## 3 Methodology
### Optical Flow Estimation Module
#### 3.4.1 Design of the Optical Flow Estimation Module  
The design of the optical flow field model in this study is inspired by the classical RAFT architecture [29] but differs significantly in how image features are handled. Unlike RAFT, where features are extracted independently, the image features in our method are derived directly from the encoder output. These encoded features are passed into a feature correlation layer, which computes a four-dimensional (4D) correlation volume of size \(W\times H\times W\times H\), representing the similarity between all pairs of pixels in the source and target features. This 4D correlation volume is constructed by computing the dot product between every pair of feature vectors. To improve efficiency and enable scale-aware matching, multi-scale pooling is applied along the last two dimensions of the 4D volume to construct a hierarchy of multi-scale  
Figure 6: Output control point map generated by the decoder. correlation volumes. The 4D volume \(C\) can be efficiently computed as a single matrix multiplication, as defined in Equation (3). \[C(g_{\theta}(I_{1}),\ g_{\theta}(I_{2}))\in R^{H\times W\times H\times W},C_{ijkl }=\sum_{h}g_{\theta}(I_{1})_{ijh}\cdot g_{\theta}(I_{2})_{klh} \tag{3}\]  
The constructed 4D correlation volume is fed into a recurrent GRU-based update module, which iteratively generates the final optical flow field. The flow estimation process is initialized with a zero-flow field, \(f^{(0)}=0\), and a series of flow predictions \(\left\{f^{(1)},f^{(2)},f^{(N)}\right\}\) are progressively refined through iteration. At each iteration step, the update operator computes a flow increment \(\Delta f\), which is added to the previous estimate to obtain the updated flow field: \(f^{(t+1)}=f^{(t)}+\Delta f\). The inputs to the update operator include the current optical flow estimate, the 4D correlation volume, and a latent hidden state. Its output consists of the updated flow increment \(\Delta f\) and a new hidden state. The design of this structure is inspired by the iterative optimization processes commonly found in traditional numerical solvers, aiming to emulate convergence behavior. To this end, the update operator employs parameter sharing (i.e., tied weights) and bounded activation functions to enhance convergence stability during training, ensuring that the predicted optical flow satisfies the convergence condition \(f^{(k)}\to f^{*}\). The core architecture of the update operator is based on a Gated Recurrent Unit (GRU), where the fully connected layers are replaced with convolutional layers to enhance local spatial awareness. This modification enables more effective processing of spatial features. The specific computation process is defined in Equation (4). \[ \begin{split} z_{t}&=Conv_{3\times 3}([h_{t-1},x_{t}],W _{z})\\ r_{t}&=Conv_{3\times 3}([h_{t-1},x_{t}],W_{r})\\ h_{t}&=(1-z_{t})\otimes h_{t-1}+z_{t}\otimes\tanh( Conv_{3\times 3}(r_{t}\otimes h_{t-1},x_{t}),W_{h})\end{split}  \tag{4}\]  
In this structure, \(z_{t}\) and \(r_{t}\) represent the update gate and reset gate of the GRU, respectively, both computed from feature maps using two distinct convolutional kernels. The convolution operations within the GRU are implemented via learnable kernels \(Wx\) and the nonlinear activation function used is \(tanh(\cdot)\). The current hidden state \(h_{t}\) and the previous state \(h_{t-1}\) encode the temporal dynamics of the optical flow estimation process. The input \(x_{t}\) is formed by concatenating the initial flow estimate, the 4D correlation information, and the context features. To enlarge the receptive field without significantly increasing model complexity, this study replaces the traditional 3 \(\times\) 3 convolution with two directionally separable GRUs: one utilizing a 1 \(\times\) 5 convolutional kernel to capture horizontal context and the other using a 5 \(\times\) 1 kernel to extract vertical contextual information.