# 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  
In order to perform the reconstruction of the holographic stereogram without a distortion, the perspectives must be re-arranged to form parallax-related images corresponding to different volume holograms in the hologram plane. The conventional pixel re-arrangement technique creates a sequence of hogels reflecting the angular intensity distribution of the 3D object, which is related to the smooth motion-parallax information available at the viewer's position. The holographic stereogram synthesizes a small volume hologram of the re-arranged hogels. When the hologram is illuminated, each hogel is simultaneously reconstructed. The changes in both intensity and parallax are made by reference to the hogel angular intensity distribution. The pixel re-arrangement technique is employed to generate the array of hogel images \(H_{i\bar{y}}\) from a sequence of \((M\times N)\) perspectives captured in the horizontal and vertical directions. All pixels at the same perspective location are extracted in the horizontal and vertical directions to form a new matrix \(H_{i\bar{y}}\) composed using the pixels of all perspectives. \[H_{i\bar{y}}(s,t)=P_{st}(i,j),\ s=1\ldots M;\ t=1\ldots N;\ i=1\ldots m;\ j=1 \ldots n; \tag{3}\]  
In Equation (3), each perspective in image matrix \(P\) has \((m\times n)\) pixels and their number is given by \(M\times N.\) On the contrary, the resolution of re-arranged hogel image is defined by \(M\times N,\) and the number of images is \(m\times n,\) according to Equation (3). This hogel-making process shows that the hogel image resolution is defined by the number of perspectives. Thus, the resolution of the hogel image is limited by the number of acquired perspectives subjected to a conventional pixel re-arrangement algorithm. The angular resolution of the reconstruction is determined by the number of hologram sampling points in each hogel. To re-arrange a higher resolution hogel, many perspectives are required to capture, according to Equation (3). Figure 3 presents the transformation of the perspectives to the parallax-related images by the conventional pixel re-arrangement technique based on Equation (3). Figure 3: Conventional pixel re-arrangement process from captured perspectives. One drawback of the conventional hogel re-arrangement algorithm in Figure 3 is that a holographic stereogram improves angular resolution at the cost of spatial resolution as the hogel size increases. One important question for HS algorithms is how to choose the hogel size. A larger hogel size increases angular resolution and a smaller hogel size increases spatial resolution. When the hogel re-arrangement algorithm in Equation (3) is applied, there exists a trade-off problem between these resolutions. Therefore, it does not require a high memory usage and computation time to acquire a large number of perspectives at lower resolution because of the hogel re-arrangement process. In the proposed method, fewer perspective images with higher resolution are captured by re-centering the camera model (Figure 2) and then further processed to generate a plane wave moving in a certain direction. This wave is diffracted by hogel (\(i,j\)) onto the HS plane. The pixel (\(i,j\)) of each hogel image is diffracted by the hogel (\(m,n\)), corresponding to different locations in the HS. Then, a novel pixel re-arrangement algorithm is proposed to rearrange the higher resolution hogels from a reduced number of perspectives with high resolution compared to the conventional method. Instead of capturing many perspectives with lower resolution, fewer perspectives are pre-defined and captured with higher resolution. Based on fewer captured perspective images, the pixel re-arrangement algorithm in Equation (3) is modified into Equation (4) to satisfy the condition of the same number of hogels, which are re-arranged by the conventional method.