## 2 Methodology
### Bidirectional Cross Fusion
#### 2.2.1 Time Direction  
The time direction primarily focuses on optimizing the fused features through the temporal changes across time points. In the feature fusion of the time direction, the idea from STARFM is adopted. By fusing the fine features from the reference time with the difference features generated by temporal changes, the fused spatiotemporal features are obtained. The mathematical expression is  
\[F_{t}^{(i)}=H_{0}^{(i)}+f_{time}\Big{(}H_{0}^{(i)},L_{0}^{(i)},\Delta L^{(i)} \Big{)} \tag{9}\]  
where \(F_{t}^{(i)}\) is the spatiotemporal feature fused along the time direction and \(f_{time}(\cdot)\) represents the feature fusion operation in the time direction. This operation includes a combination of short-range and long-range attention mechanisms, followed by nonlinear feature mapping through a multi-layer perceptron (MLP) to enhance interactions between features.  
The function \(f_{time}(\cdot)\) is introduced in detail below. The input feature maps are \(H_{0}^{(i)},L_{0}^{(i)}\), and \(\Delta L^{(i)}\). \(\Delta L^{(i)}\) represents the temporal variation information. Specifically, \(\Delta L^{(i)}=L_{0}^{(i)}-L_{0}^{(i)}\), which denotes the feature difference between the reference time and the predicted time. It is used to guide the recovery and enhancement of fine-difference features in subsequent processes. A linear layer is applied to \(H_{0}^{(i)}\), \(L_{0}^{(i)}\), and \(\Delta L^{(i)}\), mapping them to the query vector \(Q_{t}^{(i)}\), key vector \(K_{t}^{(i)}\), and value vector \(V_{t}^{(i)}\).  
\[Q_{t}^{(i)}=H_{0}^{(i)}W_{Q_{t}}^{(i)},K_{t}^{(i)}=L_{0}^{(i)}W_{K_{t}}^{(i)}, V_{t}^{(i)}=\Delta L^{(i)}W_{V_{t}}^{(i)} \tag{10}\]  
Figure 2: Bidirectional Cross Fusion module.  
The matrices \(W_{Q_{i}}^{(i)}\), \(W_{K_{i}}^{(i)}\), and \(W_{V_{t}}^{(i)}\) are the learnable weight parameters. The calculation formula for the attention mechanism is  
\[Attention(Q_{t}^{(i)},K_{t}^{(i)},V_{t}^{(i)})=SoftMax\left(\frac{{Q_{t}^{(i)}{K_ {t}^{(i)}}^{T}}}{\sqrt{d}}+B\right)V_{t}^{(i)} \tag{11}\]  
The attention mechanism is a crucial part of \(f_{time}(\cdot)\). The inner product of \(Q_{t}^{(i)}\) and \({K_{t}^{(i)}}^{T}\), normalized by \(\sqrt{d}\) (where \(d\) is the dimensionality of the feature vectors) determines the attention weights. These weights are then passed through \(SoftMax(\cdot)\) to ensure they sum to one, effectively distributing attention across the features. A bias term \(B\) is added to introduce positional information. The weighted sum of \(V_{t}^{(i)}\) is computed, representing the aggregated temporal information enhanced by attention. The core idea of the entire process is to leverage the latent scale relationship between coarse image \(H_{0}^{(i)}\) and fine images \(L_{0}^{(i)}\) to dynamically adjust the coarse change features \(\Delta L^{(i)}\), thereby enhancing the model's ability to capture fine temporal variation information.  
To reduce the computational cost of attention, a window-based attention mechanism is employed to achieve global interactions among pixels within each window. The window grouping methods for SDA and LDA are illustrated in Figure 3, where only two groups are shown for simplicity. Pixels enclosed by the red rectangles form one group, while those within the purple rectangles form another. As depicted in Figure 3a, the window grouping in SDA ensures that pixels within the same group are spatially adjacent. In contrast, the grouping method in LDA, shown in Figure 3b, follows a principle that enables non-adjacent pixel embedding. By concatenating SDA and LDA, both short-distance and long-distance feature interactions are effectively captured.