# Efficient Hogel-Based Hologram Synthesis Method for Holographic Stereogram Printing
## 2 Method  
### Acquisition of Perspective Images  
A holographic stereogram is synthesized conventionally based on three procedures: acquisition of perspectives, the transformation from perspective images to parallax-related images, and hogel printing. Figure 1 shows a schematic procedure of hogel-based holographic stereogram synthetization approach. The perspective projections of a 3D scene are usually acquired by an array of virtual cameras based on a re-centering camera model in Blender software. In recording of a full-parallax HS, we assume that (\(M\times N\)) view projections of resolution (\(m\times n\)) are captured along the horizontal and vertical axes, respectively. The perspectives are numbered (\(M\times N\)) in the camera plane such that the upper-left projection is (\(I\), \(I\)) and the lower-right projection is (\(M\), \(N\)).  
The array of these captured perspectives is stored in image matrix \(P\) as:  
\[P=\left(\begin{array}{cccc}P_{1,1}&P_{1,2}&\cdots&P_{1,N}\\ P_{2,1}&P_{2,2}&\cdots&P_{2,N}\\ \vdots&\vdots&\ddots&\vdots\\ P_{M,1}&P_{M,2}&\cdots&P_{M,N}\\ \end{array}\right) \tag{1}\]  
For each hogel, the object field for HS is a superposition of rectangularly propagated plane waves with amplitudes defined by the corresponding discrete light field samples. For the simplicity, the object field at the hologram plane is defined in one dimensional as:  
\[O_{HS}(x)=\sum_{h}rect(\frac{x-h\Delta_{x}}{\Delta_{x}})\times\sum_{r}\sqrt{P _{M,N}}\exp(j2\pi f_{x}^{hr}x) \tag{2}\]  
where \(f_{x}^{hr}\) is the spatial frequency of hogel \(h\) and ray \(r\) on the \(x\) axis and \(\Delta_{x}\) is the hologram plane sampling distance. The inner sum in Equation (2) is the spatial pattern to be written in the corresponding hogel and can be obtained by inverse Fourier transformation of parallax-related images. Figure 2 shows the acquisition geometry of the perspectives obtained by the re-centering camera model.  
Figure 1: Schematic configuration of hogel-based holographic stereogram synthetization.  
Figure 2: Geometry used for capturing perspectives by the re-centering camera model.  
The perpendicular re-centering image plane lies at a distance D from the camera array \(C_{c}\) where the perspectives are captured at the centre of projection for each camera that crosses the centre of the re-centering plane. The advantage of re-centering camera model is that, less of the field of view is wasted when capturing rays emitted by the object. One limitation of the holographic stereogram is its poor imaging resolution for 3D scenes with a big depth range or located far from the hologram plane. When objects located far from the camera plane may be under-sampled, resulting in a blurred reconstruction. However, these images will be algorithmically distorted at each image taken at a different angle. The virtual cameras in the re-centering camera method were calibrated such that the its field of view (FOV) was equal to the FOV produced by the imaging lens system. Therefore, we place the virtual cameras close to the 3D object and rendered directional view images to avoid this problem. Given the properties of the geometric model, each pixel at the image plane corresponds to the same pixel in the captured image. The camera \(C_{1}\) captures the leftmost top position, and the camera \(C_{c}\) captures the rightmost bottom position (Figure 2).