# Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography
## 3 Tilt Phase Error Compensation  
The angle between the object and reference wave brings about the tilt phase error, though the off-axis configuration separates the zero-order and twin-images spatially, which will result in the appearance of tilt in the topography of the reconstructed image. In recent years, many scholars have conducted research on the compensation of the tilt phase error to improve the imaging quality. In 1999, [PERSON] et al. [6] proposed a digital reference wave method. The new hologram \(I_{H}\)' only includes original image information and a reference wave's conjugate after frequency filtering, and it multiplies by a digital reference wave that is the same as a physical reference wave to eliminate the tilt phase error. The method has been widely applied [5; 23; 56; 57]. Tilt phase error compensation was discussed initially in interferometry. In 1982, [PERSON] et al. [58] proposed the Fourier transform method, in which the input fringe pattern was represented by:  
\[ \begin{array}{c}g(x,y)=a(x,y)+c(x,y)\exp\left[j\big{(}2\pi xf_{x0}+2\pi yf_{ y0}\big{)}\right]\\ +c^{*}(x,y)\exp\left[-j\big{(}2\pi xf_{x0}+2\pi yf_{y0}\big{)}\right],\end{array}  \tag{11}\]  
where \(a(x,y)\) was the background intensity, \(b(x,y)\) was the visibility of the fringes,\(f_{x0}\) and \(f_{y0}\) were the spatial carrier frequencies in the \(x\) and \(y\) directions, respectively, \(\varphi(x,y)\) was the modulating phase, and \(c(x,y)\) was given as:  
\[c(x,y)=\frac{1}{2}b(x,y)\exp[j\phi(x,y)]. \tag{12}\]  
The Fourier transform of Equation (11) was given as:  
\[G(f_{x},f_{y})=A\big{(}f_{x},f_{y}\big{)}+C\big{(}f_{x}-f_{x0},f_{y}-f_{y0} \big{)}+C^{*}\big{(}f_{x}+f_{x0},f_{y}+f_{y0}\big{)}, \tag{13}\]  
where \(A\big{(}f_{x},f_{y}\big{)}\), \(C\big{(}f_{x}-f_{x0}f_{y}-f_{y0}\big{)}\) and \(C^{*}\big{(}f_{x}+f_{x0}f_{y}+f_{y0}\big{)}\) were the Fourier transforms of \(a(x,y)\), \(c(x,y)\) and \(c^{*}(x,y)\), respectively. Since the spatial variations of \(a(x,y)\), \(b(x,y)\) and \(\varphi(x,y)\) were slow compared with the spatial carrier frequency \(f_{x0}\) and \(f_{y0}\), the Fourier spectra in Equation (13) were separated by the carrier frequency \(f_{x0}\) and \(f_{y0}\). Either of the two spectra on the carrier was translated by \(f_{0}\) on the frequency axis toward the origin to realize the tilt phase error elimination. After that, [PERSON] et al. [16] utilized the spectrum translation method around year of 2000. The method corresponds to the multiplication of a plane wave in the digital reference wave method [23; 26; 59]. In 2003, [PERSON] et al. [22] removed the tilt phase error by means of the phase mask. Compared with other methods, the proposed method first found the correcting wave front in the hologram plane, not directly at the reconstructed plane. The reconstructed phases, relating to the recorded hologram and correcting wave front, were subtracted to obtain the compensated contrast-phase image, in which the correcting wave for tilt error compensation was given by the adjustment of the mask's parameters. The surface fitting method was presented by [PERSON] et al. [23; 60] in 2006 to compensate phase error. A model of 2D standard or Zernike polynomials was selected to fit a blank region of the reconstructed image, and then a surface fitting could obtain their parameters, thus realizing the phase error compensation. However, the fitted blank region does not include the total phase information, which would result in the residual error. In 2009, [PERSON] et al. [56] presented a numerical procedure of a tilt phase aberration compensation by interference patterns in in-line configuration. Because the tilt aberration had no effect on the phase of the sample, the parameters of the tilt phase may be extracted from the period of interference patterns in the reconstructed image. The corrected phase was computed and the real phase was recovered. The method uses only one hologram to minimize the tilt aberration in in-line DH.