# Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography
## 3 Tilt Phase Error Compensation
But for the interference fringe with a higher density, the application of this method would be restricted. In a word, the digital reference wave method and the spectrum translation method are the two most common approaches to compensate the tilt phase aberration, in which they need not obtain the mathematic model of phase distortion in advance [5; 22] and they avoid the problems of inaccurate fitting results due to the surface profile [23; 60]. However, the two methods first require the hologram spectrum's centering judgment of side band. One obtains the tilt parameters by means of spectrum centering, another uses the spectrum centering as the basis for its translation. Hence, the correct hologram spectrum's centering judgment of side band plays a critical role in the two approaches. Its realization is mainly to select the spectrum's amplitude maximum [16; 23; 26; 57; 59]. In effect, it is difficult to determine the specimen spectrum's center location of side band simply by the amplitude maximum in the pre-amplification configuration [23; 26; 59]. Figure 2 gives the procedure of spectrum centering in the spectrum translation method. The correct spectrum centering, labeled by the white cross, was obtained by the filtered spectrum's amplitude maximum, as shown in Figure 2a. The arrow represents the shift between the amplitude maximum of the frequencies associated with the virtual image and the center of the entire spectrum. The tilt phase error can be suppressed by translating the filtered spectrum in the frequency domain, as shown in Figure 2b. Due to the presence of the quadratic phase aberration induced by the MO, the amplitude distribution of the spectrum appeared to spread, as shown in Figure 2c. The nonpunctual central frequency resulted in the failure to judge the spectrum center, which differed from the judging result in Figure 2a. However, it was pointed out by [PERSON] et al. [23] that the judgment of the spectrum's amplitude maximum was adversely affected by the shifting accuracy and the central frequency spreading. They did not provide the solution to obtain the correct spectrum's centering of side band in pre-amplification configuration. In order to obtain the correct spectrum centering, some scholars have developed the research on the carrier frequency elimination methods. In 2010, [PERSON] et al. [61] analyzed that if the carrier frequencies were not an integer multiple of the frequency interval, the peak coordinates of \(C\) and \(C^{*}\) in Equation (13) would not be equal to the carrier frequencies. Aimed at the problem, a carrier frequency judgment method by means of a spectral centroid was presented to suppress the carrier removal error. The method was applied for the case that the carrier frequencies were not an integer  
Figure 2: Procedure of spectrum centering in the spectrum translation method. (**a**) Initial filtered spectrum; (**b**) Spectrum centered; (**c**) Spectrum of a hologram for which the curvatures of the reference and object waves were different, inducing a nonpunctual central frequency in the spectrum. (Reproduced with permission from [23], Copyright OSA publishing, 2006). multiple of the frequency interval. Yet for the asymmetric spectrum, the spectral centroid judgment shows randomness and instability because the carrier frequencies estimated are affected by the noise distribution in the fringe pattern. The argument judgment method, proposed by [PERSON] et al. in 2011 [PERSON] et al. (2011), selects the argument maximum of the hologram spectrum as the spectrum center to remove the tilt aberration. It was found from the experimental results that the argument maximum corresponded to the carrier frequencies, as shown in Figure 3. Figure 3b-d are the modulus image, argument image, and the three-dimensional argument distribution of the hologram spectrum, respectively. Fortunately, there always existed a maximum on the argument distribution, as shown in Figure 3c,d. Selecting it as the center, the filtered hologram was translated down to the origin of coordinates.