
Figure 1: Overall model structure of the proposed BCSR-STF. \[H_{0}^{(i)}=P_{\theta_{2}}^{(i)}\Big{(}E_{\theta_{2}}^{(i)}\Big{(}H_{0}^{(i-1)} \Big{)}\Big{)} \tag{5}\]  
where \(t\) represents the image acquisition time. \(E_{\theta_{1}}^{(i)}(\cdot)\) represents the low-resolution feature extraction module parameterized by \(\theta_{1}\) at level \(i\). \(E_{\theta_{2}}^{(i)}(\cdot)\) represents the high-resolution feature extraction module parameterized by \(\theta_{2}\). \(P_{\phi}(\cdot)\) denotes the parameterized Patch Merging operation [27], which performs downsampling by concatenating neighboring patches and applying a learnable linear transformation to reduce the dimensionality of the resulting features. The feature extraction module employs the CrossFormer Block [41], which facilitates multi-scale interactions between neighboring embeddings and distant embeddings, effectively capturing local details and global contextual information. To address the scale variability in remote sensing imagery and the resolution differences between coarse and fine images, a multi-level distributed fusion module is proposed for the feature fusion stage.