# Efficient Hogel-Based Hologram Synthesis Method for Holographic Stereogram Printing
## 2 Method
### Hogel Generation and Enhancement of Angular Resolution by the Pixel Re-Arrangement Algorithm
Then, rather than extracting a single pixel from each perspective as shown in Figure 3, our algorithm extracts 20 pixels (the number of extraction factor can vary by the number of hogels to be recorded) from a reduced number of perspectives. This pixel-to-pixel distribution is defined as shown in Equation (4). \[H_{u/20,v/20}(20\times U,20\times V)=P_{UV}(u,v),\ U=1\ldots K;V=1\ldots L;u=1 \ldots k;v=1\ldots l; \tag{4}\]  
where \(U\), \(V\) denotes the horizontal and vertical camera position with \(K\times L\) captured images and \(u\), \(v\) denotes the horizontal and vertical pixel position for each perspective with \(k\times l\) image resolution. This process increases the re-arranged hogel resolution without any need for interpolation or pixel resizing, unlike the conventional method. It also improves the angular resolution of the reconstruction while retaining the hogel spatial resolution. This process is presented in Figure 4.  
In this case, the hogel resolution is four times greater than that of a conventional re-arrangement. No pixel resizing, or interpolation, is required to match the resolution of the micro display. Thus, the total number of perspectives is reduced 25 times, and the proposed method avoids spatial resolution loss because of not applying a pixel re-sizing technique. The angular resolution of the reproduced light can be determined by the diffraction limit of the size of hogel and given by the following equation. \[\Delta\theta_{\rho}=\frac{\Delta\lambda}{\rho} \tag{5}\]  
Figure 4: Proposed pixel re-arrangement algorithm from captured perspectives. where \(\rho\) denotes the size of the elemental hologram and \(\lambda\) is the wavelength of the reconstructed light. To evaluate the angular resolution of diffracted light of HS for both methods, we obtain a different size of elemental holograms based re-arranged hogels by both methods. The size of the elemental hologram is \(0.32\times 0.32\) mm for the conventionally re-arranged hogel, whereas the size of the elemental hologram is \(1.28\times 1.28\) mm by the proposed approach based on a 6.4-um pixel pitch of the SLM display. Calculated results for angular resolution in Equation (5) were 1.66 and 4.15 degrees for the conventional and proposed method, respectively, when the wavelength \(\lambda\) is monochromatic light of 532 nm. As a result, the angular resolution was observed to be increased since the hogel size is raised by the proposed pixel re-arrangement algorithm. One holopixel on SLM corresponds to 4.66-degree angular resolutions. However, the number of perspectives is reduced by the proposed algorithm. It increases computation time for rendering and for a further hogel generation process, as shown in Table 1. This is one of the side effects on reducing perspective images and another effect exists on motion parallax information. Even fewer perspective images are captured within the same field of view (FOV) in the horizontal and vertical directions, respectively. The angular step of the virtual camera is increased in both directions, and the parallax changes rapidly within angular intensity distribution, which is not smooth when the observer's eye moves hogel to hogel. Another drawback of the proposed method is that it leads to a deviation error in the reconstruction due to the proposed pixel re-arrangement algorithm. It is hard to correct this error and it is the physical nature of the method. Instead, hogels need to be captured in a direct way. When parallax-related images are generated in the hologram plane, each hogel is converted into a small volume hologram recorded on a holographic plate by our developed holographic printer. Holograms can be synthesised for the HS plane sampled from phase holograms by using phase-modulated (\"phase-only\") SLM. Volume hologram of each hogel image is displayed on SLM sequentially and recorded on the holographic plate.