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Abstract
We propose a numerical and automatic quadratic phase aberration elimination method in digital holography for phase-contrast imaging. A histogram segmentation method based on Gaussian 1σ-criterion is used to obtain the accurate coefficients of quadratic aberrations using the weighted least-squares algorithm. This method needs no manual intervention for specimen-free zone or prior parameters of optical components. We also propose a maximum-minimum-average-standard deviation (MMASD) metric to quantitatively evaluate the effectiveness of quadratic aberration elimination. Simulation and experimental results are demonstrated to verify the efficacy of our proposed method over the traditional least-squares algorithm.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Digital holography (DH) allows full-field and non-destructive imaging, it relies on the interference pattern called the digital hologram produced by a reference beam and an object beam. A digital hologram can record the amplitude and phase information of an object, and then both of the information can be retrieved numerically. In DH, the digital hologram records not only the specimen phase information but also quadratic or high-order phase aberrations in off-axis optical setup [1,2], which affects the accurate measurement of the object’s true thickness/height. For example, the angle between the object beam and the reference beam adds a linear phase ramp into the retrieved phase. A quadratic phase will be included if the laser beam is not well collimated after being expanded into a divergent spherical wave. Also, the optical paths of the object beam and the reference beam are not usually the same, thereby introducing some additional phase shift. In addition, the microscope objective in general introduces aberrations, if there is no matching telecentric optical components for compensation [3]. Therefore, quadratic phase aberrations or high-order phase aberrations will be inevitably introduced in digital holographic systems.
Various approaches have been proposed to eliminate the quadratic phase aberrations in DH, which could be categorized into two groups: physical methods and numerical methods. In the physical methods, practical solutions include introducing additional microscope objectives (MO) in the optical path [4,5], such as the use of a telecentric lens structure [6–8], or the insertion of an electrically tunable/adjustable lens in the illumination path or in the reference optical arm [9,10].
The main drawback of the physical methods is the involvement of a complicated recording setup and difficult experimental procedures. Numerical compensation methods could also correct the aberrations without any additional physical devices [11]. Typically, the phase separation techniques and the polynomial coefficients optimization techniques are used for this purpose. The phase separation technique is based on the assumption that the object's phase and quadratic phase aberrations are independent. Quadratic term phase errors can be eliminated using double exposure and phase shift [12,13], and the tilt in the specified direction can be eliminated by rotating the hologram [14]. In particular, based on the mutual independence of the object phase and quadratic phase aberrations, the use of lateral shear, transport of equation (TIE) or spectral analysis, can be used to eliminate the parabolic or central spherical aberrations [15–17].
A polynomial coefficient optimization technique commonly selects a region without the object as the starting point to iteratively improve the accuracy of phase reconstruction. Based on the initial phase map, the object-free region is selected and the curvature and spherical center of the quadratic phase are calculated according to the distance of the optical path components [18–24]. A least-squares algorithm is typically used to resolve the quadratic term coefficients [25]. However, it may introduce the overfitting problem and limits its potential uses. Based on the constrained matrix inversion and Zernike polynomials, the quadratic aberrations coefficients can also be obtained by using the ${l_1}$-norm or ${l_2}$-norm [11,26–29]. By determining the matrix sampling area, weighted Zernike polynomials fitting method has been used to improve the accuracy of aberration compensation [30,31]. Recently, deep learning compensation based on convolutional neural networks has also achieved great success, but the method is known to be computationally heavy [32].
We propose a histogram segmentation method based on the Gaussian $1\sigma $-criterion to obtain accurate coefficients of the quadratic aberration using the weighted least-squares algorithm. We also propose a metric called the maximum-minimum-average-standard deviation (MMASD) metric to evaluate the effectiveness of quadratic aberration elimination. Experimental results indicate that the MMASD evaluation metric can accurately describe the flatness of the object phase. The proposed method is inspired by the advantages of both polynomial coefficient optimization and phase separation techniques [33,34]. Our method takes into account the influence of the position, shape, size and height of the object on the solution of the quadratic coefficients. Compared to the traditional least-squares algorithm [35,36], our method can also achieve a high accuracy of phase reconstruction and effectively eliminate the quadratic phase errors.
