# Improved Localization and Recognition of Handwritten Digits on MNIST Dataset with ConvGRU
## 3 Methodology
### Video Prediction Model  
Due to the spatiotemporal nature of videos, we replace the linear operations in the gating units of GRUs with convolutional operations to optimize parameters and accelerate convergence. ConvGRU achieves this by substituting fully connected operations in the \"cell\" units with convolutional operations, allowing the model to retain important spatial information that changes over time. The inputs to ConvGRU include the data from the current time step and the hidden state from the previous time step. ConvGRU leverages the GRU's capability to handle long-term dependencies in sequences while incorporating early fusion architectures to learn the evolution of appearance and motion features. Unlike traditional methods, ConvGRU further extracts spatial information from the abstract features during video prediction. Although the structure of ConvGRU appears similar to the basic ConvGRU architecture (as shown in Figure 2), its internal structure differs: the \"cell\" units use convolutional operations rather than fully connected operations to process input signals. The proposed ConvGRU-based architecture  
Figure 1: Enhanced discriminator architecture for spatiotemporal feature learning. incorporates several key mathematical components. The core ConvGRU operations are defined as follows:  
\[ \begin{split} r_{t}&=\sigma(W_{xr}*x_{t}+W_{lr}*h_{t-1}+ b_{r})\\ z_{t}&=\sigma(W_{xz}*x_{t}+W_{liz}*h_{t-1}+b_{z})\\ \tilde{h}_{t}&=\tanh(W_{xh}*x_{t}+r_{t}\odot(W_{hh}* h_{t-1})+b_{h})\\ h_{t}&=z_{t}\odot h_{t-1}+(1-z_{t})\odot\tilde{h}_{t} \end{split}  \tag{5}\]  
where \(r_{t},z_{t},h_{t}\in hbbR^{H\times W\times C}\) represent the reset gate, update gate, and hidden state, respectively, at time \(t\). \(x_{t}\in hbbR^{H\times W\times C_{in}}\) is the input tensor. \(W_{(\cdot)}\) denote the learnable weight matrices. \(b_{(\cdot)}\) represent the bias terms. \({}^{*}\) denotes the convolution operation. \(\odot\) represents the Hadamard (element-wise) product. \(\sigma\) is the sigmoid activation function. The feature fusion mechanism integrates temporal and spatial information through  
\[F_{f}=\Phi(hcal\,A_{t}(F_{t})\oplus hcal\,A_{s}(F_{s})) \tag{6}\]  
where \(hcal\,A_{t}(\cdot)\) and \(hcal\,A_{s}(\cdot)\) are the temporal and spatial attention functions:  
\[ \begin{split} hcal\,A_{t}(F_{t})&=\sigma(W_{t}F_{t }+b_{t})\odot F_{t}\\ hcal\,A_{s}(F_{s})&=\sigma(W_{s}F_{s}+b_{s}) \odot F_{s}\end{split}  \tag{7}\]  
We propose a new video prediction framework based on ConvGRU, which uses DCGAN to build the background feature encoder, target feature encoder, and decoder. The ConvGRU video prediction process is shown in Figure 3.  
* Data preparation: Before building the video prediction framework, we need to collect a video dataset. We are using the MNIST dataset and performing simple preprocessing to adapt it for video prediction purposes. * Training the decoupling model: We use the training dataset to train a decoupling model that extracts background features and motion features from the sequences. We utilize the two abstract matrices extracted by the decoupling model as inputs for the prediction model. * Preparing inputs for the prediction model: In a loop, the model concatenates the background feature matrix of the first frame with the motion feature matrix of the i  
Figure 2: Structure diagram of ConvGRU. frame along the first dimension. We use the concatenated matrix as the input to the prediction model based on ConvGRU. * Training the prediction model: The prediction model uses the input features to learn the temporal dependencies between sequences. It outputs the **motion information** of subsequent sequences. * Generating future sequence frames: The model uses a decoder constructed with a deep transposed convolutional network to fuse the predicted motion features with the background features output by the decoupling model. It generates abstract matrices for future sequence frames. * Visualizing results: Finally, the model visualizes the abstract matrices output by the fusion module.