# Systematic Approach for Alignment of Light Field Mirage
## 2 Light Field Mirage  
First, we briefly explain the light field Mirage that was proposed in our previous study [1]. We previously proposed a concept of a digital implementation system in which LFDs, which substitute the existing Mirage parabolic mirrors, generate a 360-degree 3D image in the top hole as shown in Figure 1. The LFDs are alternately configured between upper and lower positions, and the optical distance from the upper LFDs to the top hole is the same as that from the lower LFDs by using a half mirror. The LFDs not only generated an aerial 3D image at the top hole leveraged by the resolution-priority technique [20; 21], but also provided higher-resolution 3D images than the conventional depth-priority technique [22]. The plane at the top hole where the aerial images are produced is called the \"central depth plane\" (CDP) in the resolution-priority technique. The lens array of LFDs uses an ordinary lens array, since slanted LFDs are circularly aligned to redirect rays across the whole circumference. The number of display devices for the light field Mirage were significantly less than that of multi-projection systems [10; 11; 12; 13; 14; 15; 16] to utilize LFDs. In addition, as the light field Mirage consists of multiple LFDs, flat-panel displays for LFDs enable the use of middle-resolution displays in contrast to conventional flat-panel systems. Because flat-panel type LFDs generate repeated 3D images, we have developed two techniques to avoid the observation of repeated 3D images by viewers. The first technique, \"non-tracking\", eliminates repeated 3D images by using the top hole, since desired 3D images from all LFDs are overlaid on one another in the CDP, as shown in Figure 2a. The second technique, \"tracking\", divides the rays from the desired 3D image on the viewing plane. Therefore, repeated 3D images can be bypassed by moving the correct viewing regions, as shown in Figure 2b. Although the first technique does not require tracking  
Figure 1: Schematic of light field Mirage: (**a**) side view and (**b**) top view. of viewers, the 3D image size was constrained, and even though the second technique requires tracking of viewers, large 3D image sizes could be produced. When the LFD width is denoted by \(w_{d}\), the radius from the central axis of the display system to the center of each screen placed on the circumference is denoted by \(r\), because the angle of view for the LFD width from the center, denoted by \(\varphi\), is given by \(\varphi=2\ \tan^{-1}(w_{d}/2r)\), and the number of LFDs required to assemble the system, denoted by \(N\), is given by:  
\[N=2\pi/fullwidth\varphi=\pi/\tan^{-1}(w_{d}/2r) \tag{1}\]  
When the distance from the lens array to the CDP is denoted by \(l_{l}\), the inclination angle is given by \(\alpha=\sin^{-1}\ (r/l_{l})\). For the non-tracking technique, the lens pitch of the lens array is denoted by \(p_{l}\), and the number of the rays for one LFD is denoted \(N_{r}\) and is equal to the number of lenses in the array as \(N_{r}=w_{d}/p_{l}\). The pixel pitch of the flat-panel displays is denoted by \(p_{p}\), and the 3D image resolution along the diameter of the top hole, denoted by \(X_{\text{3D}}\), is equal to the number of elementary images:  
\[X_{\text{3D}}=p_{l}/p_{p} \tag{2}\]  
For the tracking technique, the number of rays for one LFD is equal to the resolution of the elementary images by \(N_{r}=p_{l}/p_{p}\). The pixel number of the flat-panel displays is represented by \(X\times Y\), and the 3D image height relates to the flat-panel display height. Taking into account the inclined screen, the 3D image height is given by:  
\[W_{\text{3D}}=Y\ p_{p}/\cos\alpha. \tag{3}\]  
For the resolution-priority technique, the focal length of the lens array is denoted by \(f\), that between the lens array and the pixels of the flat-panel display is denoted by \(l_{o}\), and the 3D image pixel pitch is given by \(p_{p}\ l_{l}/l_{o}\simeq p_{p}\ l_{f}\).