# Robust Evaluation of Reference Tilt in Digital Holography
## 2 Principles
The minimum unit for \(u\), \(u_{0}\) and \(v\), \(v_{0}\) are denoted as \(\Delta u\) and \(\Delta v\), respectively  
\[\Delta u=\frac{1}{M\Delta x},\Delta v=\frac{1}{N\Delta y} \tag{7}\]  
where \(M\), \(N\) are the pixel numbers along the horizontal and vertical directions in the recording plane with the pitch of \(\Delta x\) and \(\Delta y\). So the precision here is affected by both the CCD pixel size and the CCD matrix size. In the computer, the variants \(u\) and \(v\) or parameters \(u_{0}\) and \(v_{0}\) are expressed as  
\[u=m\Delta u,v=n\Delta v,u_{0}=m_{0}\Delta u,v_{0}=n_{0}\Delta v \tag{8}\]  
where \(m\), \(n\), \(m_{0}\), \(n_{0}\) are all integral numbers in order on the SSP to represent the location of the spectrum. After the spectrum with the maximum value detected, the numbers \(m_{0}\) and \(n_{0}\) are determined. Then using Equations (7) and (8), the RTA can be calculated by  
\[\theta_{x}=sin^{-1}[\lambda m_{0}/(M\Delta x)],\theta_{y}=sin^{-1}[\lambda n_{ 0}/(N\Delta y)] \tag{9}\]  
The precision of the calculated RTAs are decided by the sizes of the frequency units (FPs) \(\Delta u\) and \(\Delta v\) on SSP. Smaller \(\Delta u\) and \(\Delta v\) predict higher precision of RTA. Figure 1a shows the symmetric distribution of one Gaussian function normalized by the maximum value at its center in the spatial domain. One of the frequency peaks from the Fourier spectra of two plane waves interference fringe is depicted in Figure 1b. Since the two distributions are similar, the best choice should be a Gaussian function to fit the real distribution of a frequency peak on the SSP. For the spectra regime of the off-axis holography, the distribution should be a delta function of Equation (6) in theory prediction, which should be two narrow peaks without any width. But for a specific practice, the spectra become a round spot with a certain area because of the diffraction effect from the limited aperture of the recording system under the constraint of the CCD sensor. Naturally, the spot should be a symmetrical distribution. If the distribution is a continuous function, the center point of the spot can be detected by searching its maximum value. Unfortunately,  
the spectra can only be expressed as a discrete distribution since it is the result of a discrete fast Fourier transform in a digital computer. The size of the minimum unit \(\Delta u\) and \(\Delta v\) is determined by the size of the hologram and the size of the pixels on it in Equation (5). The absolute value of the spectrum shown in Figure 1b is, in fact, averaged in one block or FP (\(\Delta u\), \(\Delta v\)). The FP with the biggest absolute value can be located easily by the corresponding count number, but it is impossible to calculate the exact locations of the center of the delta function in Equation (6) unless the points (\(\pm u_{0}\), \(\pm v_{0}\)) fall exactly right at the center of one FP. In the specific experiment, it is always impossible to fix the RTA values exactly to satisfy this condition. Mostly, the location of the spectrum decided by RTA mismatches with any FP center. This misplacement renders an error in the calculation result of RTA and then degrades the image quality by the residue phase error. Of course, the smaller the block size denoted by \(\Delta u\) and \(\Delta v\), the less phase error from the mismatch. So many endeavors have been made to decrease the FP size on the SSP. These up-sampling methods, including extrapolation [27] and zero padding [28], increase the data array of the hologram or SSP and then impose heavy storage and computation loads on computers very quickly [27; 28]. On the other hand, the distribution of the Fourier frequency spectrum has its own law so that a proper function target can increase the fitting efficiency and improve the accuracy of the Fourier frequency spectrum location.