# Full-Parallax Multiview Generation with High-Speed Wide-Angle Dual-Axis Scanning Optics
## 2 Interpretation of the VZ Using WDF
The position of SVZ is changed by angle \(\varphi_{y}\) around the y-axis, where \(\varphi_{y}\) is the rotation angle of the image at the center of the ball lens. It is slightly smaller than the optical scanning angle of the dual-axis scanner because the image at the scanner is enlarged via magnifying optics. The number of views in the system is determined using the maximum rotation angle \(\varphi_{y}\) and the field of view of SVZ, \(\theta_{SVZ}\), as  
\[N=1+\varphi_{y}/\theta_{SVZ}. \tag{7}\]  
Figure 3 depicts the formation of a VZ from SVZs. Because the observer can watch the view integrated from different views outside of the SVZ, the area of VZ exceeds the summation of all SVZs. The width of the VZ is expressed as follows  
\[w_{VZ}=N\times w_{SVZ}. \tag{8}\]  
By contrast, the viewing range along the longitudinal direction becomes enlarged. The minimum and maximum viewing distance are given by  
\[\textit{where}\quad\alpha=\sin^{-1}\frac{\pi_{\alpha}}{R} \tag{2}\]0 \[\textit{where}\quad\alpha=\sin^{-1}\frac{\pi_{\alpha}}{R} \tag{2}\]1  
\[\textit{where}\quad\alpha=\sin^{-1}\frac{\pi_{\alpha}}{R} \tag{2}\]2 Formation of viewing zone (VZ) from SVZs in a multiview display. Figure 4 shows the front and rear IVZs for the integrated view and nine SVZs for individual views. The front and rear IVZs are separated by a set of SVZs. The SVZ is the area defined by a single view and the IVZ is the area outside of SVZ. Even though each view is vignetted in the IVZ, multiple vignetted views in the IVZ are overlapped with each other in the SVZ, and the entire scene is integrated. In Figure 4, five eye positions for observation are indicated by \(e_{1}\)-\(e_{5}\). The properties of IVZ are investigated using WDF, which represents space and spatial frequency information visually [25, 26, 27]. The general definition of WDF, \(W\big{(}x,y;v_{x},v_{y}\big{)}\) is  
\[\textit{where}\quad\alpha=\sin^{-1}\frac{\pi_{\alpha}}{R} \tag{2}\]3  
where \(*\) denotes the complex conjugate; \((x,y)\) and \((x^{\prime},y^{\prime})\) are spatial coordinates and \((v_{x},v_{y})\) are the spatial-frequency coordinates. The WDF analysis is performed using MATLAB (MathWorks). Figure 5 depicts the WDF analysis of the observations based on five eye positions. Figure 5a shows an initial WDF of nine views assigned to SVZs inside the ball lens, where \(SVZ_{0}\) indicates the WDF of the view assigned to the SVZ located on the optical axis. The SVZs are numbered according to the rotation angle. Accordingly, as the rotation angle increases, the number \(k\) of the corresponding \(SVZ_{k}\) also increases. As the WDFs of the whole views are stacked along the spatial frequency \(v_{x}\), each view provides the information in a different direction. In the plots, the red dashed lines represent the size of the image inside the ball lens. Figure 5b shows the change in WDF at the reference viewing distance; the WDF of the observations based on five different eye positions are indicated by blue lines. These five eye positions are indicated in \[\textit{where}\quad\alpha=\sin^{-1}\frac{\pi_{\alpha}}{R} \tag{2}\]4 Here, the red dashed lines represent the boundaries transformed from those in Figure 5a. Because the reference viewing distance is approximately equal to the focal plane, the WDFs of the views are stacked horizontally. The second eye position, \(e_{2}\), is placed on the SVZ plane, and the WDF of the observation covers the entire \(v_{x}\) range of \(SVZ_{0}\). This implies that at the second eye position, the entire single view can be observed. The upper and lower limits of the blue lines are not indicated because the field of view of the human eye is large and extends beyond the range of this plot. At the third eye position, \(e_{3}\), three views corresponding to \(SVZ_{+2}\), \(SVZ_{+3}\), and \(SVZ_{+4}\) can be observed. Furthermore, the WDF of the third eye position indicated by the blue lines contains the upper portion of \(SVZ_{+2}\), middle portion of \(SVZ_{+3}\), and lower portion of \(SVZ_{+4}\).