2. Segmentation method
2.1 Overfitted phase by using traditional least-squares algorithm
For DH phase reconstruction, the wrapped phase map is obtained by Fresnel diffraction reconstruction or angular spectrum reconstruction. We express the reconstructed unwrapped total phase ${\phi _{tot}}(x,y)$ as
where ${\phi _{obj}}(x,y)$, ${\phi _{abe}}(x,y)$ and ${\phi _{noi}}(x,y)$ represent the error-free object phase (or sample phase), quadratic phase aberrations and phase noise, respectively. Figures 1(a)–1(c) also show the profile features of ${\phi _{tot}}(x,y)$, ${\phi _{obj}}(x,y)$ and ${\phi _{abe}}(x,y)$, respectively from the traditional least-squares algorithm. Since the phase noise ${\phi _{noi}}(x,y)$ is small enough in magnitude and it randomly fluctuates around 0, it has little effect on the calculation results based on the phase integration operation in the matrix inverse transport sequence process. As mentioned earlier, least-squares algorithm is typically used to eliminate the quadratic phase aberration ${\phi _{abe}}(x,y)$. However, an overfitting makes it impossible to obtain accurate error-free phase ${\phi _{obj}}(x,y)$[26]. Figure 1(d) illustrates the effect of overfitting from traditional least-squares algorithm and we term the incorrectly compensated phase as the roughly corrected phase ${\phi _{ls}}(x,y)$. Compared to the phase distributions in Fig. 1(b), the phase background (or base-phase) in Fig. 1(d) has an obvious bending, especially around the edges of the map.
Fig. 1. (a) Reconstructed unwrapped total phase; (b) true object phase; (c) quadratic phase aberration; (d) roughly corrected phase from traditional least-squares algorithm.
We can characterize the differences of the phases shown in Figs. 1(b) and 1(d) in terms of their histograms and Gaussian distributions (HGDs). We show the HDG plots ${\phi _{obj}}(x,y)$ and ${\phi _{ls}}(x,y)$ in Figs. 2(a) and 2(b), respectively. The horizontal coordinate represents the phase value in radians. The vertical coordinate represents the number of pixels. From the observed data, we employ the technique used by Guo [31] and plot the Gaussian distributions for ${\phi _{obj}}(x,y)$ and ${\phi _{ls}}(x,y)$. The Gaussian curves are shown as red curves on the histogram plots, and for this reason we call Figs. 2(a) and 2(b) as the HGD plots. $\mu $ and $\sigma $ are the average value and standard deviation of the Gaussian function.
Fig. 2. The HGD comparison plots. (a) HGD plot of the object phase; (b) HGD plot of the phase result from traditional least-squares algorithm.
From the HGDs in Figs. 2(a) and 2(b), we see that ${\phi _{obj}}(x,y)$ with a flat base-phase presents a symmetrical distribution around the zero phase. The uneven base-phase introduced by overfitting makes the histogram of ${\phi _{ls}}(x,y)$ not symmetrical but skew toward to the left indicated by the arrow.
In this paper, we propose a weighted least-squares algorithm based on $1\sigma $-criterion with the MMASD evaluation metric (to be defined in the next section) to suppress overfitting, and the “leftward shifting” in the histogram is alleviated by using a proper weighted matrix $\omega (x,y)$[33]. The crucial task is the detection and separation of the phase base and the sample phase.
2.2 Weighted least-squares based on $1\sigma $-criterion
We assume that the quadratic phase aberrations ${\phi _{abe}}(x,y)$ can be expressed as
where $x = {x_1},{x_2},\ldots ,{x_N};y = {y_1},{y_2},\ldots ,{y_M};N,M \in {\mathbb Z}$.Equation (2) expresses an over-determined equation [25]. The overdetermined system in the least-squares algorithm [29] can be expressed as
We let ${\boldsymbol S} = \left[ {\begin{array}{cccccc} 1&x&y&{xy}&{{x^2}}&{{y^2}} \end{array}} \right]$ represent the matrix of the polynomials in Eq. (2), and define the fitting coefficients ${\boldsymbol U} = {\left[ {\begin{array}{cccccc} {{u_1}}&{{u_2}}&{{u_3}}&{{u_4}}&{{u_5}}&{{u_6}} \end{array}} \right]^T}$, where the superscript T represents the transpose of a matrix or vector. In order to obtain the fitting coefficients ${\boldsymbol U}$, a matrix inverse operation for Eq. (3), as deduced in Ref. [31], is calculated as follows:
Equation (4) shows that overfitting is introduced because all of the phase values including the sample phase are used to fit by the traditional least-squares algorithm. The sample phase should have been removed as they represent invalid fitted data. Hence, we will apply a weighted least-squares algorithm based on $1\sigma $-criterion with the MMASD evaluation metric to address the overfitting issue by removing the sample phase values.
