# High-Resolution Hologram Calculation Method Based on Light Field Image Rendering
## 2 Methods  
Figure 1 shows the rendering of EIs with a pinhole array for hologram calculation. PA is a pinhole array behind the holographic plane H. The distance between PA and H is \(f\). Obj is a 3D object in space, and \(p(y,z)\) and \(p_{2}(y_{2},z_{2})\) are two object points of obj. The EI is an image rendered from the 3D object to a pinhole. The \(EI_{j}\), for example, is rendered from the pinhole \(O_{j}\) by calculating the projected image from all 3D objects to \(O_{j}\) within the opening angle from the pinhole to this EI. This \(EI_{j}\) is coded into the \(Hogel_{j}\). In this situation, the size of EI is the same as the size of hogel, as well as their coordinates. The EI on the holographic plane has a width of h, and the FOV is dependent on the opening angle \(\theta_{y}\) of the EI to the corresponding pinhole in the \(y\) direction. Each EI is closely adjacent to other EIs. The coordinate of \(j\)-th pinhole \(o_{j}\) is \(y_{fj}\), and the projections of point \(p(y,z)\) and \(p_{2}(y_{2},z_{2})\) within the FOV to each pinhole are demonstrated. The intersections of projection lines to each pinhole on the holographic plane are coordinates of the projected points of EI. For example, in Figure 1, \(y_{pj}\) is the coordinate of EI in the \(y\) direction, which can be expressed as follows:  
\[y_{pj}=f\frac{y-y_{fj}}{z}+y_{fj} \tag{1}\]  
This calculation can also be applied to the \(x\) direction, and the \(x_{pi}\) of the projected EI can be expressed as follows:  
\[x_{pi}=f\frac{x-x_{fi}}{z}+x_{fi} \tag{2}\]  
where \((x,y,z)\) is the coordinate of the object point P in the 3D coordinate system, and \((x_{pi},y_{pj})\) is the coordinate of EI \(im_{i,j}\) on the holographic plane. The coordinate of the \((i-th,j-th)\) pinhole is \((x_{fi},y_{fj})\). The \((i-th,j-th)\) EI can be expressed as follows:  
\[im_{i,j}(x_{pi},y_{pj})=A_{p} \tag{3}\]where \(A_{p}\) is the amplitude of object point \(p(x,y,z)\) and \((i,j)\) is the index of the EI. The phase of this hogel on the holographic plane is a phase of the sphere wave from the pinhole to the hogel, which can be expressed as follows:  
\[\mathrm{phs}_{ij}(x_{pi},y_{pj})=\frac{2\pi}{\lambda}[\sqrt{\left(x_{pi}-x_{fi} \right)^{2}+\left(y_{pj}-y_{fj}\right)^{2}+f^{2}}] \tag{4}\]  
The complex amplitude of the object wave-front corresponding to each hogel is found by multiplying Equation (3) by a divergent sphere wave with the phase illustrated in Equation (4). This concept is simple and straightforward. From the view of the local area, the local amplitude and spherical wave control the amplitude and propagation direction of this small beamlet.  
Assuming that the reference light is a plane wave, and the angle between the plane wave in the \(y\) direction and the \(z\) axis is \(\theta_{refy}\), the off-axis amplitude type \((i-th,j-th)\) hogel can be expressed as follows:  
\[H_{i,j}(x_{pi},y_{pj})=im_{i,j}(x_{pi},y_{pj})\cos[phs_{i,j}(x_{pi},y_{pj})- \frac{2\pi}{\lambda}y_{pj}\sin(\theta_{refy})] \tag{5}\]  
After the parameters of the PA and FOV of the hologram, as well as other necessary parameters are provided, each hogel can be calculated independently. All the hogels form the final high-resolution hologram.