# Action Recognition Based on the Fusion of Graph Convolutional Networks with High Order Features
## 3 Proposed Graph Convolutional Network with High-Order Features
### High-Order Temporal Features  
For temporal features in a single frame, we propose the velocity feature and the acceleration feature. From the Figure 2c, we can directly get the information contained by these two features.  
**Velocity feature:** Velocity features of an action are very crucial for action recognition. Learning velocity features can be relatively complemented with learning features of the joint and bone. For skeleton data, we calculate the motion velocity information of each vertex. The velocity of vertex \(v_{1}\) is equal to the coordinate of \(v_{1}\) in the next frame minus the current frame. We can obtain the velocity in three directions \((x,y,z)\), which is helpful for analyzing the action. Velocities of different orientations correspond to different changes. Therefore, velocity analysis in each orientation of the vertex is effective for the final prediction. \(j_{1}^{t}(x_{1}^{t},y_{1}^{t},z_{1}^{t})\) denotes the coordinates of joint \(j_{1}\) at \(t\) frame. \(j_{1}^{t+1}(x_{1}^{t+1},y_{1}^{t+1},z_{1}^{t+1})\) denotes the coordinates of joint \(j_{1}\) at \(T+1\) frame. The velocity of \(v_{1}^{t}(v_{x1}^{t},v_{y1}^{t},v_{z1}^{t})\) at \(t\) frame can be written as:  
\[v_{1}^{t}(v_{x1}^{t},v_{y1}^{t},v_{z1}^{t})=j_{1}^{t+1}-j_{1}^{t}=(x_{1}^{t+1} -x_{1}^{t},y_{1}^{t+1}-y_{1}^{t},z_{1}^{t+1}-z_{1}^{t}) \tag{3}\]  
For all joints, Equation (3) is transformed into Equation (4):  
\[v^{t}(v_{x}^{t},v_{y}^{t},v_{z}^{t})=j^{t+1}-j^{t}=(x^{t+1}-x^{t},y^{t+1}-y^{t},z^{t+1}-z^{t}) \tag{4}\]  
where \(v\) denotes the velocity of all joints in a single frame. Moreover, we calculate the velocity of the edge between the two joints, which is the velocity of the bone. The calculation method of velocity of the bone is the same as that of the joints. We use the 3D velocity of the bone as a feature and feed it into the network. More details of the training results and comparison experiments are provided in Section 4.  
**Acceleration feature:** Acceleration is a physical quantity used to describe the change in velocity. Acceleration is helpful for analyzing action. In one skeleton sequence, the velocities of joints may have different changes. Some joints move at a constant velocity, while other joints accelerate. The acceleration of the joint is equal to the velocity of the current frame minus the corresponding joint of the previous frame. Its feature dimensions are also three-dimensional. Basically, that means that thecalculation method of acceleration information is the same as that of the velocity information. Therefore, the features extracted from velocity and acceleration information are similar, while the acceleration uses more frames to calculate the high-order motion. We can calculate acceleration information based on Equation (5) as follows:  
\[a_{1}^{t}=v_{1}^{t+1}-v_{1}^{t}=(v_{x1}^{t+1}-v_{x1}^{t},v_{y1}^{t+1}-v_{y1}^{t},v_{z1}^{t+1}-v_{z1}^{t}) \tag{5}\]  
For all joints, Equation (5) is transformed into Equation (6):  
\[a^{t}=v^{t+1}-v^{t}=(v_{x}^{t+1}-v_{x}^{t},v_{y}^{t+1}-v_{y}^{t},v_{z}^{t+1}-v_ {z}^{t}) \tag{6}\]  
where \(a_{1}^{t}\) denotes the acceleration of joint \(j_{1}\) at \(t\) frame. \(v_{1}^{t+1}\) and \(v_{1}^{t}\) denote the velocity of joint \(j_{1}\) at \(t+1\) and \(t\) frames, respectively, and \(a^{t}\) denotes the acceleration of all joints in t frame.