## 3 Theoretical Derivation of the TR-PCG Algorithm  
The point cloud grid method generally consists of three steps. First, the depth camera captures point clouds containing depth information. Next, each point of the same depth is matched with a depth grid node; that is, each depth grid contains all points at the same depth. Finally, according to the coordinates of each layer in the generated grid, diffraction calculation is carried out for each layer with FFT, and the final CGH is obtained. Therefore, on the basis of PCG, we propose TR-PCG; the steps are as follows. First is the sampling stage, which samples the object's light field. The depth camera is used to obtain both depth and color information from the real scene, from which the full-color point cloud model is extracted. When sampling, we need to consider how many sample points should be used for objects and holograms, which means the resolution requirement. We need to match the size of the grid transformed from the obtained point cloud with the size of the holographic plane. Therefore, although the object is a real object, the information obtained after collection can be scaled, so can the final generated hologram. Spatial light modulator resolution size is 2048 \(\times\) 2048, each pixel interval is fixed. We match the size of the object to the resolution of the spatial light modulator. The resolution is whatever the grid is. Here, we define \(\frac{1}{\mathcal{N}}\) and \(\frac{1}{\mathcal{V}}\) as the distances between the items in the spectrum of the sampled data in the \(f_{X}\) and \(f_{Y}\) directions, respectively. In fact, the size of the object determines the bandwidth of the light field on the hologram surface, which, in turn, determines the number of samples required for the light field on the hologram surface. The resolution of normal holographic elements is 1920 \(\times\) 1080. According to the formula, as long as the number of sampling points is not less than 960 \(\times\) 960, the sampling requirement can be satisfied. The number of sampling points in this paper is more than 1000 \(\times\) 1000, which can certainly meet the requirements of sampling frequency. The following is the hologram generation stage. An FFT is performed on each grid to calculate the diffraction and obtain three computer generation holograms, one for each of the red, green, and blue channels. It is necessary to calculate the complex value of the light field at each sampling point. We perform Taylor expansion for the radial value \(r\) in the convolution based on FFT to realize relatively high-speed discrete Fourier transform of the object light field. The result is a set of discrete sample values of the complex light field, while both an amplitude and a phase are at each sample point. In the 3D Cartesian coordinate system, we assume that the wave propagates along the \(z\)-axis, so the two-dimensional plane is \(\mathbf{r}_{\perp}=(x,y)\). When the propagation distance is \(z\), the initial field becomes a new field according to the following formula:  
\[u(\mathbf{r}_{\perp},z)=u(\mathbf{r}_{\perp},0)\otimes h(\mathbf{r}_{\perp},z), \tag{1}\]  
where \(u(\mathbf{r}_{\perp},z)\) is the new field, \(u(\mathbf{r}_{\perp},0)\) is the initial field, and \(h(\mathbf{r}_{\perp},z)\) is the point spread function. Convolution in the spatial domain is equivalent to multiplication in the spatial spectrum domain:  
\[u(\mathbf{r}_{\perp},z)=\mathcal{F}^{-1}(\mathcal{F}(u(\mathbf{r}_{\perp},z_{0})) \times\mathcal{F}(h(\mathbf{r}_{\perp},z))), \tag{2}\]  
where \(\mathcal{F}/\mathcal{F}^{-1}\) is the Fourier forward/inverse transform,\(z_{0}\) represents the initial position of the object on the \(z\)-axis. According to the sampling relationship in the sampling step, we obtain the sampling frequency \(f_{X}\) and \(f_{Y}\):  
\[f_{X}=\frac{1}{N_{X}\times sam\_int}, \tag{3}\]  
\[f_{Y}=\frac{1}{N_{Y}\times sam\_int}, \tag{4}\]  
where \(N_{X}\) and \(N_{Y}\) are the number of sampling points and \(sam\_int\) is the sampling interval.