# Motion Parallax Holograms Generated from an Existing Hologram
## 3 Results  
Figure 3 shows the numerically color-reconstructed images from motion parallax holograms from left and right view positions. They are complex amplitude holograms [12]. We used the 3D model provided from [19]. We simply calculated the color holograms by applying Equations (1), (3) and (4) for red (633 nm), green (532 nm) and blue (430 nm) wavelengths, respectively. The resolution of holograms is \(512\times 512\) pixels. The pixel pitch of the holograms is 10 um. The near and far distances of 3D scenes are 3 and 5 cm from the holograms, respectively. It is an image hologram setup [21]. We used this setup to confirm the effectiveness of the proposed method, but the proposed method works well at long distances as well. Figure 3a,b show the reconstructed images of \"Antinous.\" We confirmed that the proposed method can generate motion parallax from left and right views. Likewise, the proposed method can generate motion parallax of reconstructed images of \"Greek,\" as shown in Figure 3c,d. Additionally, likewise Figure 4, we present three videos in the Supplementary Section (Video S1: reconstructed video of \"antinous,\" Video S2: reconstructed video of \"Greek,\" and Video S3: reconstructed video of \"Papillon\" whose original 3D scene are provided in [22]) to confirm the effectiveness of the proposed method. These videos present reconstructed images with motion parallax using the proposed method, while continuously changing the viewing positions and reconstruction focuses. We compared the computational complexity between the proposed method and the conventional method using Equation (1) alone. The conventional method requires the re-rendering of 3D scenes and re-calculating of holograms. If we need \(N_{v}\) holograms with different positions, the computational complexity is given by \(O((R+N_{d}N^{2}\log N)\times N_{v})\), where \(R\) denotes the computational complexity of 3D scene rendering, and \(N\) and \(N_{v}\) denote the size of a hologram, and the number of motion parallax, respectively. \(N_{d}N^{2}\log N\) indicates the computational complexity of the layer-based hologram calculation of Equation (1). The computational complexity is proportional to \(N_{v}\). By contrast, the computational complexity of the proposed method is expressed as \(O((R+N_{d}N^{2}\log N)+N_{B}N^{2}\log N+N^{2}\times N_{v})\). Notably, the computational complexity of the proposed method does not depend on the number of motion parallax \(N_{v}\). \(N_{B}N^{2}\log N\) indicates the computational complexity of the IFFT in Equation (3). \(N^{2}N_{v}\) indicates the computational complexity of the addition or subtraction for the disparity hologram. Figure 5 shows the calculation times of the proposed method and the conventional method as the function of the number of motion parallax. We used a central processing unit (CPU) and graphics processing unit (GPU) of Intel Core i7-4790 (3.60 GHz) (Santa  
Figure 3: Reconstructed images from motion parallax holograms from left and right view positions. (**a**) and (**b**) show the reconstructed images of “Antinous.” with motion parallax from left and right views. (**c**) and (**d**) show the reconstructed images of “Greek.” with motion parallax from left and right views. We used the 3D model provided from [19]. Clara, CA, USA) and Nvidia GeForce RTX 2080 SUPER (Santa Clara, CA, USA), respectively, on Windows 10 (64 bit) operating system. We generated holograms of Equation (1) on the GPU using our wave optics library [23], and the motion parallax holograms of Equations (3) and (4) on the CPU. The hologram size is \(2048\times 2048\) pixels. Other calculation parameters are the same as Figure 3.  
The horizontal axes of \"zero\" include only the hologram calculation time using Equation (1). Besides, the horizontal axes of 'one\" in the proposed method include only the calculation times of Equations (3) and (4).