# Robust Evaluation of Reference Tilt in Digital Holography
## 2 Principles
Gaussian functions are used to express some distributions like the point spread function and delta function [30]; here, a method is provided to calculate the more accurate value of RTA after constructing a function of one frequency peak distribution by Gaussian fitting. After the center of the frequency peak is detected, a local area including several \(\Delta us\) and \(\Delta vs\) is chosen around point (\(u_{0}\), \(v_{0}\)) and the values of neighboring FPs can be used to predict the local distribution of the spectra by Gaussian formula  
\[F_{l}(u,v)=A\exp\left[-\frac{(u-u_{0l})^{2}}{2\sigma_{u}^{2}}-\frac{(v-v_{0l})^ {2}}{2\sigma_{v}^{2}}\right]+C \tag{10}\]  
where \(A\) and \(C\) are two constants, \(\sigma_{u}\) and \(\sigma_{v}\) are the two standard errors variants \(u\) and \(v\). The values of \(\sigma_{u}\) and \(\sigma_{v}\) can be chosen commonly and there is no need to optimize them. In Equation (10), \(u_{0l}\) and \(v_{0l}\) are the two coordinates of the peak tip, which can be calculated by  
\[\mathbf{O}(x,y)=\mathbf{O}_{1}(x,y)+\mathbf{O}_{2}(x,y) \tag{2}\]0  
\[\mathbf{O}(x,y)=\mathbf{O}_{1}(x,y)+\mathbf{O}_{2}(x,y) \tag{2}\]1 Gaussian function and spectra distribution on an SSP (**a**) a Gaussian distribution in the spatial domain (**b**) one of the frequency peaks in the Fourier spectra domain. The values of \(u\) and \(v\) that satisfy Equation (11) are the theoretical \(u_{0l}\) and \(v_{0l}\) and they can be expressed with FP ordinal number \(m_{0l}\) and \(n_{0l}\) in two directions on the SSP as  
\[\mathbf{O}(x,y)=\mathbf{O}_{1}(x,y)+\mathbf{O}_{2}(x,y) \tag{2}\]2  
where \(m_{0l}\) and \(n_{0l}\) are the FP numbers with no integrals but decimals so that the tilt angle results are more accurate. So the precision of the LGF method is decided on not only the CCD pixel size and matrix size but also the remeshing grid size around the frequency peak. Then the tilt angles with high precision can be gained by  
\[\mathbf{O}(x,y)=\mathbf{O}_{1}(x,y)+\mathbf{O}_{2}(x,y) \tag{2}\]3  
Because \(F_{l}(u,v)\)s are delta functions, in theory, their distributions are more like Gaussian functions. Then the tilt angles can be calculated more accurately by Gaussian fitting. Figure 2 shows the flowchart of the RTA evaluation method. To describe the detailed measures of LGF, the specific steps are included in the following:  
Step 1: Carry out fast FT on the intensity distribution \(I\) in Equation (5) and detect the frequency peaks on the SSP in Equation (6) by searching the point with the maximum value with the value depressed at the original point. Step 2: Mark the points detected as \((-u_{0},\)\(-v_{0})\) and \((u_{0},\)\(v_{0})\), and keep the number order of these points \((m_{0},\)\(n_{0})\). Step 3: Choose a small area with certain FPs around \((-u_{0},\)\(-v_{0})\) or \((u_{0},\)\(v_{0})\) and store the values of these spectrum pixels in this small area. Divide the little area and mesh it into a new grid, so the computational grid size becomes smaller. Correspondently, \(\Delta u\) and \(\Delta v\) are represented as \(\Delta u_{l}\) and \(\Delta v_{l}\), respectively, in this small area. Step 4: Fit the spectra values in the small piece by the Gaussian formula in Equation (10) and the known values stored in step 3. Calculate the new spectrum coordinates \((-u_{0l},\)\(-v_{0l})\) or \((u_{0l},\)\(v_{0l})\) of the peak tip in the more refined grid by Equation (11). Step 5: Calculate the RTA by Equation (13) with the coordinates \((u_{0l},\)\(v_{0l})\) found in step 4. The steps in Figure 2 can also be understood by referring to the description above and the output of it represents RTA values improved.