# Full-Parallax Multiview Generation with High-Speed Wide-Angle Dual-Axis Scanning Optics
## 2 Interpretation of the VZ Using WDF  
In a multiview display, a VZ is an area where 3D contents are fully observed without vignetting. To understand the properties of the VZ, it is necessary to divide the VZ into a sub-viewing zone (SVZ) for an individual view and integrated viewing zone (IVZ) for an integrated view. In the IVZ, several views are observed simultaneously, but each view is vignetted due to the limitation in the aperture size of the observation. The extent of vignetting changes according to the relative position of the observation with respect to the SVZ. Therefore, the WDF analysis is useful for estimating the integrated view because both the space and spatial frequency are considered. Considering optical structure of the system is necessary to use WDF analysis for integrated views. Figure 1 shows a schematic of the proposed system. Our proposed system has a ball lens which is employed as the optical component for generating SVZ. Therefore, the analysis is performed on the ball lens. The viewing distance of a 3D display is typically described using ray-optics-based analysis [23]. Figure 2 depicts the definition of viewing distance range and width of the SVZ when the image is projected inside a ball lens. So, ray-optics-based analysis is performed on the ball lens. The optical ray from the image is refracted at the surface of the ball lens and the collimated rays are focused on a point. The distance from the center of the ball lens to the point of focus is called the reference viewing distance and it is calculated as  
\[VD_{0}=R\left[\frac{\sin\alpha}{\tan(\beta-\alpha)}+\cos\alpha\right], \tag{1}\]  
\[\textit{where}\quad\alpha=\sin^{-1}\frac{\pi_{\alpha}}{R} \tag{2}\]  
Here, \(R\) is the radius of the ball lens and \(x_{o}\) is the position on the image plane. \(\alpha\) and \(\beta\) are the incident and refracted angles, respectively, at the surface of the ball lens. These angles are related by [PERSON]'s law, \(\mathrm{nsin}\alpha=\sin\beta\), where n is the refractive index of the ball lens. The viewing distance is limited by the field of view, \(\theta_{SVZ}\), which is determined by the optical stop in the 4-\(f\) structure positioned before the optical scanner. The viewing distance has a distance range, and the minimum and maximum viewing distances are determined based on half of the field of view [24]. The distance range is obtained using [PERSON]'s law in the same manner as that of the viewing distance. Therefore, the minimum and maximum viewing distances of SVZ are, respectively, expressed as  
Figure 1: Schematic of the full-parallax multiview display with spherical symmetry. Figure 2: Formation of the sub-viewing zone (SVZ) in the proposed system. (**a**) Viewing distance range and width of the SVZ for the image inside of ball lens, and (**b**) change in location of the SVZ when the image is rotated around the y-axis. \[SVD_{min} =3R\bigg{[}\cos\theta_{SVZ}-\sin\theta_{SVZ}\frac{3\sin\theta_{SVZ}+ \sin(\alpha-\theta_{SVZ})}{3\cos\theta_{SVZ}-\cos(\alpha-\theta_{SVZ})}\bigg{]} \tag{3}\] \[SVD_{max} =3R\bigg{[}\cos\theta_{SVZ}-\sin\theta_{SVZ}\frac{3\sin\theta_{SVZ}- \sin(\alpha-\theta_{SVZ})}{3\cos\theta_{SVZ}-\cos(\alpha-\theta_{SVZ})}\bigg{]}. \tag{4}\]  
The viewing distance range of SVZ increases as the field of view increases. The width of SVZ is directly related to the field of view and is approximately the same as the arc length; it is determined as  
\[w_{SVZ}=2\theta_{SVZ}VD_{0}. \tag{5}\]  
When an image is observed within the SVZ, the height of the virtual image, \(x_{i}\), becomes higher than the initial position, \(x_{o}\), and it can be derived using backtracking from the viewing distance to the center of the ball lens. It is obtained as  
\[x_{i}=VD_{0}\tan(\beta-\alpha). \tag{6}\]  
To generate multiple views in the proposed system, each SVZ is placed next to each other in a circle with a radius equal to the reference viewing distance.