## 3 Theoretical Derivation of the TR-PCG Algorithm
We improve the computational efficiency by using FFT. Holograms of the three channels are generated by using 2D FFT on the depth grids. The total computation time can be effectively reduced by computing the hologram from a 2D multi-depth grid rather than a single point in a 3D point cloud. Then computer generated holograms can be obtained by performing the light-field diffraction calculation in each layer:  
\[U_{M}(f_{X},f_{Y})=\mathcal{F}[U(x,y)], \tag{5}\]  
where \(f_{X}\) and \(f_{Y}\) are the frequencies in the spatial spectrum domain, \(U(x,y)\) is the object field in the spatial domain. According to the geometric relationship, the radial value \(r\) in the spatial domain satisfies the following formula:  
\[r=(f_{X}{}^{2}+f_{Y}{}^{2})\times(\frac{\lambda}{pph})^{2}, \tag{6}\]  
where \(f_{X}\) and \(f_{Y}\) are the frequencies in the spatial spectrum domain, \(\lambda\) is the wavelength and \(pph\) is the spacing of pixels. According to the Rayleigh-Sommerfeld diffraction formula, the point spread function \(h(\mathbf{r}_{\perp},z)\) is:  
\[h(\mathbf{r}_{\perp},z)=\frac{1}{2\pi}\frac{z}{\mathbf{r}}\frac{(1-\mathrm{j}\mathbf{k} \mathbf{r})}{r^{2}}\exp(\mathrm{j}\mathrm{k}\mathbf{r}), \tag{7}\]  
where the radial value \(r=\lvert\mathbf{r}\rvert\), the vector \(\mathbf{r}=(x,y,z)\) and \(k\) is the wave vector. Here we use single-precision floating-point numbers. However, pure single-precision diffraction may lead to inaccurate results due to phase error. Therefore, for the single-precision floating-point numbers, by proper Taylor expansion of the radial value \(r\) in the fast Fourier transform and mixing with the angular spectral diffraction (ASM), we can achieve a relatively accurate and fast numerical diffraction propagation. The single-channel data need to be looped separately. The red channel is used as an example, and the same goes for the green and blue channels. Fourier transform is applied to the point spread function \(h\) to obtain the transfer function \(H\) in the spatial spectrum domain, as shown in the following formula:  
\[H=\exp(\mathrm{j}k_{1}d_{1})\exp(-\mathrm{j}\pi\lambda d_{1}r), \tag{8}\]  
where \(k_{1}\) is the wave vector of the red wave, \(\lambda\) is the wavelength of the red wave and \(d_{1}\) is as shown in the following equation:  
\[d_{1}=d-Cut\times\frac{sam\_int}{2}, \tag{9}\]  
where \(d\) is the distance to generate the hologram, \(sam\_int\) is the sampling interval and \(Cut\) is the segmented point cloud grids. Furthermore, \(r\) is as follows:  
\[r=\sqrt{1-\lvert\rho_{\perp}\rvert^{2}(\frac{\lambda}{pph})^{2}-1} \tag{10}\]where \(\Big{|}\rho_{\perp}\Big{|}_{2}=\left(f_{X}{}^{2}+f_{Y}{}^{2}\right).\)  
For convenience, here we define \(r^{\prime}\) as follows:  
\[u(\mathbf{r}_{\perp},z)=\mathcal{F}^{-1}(\mathcal{F}(u(\mathbf{r}_{\perp},z_{0})) \times\mathcal{F}(h(\mathbf{r}_{\perp},z))), \tag{2}\]0  
Therefore, the relationship between \(r\) and \(r^{\prime}\) is:  
\[u(\mathbf{r}_{\perp},z)=\mathcal{F}^{-1}(\mathcal{F}(u(\mathbf{r}_{\perp},z_{0})) \times\mathcal{F}(h(\mathbf{r}_{\perp},z))), \tag{2}\]1  
According to the [PERSON] expansion formula:  
\[u(\mathbf{r}_{\perp},z)=\mathcal{F}^{-1}(\mathcal{F}(u(\mathbf{r}_{\perp},z_{0})) \times\mathcal{F}(h(\mathbf{r}_{\perp},z))), \tag{2}\]2  
\(r^{\prime}\) in Equation (12) and \(x^{2}\) in Equation (13) are equivalent, so \(r\) is expanded as follows:  
\[u(\mathbf{r}_{\perp},z)=\mathcal{F}^{-1}(\mathcal{F}(u(\mathbf{r}_{\perp},z_{0})) \times\mathcal{F}(h(\mathbf{r}_{\perp},z))), \tag{2}\]3  
According to the findings of the experiment, the reconstruction image with shorter computation time and higher quality can be obtained when retaining to the quadratic term. When retaining to the cubic term, the reconstruction quality is similar to that of the quadratic term. Relative to the increase in reconstruction time, the reconstruction quality is not significantly improved.