# Motion Parallax Holograms Generated from an Existing Hologram
## 2 Proposed Method  
The proposed method generates motion parallax holograms from an existing hologram. Here, we employ a layer-based hologram calculation [16; 17; 18], given as  
\[u(m_{2},n_{2})=\sum_{z}^{N_{d}}\mathcal{P}_{z}[u(m_{1},n_{1})\mathcal{M}_{z}(m _{1},n_{1})], \tag{1}\]  
where \(u(m_{2},n_{2})\) is the complex amplitudes on a hologram plane, \(\mathcal{P}_{z}\) is the operator of a diffraction calculation at a depth \(z\), \(N_{d}\) is the number of layers, \(u(m_{1},n_{1})\) is one color component of an RGB image, and \(\mathcal{M}_{z}(m_{1},n_{1})\) is a binary function [16] for extracting \(u(m_{1},n_{1})\) pixels at the depth \(z\). This study employed the angular spectrum method [12] for diffraction calculation. First, the hologram calculation divides the 3D scene into multiple layers using depth image information. Then, we perform diffraction calculations from the multiple layers to the hologram; subsequently, we accumulate all diffracted results to obtain the final hologram. The proposed method is based on [13], called Hidden Stereo, which yields motion parallax images directly from RGB and depth images. This technique has been developed for switchable stereoscopic displays of 2D and 3D display modes, not for holographic displays. The principle is simple. As well-known, an image can be decomposed into a combination of sinusoidal waves according to the Fourier th  
Figure 1: Outline of the proposed method. It yields left- or right- viewed holograms by adding or subtracting a disparity hologram, respectively. We used the 3D model provided from [19]. sinusoidal wave \(\sin(\omega x)\), where \(\omega\) is an angular frequency and \(x\) is a spatial variable. If we can generate the orthogonal sinusoidal wave of the original sinusoidal wave, we can derive the following simple relation:  
\[\sin(\omega x)\pm A\sin(\omega x+\pi/2)=\sqrt{1+A^{2}}\sin(\omega x\pm\phi), \tag{2}\]  
where \(\tan\phi=A\)[13]. This relation indicates that we can generate a right- or left-shifted image with the phase \(\phi\) by adding or subtracting the orthogonal image with an amplitude \(A\) to the original image. Although, in [13], a complex steerable pyramid [20] was used to generate orthogonal images, it is computationally complex. Thus, our proposed method uses simple Fourier band-pass filtering. Figure 1 shows the outline of the proposed method. First, We generate a hologram from 3D scenes with an RGB image and depth image using Equation (1); subsequently, we generate a disparity hologram from the calculated hologram using Fourier band-pass filtering. By adding or subtracting the disparity hologram to the calculated hologram, we obtain new holograms viewed at a right or left position. The newly generated holograms do not experience computational complexity because it requires only addition or subtraction operations. As the number of the bands increases, we obtain finer motion parallax. Figure 2 shows the generation of disparity holograms. We first perform Fourier band-pass filtering to obtain quasi-orthogonal signals of the original hologram. The band-pass filtering consists of a fast Fourier transform (FFT) and band-pass masks \(\mathcal{B}_{j}\). Subsequently, we perform the inverse FFTs (IFFTs) of the filtered spectra and take only the imaginary parts of the IFFT results to obtain the quasi orthogonal signals. That is, we calculate the following equation:  
\[\mathcal{O}_{j}(m_{2},n_{2})=\Im\big{[}\text{IFFT}\big{[}\text{FFT}[u_{2}(m_{2},n_{2})]\mathcal{B}_{j}(m,n)\big{]}\big{]}, \tag{3}\]  
where \(\Im\) denotes the operator to take the imaginary part of the IFFT and then multiply the results with \(\pm 1\). Holograms are in genral expressed as \(\cos(\theta)\) where \(\theta\) is arbitrary phases. Due to the relation of \(\cos(\theta+\pi/2)=-\sin(\theta)\), the quasi-orthogonal signal can be obtained by taking the imaginary part of the IFFT result. \(\mathcal{O}_{j}\) corresponds to \(\sin(\omega x+\pi/2)\) in Equation (2). Figure 2: Generation of a disparity hologram.