## 2 The Proposed Network
### 3D Face Representation  
Figure 3a shows the 3D face model. In feature classification, classical methods usually consider extracting specially designed global features or local features from 3D faces. However, handcrafted features have been proven to be suboptimal compared to depth features [45]. The development of CNN networks has made it possible to generalize well for various types of vision tasks, so that the conversion of 3D faces to 2D can borrow mature network structures to achieve recognition tasks and uses transfer learning techniques to speed up the efficiency of network training. The use of multiple facial encoding strategies enhances the discrimination between classes [46], so to improve the recognition performance of the network. Inspired by [10; 36; 47], we compute and extract the depth information, elevation, and azimuth angle of the normal vector of each 3D coordinate point as the new 3D facial representation. The three geometric data points are generated and interpolated into the three channels of the 2D image using a grid-fitting algorithm (as shown in Figure 3b). Specifically, the 3D face of the pose variation is first aligned to the reference coordinate space in order to calculate the facial feature information; then, the depth information of each point of the 3D face can be directly determined by the spatial coordinate points \((x,y,z)\). Next, the discrete 3D face points in space are meshed by the Delaunay algorithm, and the face point cloud is transformed into a mesh model consisting of small multivariate objects. Finally, we calculate the normal angle information using grid points. The norm \(N_{f_{K}}\) of the polygon surface formed by the mesh vertex \(P_{i}\) and the adjacent points \(P_{i}\), \(P_{j+1}\) is obtained using Equation (3).  
\[N_{f_{K}}=\frac{\big{(}P_{i}-P_{j+1}\big{)}\times\big{(}P_{i}-P_{j}\big{)}}{\| \big{(}P_{i}-P_{j+1}\big{)}\times\big{(}P_{i}-P_{j}\big{)}\|} \tag{3}\]  
\[N_{P_{i}}=\frac{\sum A_{f_{K}}N_{f_{K}}}{\|\sum A_{f_{K}}N_{f_{K}}\|} \tag{4}\]  
\[azimuth_{N_{P_{i}}}=\arctan\big{(}N_{P_{i}Y},N_{P_{i}X}\big{)} \tag{5}\]  
\[elevation_{N_{P_{i}}}=\arctan\bigg{(}N_{P_{i}Z},\sqrt{\big{(}N_{P_{i}X}\big{)} ^{2}+\big{(}N_{P_{i}Y}\big{)}^{2}}\bigg{)} \tag{6}\]  
Figure 3: (**a**) A three-dimensional face model displayed by colormap. (**b**) A multi-feature face whose three image channels are replaced by (**c**–**e**); (**c**) a depth map; (**d**) an elevation map of the surface normal; (**e**) an azimuth map of the surface normal.  
The normal \(N_{P_{i}}\) of the face's surface point can be obtained by averaging the normal of all small polygons sharing the point \(P_{i}\) using Equation (4) [48], where \(A_{f_{K}}\) is the area of polygon \(f_{K}\). Thus, the azimuth angle \(azimuth_{N_{P_{i}}}\) and elevation angle \(elevation_{N_{P_{i}}}\) of the facial surface can be obtained by converting the Cartesian coordinates \((N_{P_{2}X},N_{P_{2}Y},N_{P_{2}Z})\) of the normal \(N_{P_{i}}\) to spherical coordinates using Equations (5) and (6). After processing the z-value, azimuth, and elevation angles of each discrete point in the space by grid interpolation, the desired depth map (see Figure 3c), elevation map (see Figure 3d), and azimuth map (see Figure 3e) can be obtained.