We define a MMASD evaluation metric as:
According to Wang et al. [34], in Eq. (5), the term max{$\phi (x,y)$}-min{$\phi (x,y)$} decreases first and then increases in the process of eliminating the quadratic aberrations. The change of $\mu \{{\phi (x,y)} \}+ \sigma \{{\phi (x,y)} \}$ represents the tendency of the histogram to shift sideways in the HGD. It is clear that flatness of the overfitted phase with a large bending edge should be different from that of the error-free object phase. That means MMASD evaluation metric also indicates a flatness of phase background. For example, phase ${\phi _{ls}}(x,y)$ consists of an uneven phase background and a sample-phase, where the MMASD evaluation metric of phase background is much lower than that of the sample-phase. Therefore, if a proper grid is applied [30], the sample-phase can be detected and separated from the phase background by the MMASD evaluation metric of each sub-region controlled by the grid size.
We set a grid phase ${\phi _{grid}}(x,y)$ of the roughly corrected phase ${\phi _{ls}}(x,y)$ based on the MMASD metric as
Thus, putting ${\phi _{grid}}(x,y)$ into Eq. (4) yields the fitted coefficients ${{\boldsymbol U}_{grid}}$, and the pre-compensated phase ${\phi _{pc}}(x,y)$ is obtained as follows:
where ${\boldsymbol S} \cdot {{\boldsymbol U}_{grid}}$ represents the fitted phase constituted by ${{\boldsymbol U}_{grid}}$.Then, for Eq. (7), the threshold segmentation method is applied to obtain the weighted matrix $\omega (x,y)$ based on the $1\sigma $-criterion as follows:
By introducing the weighted least-squares algorithm, we rewrite Eq. (4) as
We let $\phi ({x,y} )= {\phi _{tot}}(x,y)$ in Eq. (9) to obtain the vector of quadratic estimation coefficients ${{\boldsymbol U}_\omega }$. Since we apply the dot product operation to ${\phi _{tot}}(x,y)$, the zero values in the $\omega (x,y)$ matrix will make those invalid phases zero, thereby suppressing the overfitting. Thus, the fitted surface formed by ${\boldsymbol S} \cdot {{\boldsymbol U}_\omega }$ approximates the true aberrations ${\phi _{abe}}(x,y)$. Finally, the corrected phase is
The effectiveness of the proposed method is demonstrated by simulations and experimental work in Section 3 and Section 4, respectively.
3. Simulation and evaluation
The USAF 1951 resolution plate with resolution of 2048 × 2048 pixels is used as a test sample. The phase plate was placed on a flat plane (it is the base-phase, being set to 0 rad), and the sample phase is set to be a complex structure consisting of inclined, parabolic and stepped surfaces. Based on the simulation study of noise and window size by Liu et al [37], Gaussian noise with zero average value ($\mu = 0$) and 0.02 rad standard deviation ($\sigma = 0.02$) is added to the sample as shown in Fig. 3(a). Due to the complex morphology of the object phase, we also set ${\rho _x} = {\rho _y} = 10$ for grid phase distribution to ensure that the grid phase can describe the characteristic of the object phase as much as possible. According to the most practically used quadratic term phase distribution, the quadratic aberration coefficients are set to be ${{\boldsymbol U}_{abe}}$ = [-3.419, 2·10−3, 3.3·10−3, -8·10−7, -1·10−6, -1.6·10−6].
Fig. 3. Flow chart of the proposed method. (a) Simulated object phase map; (b) simulated quadratic aberration phase map; (c) synthetic total phase map; (d) phase map after traditional least-squares algorithm; (e) the grid phase; (f) phase after pre-compensation; (g) the distribution of weighted matrix with $\omega (x,y)$ equal to 1; (h) fitted surface calculated by the proposed method; (i) corrected phase obtained from proposed method.
All simulation results are shown in Figs. 3(b)–3(i). The fitted quadratic phase aberration ${\phi _{abe}}(x,y)$ is shown in Fig. 3(b). The sum of the object phase and the aberration phase ${\phi _{tot}}(x,y)$ is shown in Fig. 3(c). The roughly corrected phase ${\phi _{ls}}(x,y)$ then is obtained using traditional least-squares algorithm, as shown in Fig. 3(d), in which an obvious distortion at the edge location is evident. By using the proposed MMASD evaluation metric, the grid phase ${\phi _{grid}}(x,y)$ can be calculated and is shown in Fig. 3(e). Figure 3(f) provides the pre-compensated phase ${\phi _{pc}}(x,y)$ obtained by the fitted coefficients ${{\boldsymbol U}_{grid}}$. Figure 3(g) shows that the distribution of all the weighted values equal to 1 in the weighting matrix. The fitted surface formed by ${\boldsymbol S} \cdot {{\boldsymbol U}_\omega }$ approximates the true aberrations ${\phi _{abe}}(x,y)$, as shown in Fig. 3(h). Clearly we can see that the proposed algorithm can extract the base phase as Fig. 3(h) is “identical” to Fig. 3(b) visually, indicating that the true quadratic aberration can accurately be fitted by the proposed method. The final corrected phase is obtained as shown in Fig. 3(i).
