## 2 The Model of Stereo Depth for Lenticular-Based Autostereoscopic Displays  
We implement a light propagation model of a typical AD system shown in Figure 1 in this section and analyze the influence of the lens aberration to the SD, accordingly in the following part. As shown in Figure 1, two pixels in the display panel, represented by two orange spots, are loaded a stereo pair of points, which can be considered to be two point light sources. The display panel is in the \(x_{1}-y_{1}\) plane. The lenticular sheet is in the \(\varepsilon-\eta\) plane. The viewing plane is the \(x_{0}-y_{0}\) plane. Light emitted by the two-point light sources are redirected by the lenticular sheet and form two spots at the viewing plane, which are represented by the two orange spots in the \(x_{0}-y_{0}\) plane. The positions of the viewer's eyes must be superposed with the two orange spots in the \(x_{0}-y_{0}\) plane to let the viewer perceive 3D effect. The distance between the panel and the lenticular sheet is equal to the focal length \(f\) of the lenslets in the lenticular sheet, and the distance between the lenticular sheet and the viewing plane is the designed viewing distance of the AD, which is defined as \(D\). The pitch of the lenslets is \(s\). A lenslet in the lenticular sheet can be located by the center point of the lenslet, named (\(ms\), \(ns\)), wherein \(m\) and \(n\) are integers. A pixel in the display panel can be defined by its position, named as (\(kt\), \(lt\)), wherein \(k\) and \(l\) are integers, and \(t\) is the pitch of the display panel. The viewing angle \(\theta_{view}\) is defined as the angle between the line through the pixel and the viewer's eye and the line perpendicular to the display panel. Figure 1: (a) A typical autostereoscopic display system using lenticular sheet; (b) the display panel plane; and (c) the lenticular sheet plane. A pixel (\(kt\), \(lt\)) emits a spherical wave with the wavelength of \(\lambda\) through a lenslet (\(ms\), \(ns\)) in the lenticular sheet, and the light wave reaches the viewing plane and forms a light spot with the intensity distribution \(E_{mn}\). According to the scalar diffraction theory [23], \(E_{mn}\) can be expressed as:  
\[ \begin{array}{c}E_{mn}(x_{0},y_{0})=\frac{1}{\lambda^{2}/D}\iint\exp\Bigl{\{} -\frac{i\pi}{\lambda f}\Bigl{[}(\varepsilon-x_{1})^{2}+(\eta-y_{1})^{2}\Bigr{]} \Bigr{\}}\times T_{mn}(\varepsilon,\eta)\\ \times\exp\Bigl{\{}\frac{i\pi}{\lambda D}\Bigl{[}(x_{0}-\varepsilon)^{2}+(y_{ 0}-\eta)^{2}\Bigr{]}\Bigr{\}}d\varepsilon d\eta,\end{array}  \tag{1}\]  
where is the complex amplitude transmittance function of the lens (\(ms\), \(ns\)), with the expression of  
\[T_{mn}(\varepsilon,\eta)=\frac{1}{\lambda}P(\varepsilon-ms,\eta-ns)Q( \varepsilon-ms,\eta-ns), \tag{2}\]  
and \(P(ms,\ ns)\) is the pupil function of the lens (\(ms\), \(ns\)) and is defined as:  
\[P(ms,ns)=\left\{ \begin{array}{c}R,|\varepsilon|,|\eta|\leq\frac{s}{2},\\ 0\ otherwise,\end{array} \right. \tag{3}\]  
\(R\) is the amplitude transmittance of the lens substrate at the wavelength of \(\lambda\), and \(Q(ms\), \(ns)\) is the surface function of the lens. If the shape of the lenslet is spherical, there is  
\[\frac{1}{\lambda}Q(ms,ns)=exp\biggl{\{}-\frac{i\pi}{\lambda f}\Bigl{[}( \varepsilon-ms)^{2}+(\eta-ns)^{2}\Bigr{]}\biggr{\}}. \tag{4}\]  
Equation (1) only considered the paraxial incident light beam to the viewing point. However, in most cases, the wave from the pixel to the viewing point is slant, which makes Equation (1) invalid. To solve this problem, an inclination factor [24] is introduced into Equation (1), which is related to the viewing angle \(\theta_{view}\), as expressed by:  
\[K(\theta_{view})=\frac{1+\sin\theta_{view}}{2}. \tag{5}\]  
With the incline factor \(K(\theta_{view})\), Equation (1) is modified into a general form:  
\[