# Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography

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^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Principle of DHM

_{0},y

_{0}) to the CCD output plane (x,y), through the MO plane (x’,y’).The quadratic phase factor introduced by MO is expressed as [54]:

_{1}is the focal length of MO. The wave field propagation process is described based on the Fresnel diffraction. The object field O

_{0}(x

_{0},y

_{0}) goes through two Fresnel transformations, from input plane to MO over the distance d

_{1}, further to the output plane over the distance d

_{2}. The output field O(x,y) is derived as [55]:

_{i},y

_{i}) should be obtained by Fresnel transform from the CCD plane to the image plane over the distance z

_{i}, then it is expressed as:

## 3. Tilt Phase Error Compensation

_{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].

_{x}

_{0}and f

_{y}

_{0}were the spatial carrier frequencies in the x and y directions, respectively, φ(x,y) was the modulating phase, and c(x,y) was given as:

_{x},f

_{y}), C(f

_{x}− f

_{x}

_{0},f

_{y}− f

_{y}

_{0}) and C*(f

_{x}+ f

_{x}

_{0},f

_{y}+ f

_{y}

_{0}) 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 φ(x,y) were slow compared with the spatial carrier frequency f

_{x}

_{0}and f

_{y}

_{0}, the Fourier spectra in Equation (13) were separated by the carrier frequency f

_{x}

_{0}and f

_{y}

_{0}. 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, Cuche 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].

## 4. Phase Aberration Compensation

_{0}is lateral magnification, i.e.,

_{0}is written as:

_{2}in the reference branch as the MO

_{1}in the object branch in 2005, in which the MO

_{2}’s location was adjusted to offset the phase curvature in the object branch, as shown in Figure 5. The distance of L

_{0}+ L

_{1}was equal to the distance of L

_{0}+ L

_{2}as adjustable as possible to ensure the matching of the spherical waves in two branches. In addition, Qu et al. [25,26,29] further carried out the research on the Michelson interference structure. They observed the spectrum distribution of a hologram to judge whether or not to remove the quadratic phase distortion. The method demands that only the distances between the MO and the CCD in the two branches are controlled precisely. The phase curvature can be eliminated.

_{2}and a smaller numerical aperture was located behind the MO. Its front focus plane coincided with the MO’s rear focus plane, resulting in an afocal arrangement. The interference happened between two plane waves, thus compensating the quadratic phase distortion. When the object was located at the MO’s front focus plane, the image would be produced in the rear focus plane of the collimated lens. The lateral magnification was estimated by the ratio of the lens’s focal length f

_{2}to the MO’s focal length f

_{1}. The depth of the grooves of the zone plate was measured by the telecentric DHM. It was found that it agreed with the results obtained by profilometer. The method not only compensates the quadratic phase distortion, but enlarges the available region of spatial frequency in the off-axis configuration. In 2013, the same team [30] continued to analyze the telecentric arrangement in-depth. They demonstrated that the afocal arrangement possesses the shift-invariant imaging property, and the optical arrangement can remove the quadratic phase aberration.

_{i}, respectively. The two reconstructed phases were subtracted to obtain the sample information.

_{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:

_{fit}(x,y) may be obtained according to the methods mentioned above. It is subtracted from the reconstructed phase ϕ

_{rec}(x,y) to recover the sample phase free of aberration, that is:

_{x}and l

_{y}were the parameters to describe the spherical phase curvature. Its conjugate multiplied with the filtered hologram, leads to a new aberration-free hologram. The experiments of two cells were performed, in which the subcellular features as well as the thin borders of the cells were clearly observed without any curved or tilted background. Despite this, the approach improves the compensation efficiency by transferring 2D phase unwrapping and 2D surface fitting into a 1D procedure on two orthogonal directions. The PCA itself is the time-consuming task due to its processing time. To ensure efficiency, an optimal PCA-based method, proposed by Sun et al. [75] in 2016, only extracted the reduced-sized aberration spectrum to compensate the phase aberration, thus improving the computational efficiency. In 2017, Nguyen et al. [76] proposed an automatic phase aberration compensation method that combines a supervised deep learning technique with a convolutional neural network (CNN) and the Zernike polynomial fitting. The deep learning CNN was implemented to detect the background region automatically, which resolved the problem of aberration and the speckle noise that existed in other segmentation techniques. In addition, the Zernike polynomial was allowed to compute the self-conjugated phase to compensate for most aberrations. The approach can perform the real-time and automatic phase aberration compensation in DHM.