Figure 4 presents the HGD plot of all phase. Figure 4(a) is the original HGD by calculated the phase of the test sample. From the HGD plot in Fig. 4(b), it is observed that there is an obvious leftward shift as marked by the blue arrow in the phase ${\phi _{ls}}(x,y)$. The HGD plot of the pre-compensation ${\phi _{pc}}(x,y)$ is shown in Fig. 4(c), it is found that the pre-compensation rightward shifts, and the peak of the histogram are concentrated around the mean value of ${\phi _{pc}}(x,y)$, verifying the effect of pre-compensation. Figure 4(d) shows the HGD plot of the finally corrected phase ${\phi _\omega }(x,y)$, which is being almost the same as that in Fig. 4(a). In order to demonstrate the superiority of the proposed method, we also take the phase difference along the midline from Fig. 3(a), Fig. 3(d) and Fig. 3(i), respectively, as marked by red dashed line. The results shown in Fig. 4(e) indicate the proposed method is able to improve the accuracy by 3 orders of magnitude.
Fig. 4. Shifting in HGD plots. (a), (b), (c) and (d) are the HGD from Fig. 3(a), Fig. 3(d), Fig. 3(f) and Fig. 3(i), respectively; (e) comparison of phase differences between ${\phi _{ls}}(x,y) - {\phi _{obj}}(x,y)$ and ${\phi _\omega }(x,y) - {\phi _{obj}}(x,y)$.
Fitted coefficients are shown in Table 1. We compare the relative errors from coefficients ${{\boldsymbol U}_{tot}}$ and ${{\boldsymbol U}_\omega }$, the fourth column and sixth column, and it can be seen that ${{\boldsymbol U}_\omega }$ improves by 2 orders of magnitude. Meanwhile, comparing to the traditional least-squares algorithm, the proposed method improves by 3 orders of magnitude in the RMSE (root mean square error) values, and that means the proposed method achieves higher accuracy. This is consistent with the simulation results shown in Fig. 3 and Fig. 4.
Table 1. Comparison of the simulated aberrations elimination results and evaluations between the least-squares algorithm and the proposed method
4. Experimental studies
The transmission and reflection USAF1951 resolution plates are used for experiments. Here, the holographic system is used to detect element 3 and element 4 of group 0, as shown in Fig. 5(a). The holographic microscope is also used to detect group 6 and group 7 with the linewidths ranging from 2.19µm to 7.81µm, as shown in Fig. 5(b).
Fig. 5. USAF1951 resolution plates. (a) Reflection USAF 1951 sample; (b) transmission USAF 1951 sample.
4.1 Measurement of reflective sample with complicated phase
An approximate plane wave is transmitted to the CCD with resolution of 960 × 1360 pixels and interferes with an object wave on a typical Michelson interference system shown in Figs. 6(a) and 6(b). The wavelength of the He-Ne laser source is 632.8 nm, and the pixel pitch of the detector is 2.2µm × 2.2µm. In the lens-free Michelson interference system, the optical configuration has no magnification for the reflection USAF1951 resolution plate. Since the optical path difference between the object and reference beam is difficult to adjust to zero, the reconstructed phase distribution will contain quadratic phase aberration because of the non-ideal reference plane wave used in this experiment. Figures 6(c)–6(e) are the captured digital hologram, reconstructed wrapped phase and unwrapped phase, respectively. There is also obvious distortion introduced by the spherical wave, as shown in Fig. 6(e).
Fig. 6. Experimental setup and acquisition of the total phase. (a) Diagram of digital holographic system; (b) diagram of experimental setup; (c) recorded hologram; (d) wrapped phase obtained using convolutional reconstruction; (e) obtained unwrapped phase.
