# Resolution Enhancement of Spherical Wave-Based Holographic Stereogram with Large Depth Range +
## 2 The Resolution Characteristics of RPHS  
Figure 1 shows the principle of RPHS. Based on the conventional HS which converts each EI to the corresponding hogel through FFT, the RPHS multiplies each transformed hogel with a converged spherical wave phase:  
\[H(u,v)=\exp\left[\frac{-jk}{2L}(u^{2}+v^{2})\right]\cdot A(\frac{u}{\lambda f},\frac{v}{\lambda f}),A(f_{x},f_{y})=FFT[I(x,y)] \tag{1}\]  
where \(FFT[]\) is the fast Fourier transform, \(A(f_{x},f_{y})\) is the frequency spectrum of elemental image \(I(x,y)\), \(k\) is the wave number, \(f\) is the focal length and \(L\) is the distance between hologram and the image reference plane (IRP) [25]. Assuming each elemental image has \(N\times N\) pixels with pixel size of \(\Delta x_{1}\), and the pixel size of the hogel is \(\Delta x_{2}\), the following equation is derived according to sampling theorem and FFT theory:  
\[N\Delta x_{1}\Delta x_{2}=\lambda f. \tag{2}\]  
Generally, \(\Delta x_{1}=\Delta x_{2}\), and the size of hogel is equal to the EI. Then, all the transformed hogels can be spliced seamlessly. In RPHS, each hogel emits spherical waves which are focused on the IRP to reconstruct the 3D images. The object points at the IRP are reconstructed perfectly and objects' points around the IRP are reconstructed with good quality. Thus, the IRP is also called the central depth plane (CDP), which means the center of the 3D images. That is to say, the 3D image needs to be located near the CDP to obtain the best reconstruction quality. Thus, \(L\) can also be called the capturing depth. At the CDP, the image spot size \(S\) is given by lens imaging amplification:  
\[S=\frac{L}{f}\Delta x_{1}. \tag{3}\]  
Therefore, by reducing \(L\), reducing \(\Delta x_{1}\) or increasing \(f\), the resolution at the CDP will be enhanced. However, considering the reconstructions away from the CDP, the situation becomes complicated.  
Figure 2 shows the image spot size at different reconstruction planes. At the CDP, the focused image spot is formed with the minimum size. At a reconstruction plane away from the CDP, the defocused spot is formed. The image spot size \(S\) at reconstruction depth \(z\) can be derived easily by dividing the defocused spot into two parts: the green part and the red part. The green part is related with the focused image spot (represented by the blue rectangle) in a similar triangle relation, and the red part is related with the hogel size \(p\) in another similar triangle relation. Thus, the image spot size \(S\) at reconstruction depth \(z\) can be expressed as:  
\[S=\frac{L+\Delta z}{f}\Delta x_{1}+\frac{|\Delta z|}{L}p, \tag{4}\]  
where \(\Delta z\) is the defocused depth, and \(p\) is the hogel size and is equal to \(N\Delta x_{2}\). The value of \(\Delta z\) is negative if \(z\) is smaller than \(L\), and it is positive if \(z\) is greater than \(L\). The first term in Equation (4) is indicated by the green rectangle and the second term is indicated by the red rectangle in Figure 2. The first term is related with imaging while the second term is caused by defocusing. Substituting Equation (2) into Equation (4), we can get:  
\[S=\frac{L+\Delta z}{f}\Delta x_{1}+\frac{|\Delta z|\lambda f}{L\Delta x_{1}}. \tag{5}\]  
Equation (5) shows that at a given defocused depth \(\Delta z\), the image spot size is mainly determined by three parameters: CDP depth \(L\), focal length \(f\) and pixel size \(\Delta x_{1}\). It is interesting that in Equation (5), the values of the first and second terms have opposite changing  
Figure 1: The principle of RPHS.  
Figure 2: The image spot size at different reconstruction planes.  
directions when the three parameters change. Thus, simply changing the parameters may not work for resolution enhancement. Next, we will show how to enhance resolution by changing the three parameters properly with additional techniques.