# Robust Evaluation of Reference Tilt in Digital Holography
## 1 Introduction  
Spectrum analysis in the Fourier domain [1; 2; 3] provides an alternative way to extract some important optical information, which is not available in the space domain. The Fourier domain spectrum then plays an important role either in improving the resolution in super-resolution imaging [4; 5; 6] or in solving the parameters necessary for object wave recovery in digital holography (DH) [7; 8; 9; 10; 11; 12]. Because the imaging resolution of an optical system is constrained by the cut-off frequency of the imaging system, researchers provide several kinds of methods to break through the resolution limitation. Optical devices are inserted in the experiment setup to collect the high-frequency spectrum in digital holography so that the imaging resolution can be improved. For example, [PERSON] et al. [13] introduced a grating between the test sample and the charge coupled device (CCD), and then the corresponding high-frequency component can be projected onto the small CCD chip. This method is, in fact, a kind of multiplexing technique since the high-frequency spectrum is reused in the holography imaging system. The imaging resolution improves with receiving these collected high-frequency spectra. However, using gratings to ensure the acceptance of more spectra increases the cost and operation complexity of the experiment. Structured illuminating light [4] is also used to break through the resolution bottleneck of the coherent imaging system. Nine exposures are necessary to gain more high-frequency spectra across three imaging directions. This multi-frame recording method consumes more time and only suits static imaging with super-resolution in biomedical research. A spatial-temporal light sheet [14; 15] is then used in biomedical imaging microscopy to improve image quality. In both off-axis DH (OADH) and in-line PSDH [16; 17; 18; 19] applications, the Fourier domain spectrum is also beneficial to separate the needed information for multiplexing techniques [20; 21], especially for the detection of RTA. RTA detection is significant to eliminate the negative effect of the reference tilt on the imaging quality in in-line PSDH [22] and to remove the carrier wave in the object waves reconstructed in OADH [23]. Recently, tilt reference has been suggested in OADH and PSDHwhere reference tilt information is recorded on a hologram [24] or extra fringes [25,26]. In all these methods, spectrum detection and operation are inevitably needed. During the process of the corresponding spectrum detection, the precision is limited by the size of the frequency pixels (FPs) on the SSP, which is always determined by the parameters of the recording device CCD. The CCD performance has been impeded by the slow development of modern workmanship. The discrete pixel of the recording device leads to corresponding discrete spectrum distribution in the Fourier frequency domain, and naturally, the precision of the spectrum location is limited by the size of FP on the SSP. Many researchers have attempted to improve the accuracy of the spectrum locating by decreasing the pixel size in the Fourier domain so that the data array on the SSP is enlarged in the Fourier domain [27, 28, 29]. Whereas most of these spectrum location-finding methods reported impose heavy loads on the computers, the efficiency is low in improving the precision. Here a robust method is provided to detect the RTA in DH with high precision, efficiency, and stability. By introducing local Gaussian fitting (LGF) [30], the specific equation of the spectrum distribution is established in the computer, and then the frequency peak location is fixed by it for RTA calculation. There are many merits for this frequency spectrum fitting method. Firstly, this method has the capability to detect the RTA value accurately because the fitted function predicts the analytic solution of the spectrum distribution by theory analysis and analytical simulation.