# Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography
## 4 Phase Aberration Compensation
Figure 5: Schematic of the Digital holographic microscopy (DHM) configuration where the same microscope objectives (MOs) are inserted in two branches. Figure 6: Schematic of the DHM telecentric arrangement. In 2006, [PERSON] et al. [60] adopted the phase mask to remove the phase distortion. Compared to the method presented by [PERSON] [22], the correcting parameters were estimated by directly fitting the blank region of unwrapped phase. The polynomial formulation at the reconstructed plane was introduced as:  
\[G(x,y)=\exp\left[-j\sum_{h=0}^{H}\sum_{v=0}^{V}P_{h,v}x^{h}y^{v}\right], \tag{20}\]  
where \(H\) and \(V\) defined the polynomial orders in the horizontal and vertical directions, respectively; \(P_{h,v}\) defined a set of reconstruction parameters, in which \(P_{0,0}\) represented the constant term, \(P_{1,0}\) and \(P_{0,1}\) represented the first-order term with respect to the different axis, \(P_{2,0}\) represented the quadratic term. The different physical quantities can be evaluated by the different order polynomials if necessary. The reconstructed image which is immune to the phase distortion became:  
\[U^{\prime}(x,y)=G(x,y)\cdot U(x,y), \tag{21}\]  
where \(U(x,y)\) was the reconstructed image before compensation. The numerical phase mask method must give the correcting model in advance. Its application is also restricted for the sample with high-spatial-frequency content due to fitting an area in absence of a sample. The surface fitting can obtain the phase distribution of a reconstructed image by means of a polynomial fitting. The fitted phase is subtracted from the reconstructed phase to eliminate the phase distortion. The approach may be divided into two situations. The first one is to fit a blank area in proximity to the sample, and further obtain the fitted phase distribution of the total reconstructed phase. The method requires that the total phase distortion is identical in a reconstructed plane. As examples, [PERSON] et al. adopted the high-order mathematical model [60] and [PERSON] the polynomial fitting [23] in 2006, while [PERSON] et al. [70] employed the least square ellipsoidal model in 2012. These fitting methods were all completed under the certain condition of a sample-free area in the reconstruction plane. The second of the two situations is to fit the total reconstructed phase, in which the thin sample is considered to be a modulation superimposed on the aberration distribution. The fitting of the total unwrapped phase is seen as the fitting for the phase distortion distribution, instead of a sample. It does not require a sample-free region to execute. For example, [PERSON] et al. [71] applied Zernike polynomials to fit in 2007, and [PERSON] et al. [69] carried out the surface fitting based on the second-order mathematical model of phase distortion in 2011. However, their fitting results were frequently affected by the sample's topographic distribution. The fitted phase distribution \(\phi_{fit}(x,y)\) may be obtained according to the methods mentioned above. It is subtracted from the reconstructed phase \(\phi_{rec}(x,y)\) to recover the sample phase free of aberration, that is:  
\[\xi_{2}(x,y)=\exp\biggl{[}-\frac{jk}{2f_{1}M_{0}}\biggl{(}x^{2}+y^{2}\biggr{)} \biggr{]}. \tag{19}\]0  
In 2014, [PERSON] et al. [72] proposed the total aberrations compensation method based on the Zernike surface fitting in the telecentric arrangement of DHM. In order to compensate the quadratic phase aberration, the telecentric arrangement demanded precise adjustment of the distance between the MO and the collimated lens. The adjustment was realized by way of observation, but the analyses found that observing the interference fringes' shape [28] or the Fourier spectrum's distribution of hologram [25; 26; 27; 28] may fail, and results in the residual phase curvature [25; 73]. Aiming at the situation, a posteriori numerical compensation based on the Zernike surface fitting was performed.