## 5. Coherent Noise Suppression

_{0},y

_{0}) and N(x

_{0},y

_{0}) represent the complex amplitude of the specimen and coherent noise, respectively; A is the amplitude and φ is the phase.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**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).

**Figure 3.**Principle of argument judgment method. (

**a**) Hologram of the cervical carcinoma cells; (

**b**) Modulus image; (

**c**) Argument image;(

**d**) Three-dimensional argument distribution of the hologram spectra. (Reproduced with permission from [59], Copyright Elsevier publishing, 2011).

**Figure 4.**Principles of the shift differential method. (

**a**) Region selection for conventional background subtraction method and shift differential method; (

**b**) Phase image of a sample region; (

**c**) Aberration image obtained from the background phase image; (

**d**) Improved phase image, which is the subtraction of the aberration image from the phase image; (

**e**) Horizontal shift differential phase image; (

**f**) Vertical shift differential phase image; (

**g**) Phase image retrieved from the spiral phase integration of differential phase images. (Reproduced with permission from [65], Copyright OSA publishing, 2017).

**Figure 5.**Schematic of the Digital holographic microscopy (DHM) configuration where the same microscope objectives (MOs) are inserted in two branches.

**Figure 7.**Schematic of the generation of sub-holograms in the single-shot speckle reduction approach. (Reproduced with permission from [114], Copyright OSA publishing, 2015).

**Figure 8.**Results of the single-shot speckle reduction approach. (

**a**) S = 1 sub-hologram; (

**b**) S = 4 sub-holograms; (

**c**) S = 9 sub-holograms; (

**d**) S = 16 sub-holograms. (Reproduced with permission from [114], Copyright OSA publishing, 2015).

Method | Advantage | Limitation |
---|---|---|

Double exposure method [22,63,65] | Compensating aberrations in optical components | Recording two holograms |

Two MOs compensation method in two branches [25,26,29,64] | Compensating quadratic phase aberration | Precisely adjusting the distance between MO and CCD |

Compensation method in telecentric arrangement [28,30] | Compensating quadratic phase aberration | precise adjustment of the distance between MO and collimated lens |

Phase mask method [22,26,60] | Compensating aberrations set by mask | Requiring the sample-free blank region |

Surface fitting method for a blank area [23,60,70] | Compensating aberrations set by polynomials | Requiring identical distortion in the total phase |

Surface fitting method for a total reconstructed phase [69,71,72] | Compensating aberrations set by polynomials | Affection by sample’s topographic distribution |

Reference conjugated hologram method [66,67,68] | Compensating aberrations set by polynomials | Requiring the sample-free blank region |

Method | Advantage | Limitation |
---|---|---|

Low spatial or temporal coherent source method [77,78,79,80,81,82,83,84,85,86,87,88,89,90] | Recording one hologram | Increasing the adjustment difficulty of light configuration |

OSH method [91,92,93,94] | Higher signal-to-noise ratio | Complicated setup and slower recording speed |

FINCH method [95,96,97,98] | Higher signal-to-noise ratio and dynamic measurement | Larger bias buildup for complicated objects |

Multiplexing holograms method [33,35,100,101,102,103,104,105,106,107] | Uncorrelated holograms obtained by different ways | Recording multiple holograms |

Digital processing method [31,108,109,110,111,112,113,114,115,116] | Recording one hologram | Loss of spatial resolution |

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**MDPI and ACS Style**

Liu, Y.; Wang, Z.; Huang, J.
Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography. *Appl. Sci.* **2018**, *8*, 444.
https://doi.org/10.3390/app8030444

**AMA Style**

Liu Y, Wang Z, Huang J.
Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography. *Applied Sciences*. 2018; 8(3):444.
https://doi.org/10.3390/app8030444

**Chicago/Turabian Style**

Liu, Yun, Zhao Wang, and Junhui Huang.
2018. "Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography" *Applied Sciences* 8, no. 3: 444.
https://doi.org/10.3390/app8030444