Figures 7(a)–7(g) are experimental results by using the proposed method. Figures 7(a) and 7(b) are the rough corrected phase and pre-compensated phase, respectively. Figure 7(c) shows the final corrected phase with the quadratic phase being eliminated and the resolution plate is thus located on a flat plane. The MMASD metrics for ${\phi _{ls}}(x,y)$, ${\phi _{pc}}(x,y)$ and ${\phi _\omega }(x,y)$ are 1.631, 1.596 and 1.319, respectively. Therefore, their base phase distributions gradually become flat. Note that in Figs. 7(d)–7(f), it can be found that the peak of the base phase in the HGD of ${\phi _{ls}}(x,y)$ is to the left of $\mu - 1\sigma $, i.e., out of the $1\sigma $ range. The mean value of ${\phi _{pc}}(x,y)$ rises to 0.5, and the pre-compensation is able to bring out ${\phi _{ls}}(x,y)$ according to the obtained grid phase. Thus, in threshold segmentation, the accuracy of the weighted matrix $\omega (x,y)$ can be improved. In the meanwhile, pre-compensation process shifts the histogram rightward in the HGD. Therefore, the two peaks of ${\phi _\omega }(x,y)$ shown in Fig. 7(f) are able to satisfy the condition that the base phase is within $1\sigma $ range and the sample phase is outside the $1\sigma $ range in the histogram. As the trace lines shown in Fig. 7(g), it can be clearly observed that the proposed method is able to solve the overfitting problem caused by the conventional least-squares algorithm.
Fig. 7. Comparison of HGD plots and phase value. (a) Phase map after traditional least-squares algorithm; (b) phase after pre-compensation; (c) phase obtained from proposed method; (d), (e) and (f) are the HGD plot from Fig. 7(a), Fig. 7(b) and Fig. 7(c), respectively; (g) comparison of phase profiles along the red dashed line from Fig. 6(e), Fig. 7(a) and Fig. 7(c), respectively.
4.2 Measurement of transmission sample with complex morphology
An off-axis Mach-Zehnder interferometer combined with an inverted microscopy (Olympus CKX53) has been built. The optical diagram and its experimental setup of the digital holographic microscope is shown in Figs. 8(a) and 8(b), respectively. The wavelength of He-Ne laser source is 632.8 nm. The resolution of the CCD is 2592(H) × 1944(V) with single pixel size 2.2µm × 2.2µm. The transmission USAF 1951 resolution plate is selected as the experimental sample shown in Fig. 5(b). The light beam emitted by laser propagates through the collimator and the beam splitter (BS1). The object beam reflecting off the mirror (M) is incident on the resolution plate sample and a 20x microscopic objective is used to image the diffracted field onto the CCD. Using a 10x objective, the reference beam with some phase curvature is incident on the CCD. The beam splitter (BS2) is set to be rotatable to adjust the off-axis recording angle. Since different magnifications of the microscopic objectives are selected, the curvature of the spherical waves caused by the two different objectives will cause large aberrations. Figures 8(c)–8(e) are the captured digital hologram, reconstructed wrapped phase and unwrapped phase. Note the severe distortion appears in Fig. 8(e) because of large spherical aberration. According to the obtained unwrapped phase in Fig. 8(e), the traditional least-squares algorithm and the proposed method are applied to eliminate aberration.
Fig. 8. Experimental setup and acquisition of the total phase. (a) Diagram of simplified optical path; (b) experimental setup of off-axis digital holographic microscopy system; (c) recorded hologram; (d) phase obtained using angular spectrum reconstruction; (e) obtained unwrapped phase.
Figures 9(a)–9(e) present experimental results by applying the same procedures shown in Fig. 3. Figures 9(a) and 9(b) are the corrected phases obtained from the traditional least-squares algorithm and our proposed method, respectively. From Figs. 9(c) and 9(d), it can be found that the base-phase is concentrated around the average value of${\phi _{ls}}(x,y)$. We also found that the histogram of ${\phi _\omega }(x,y)$ shifts rightward in the HGD and the peak is closer to 0 compared to that from ${\phi _{ls}}(x,y)$. The trace lines taken along the red dashed line from Fig. 8(e), Fig. 9(a) and Fig. 9(b), reveals that the proposed method is able to achieve higher accuracy.
Fig. 9. Comparison of HGD plots and phase value. (a) Phase map after traditional least-squares algorithm; (b) phase obtained from proposed method; (c) and (d) are the HGD plot from Fig. 9(a) and Fig. 9(b), respectively; (e) comparison of phase profiles along the red dashed line from Fig. 8(e), Fig. 9(a) and Fig. 9(b), respectively.
5. Conclusion
In this study, we demonstrate a histogram segmentation method based on the $1\sigma $-criterion, and a weighted least-squares algorithm to obtain accurate coefficients of quadratic phase aberration. The HGD characteristics of the phase distribution are analyzed for the phase characteristics of digital holograms. The proposed method is able to eliminate the influence of the height, area and position of the sample on the matrix inverse operation through the weighted least-squares algorithm. Pre-compensation is created to compensate the excessive leftward shifting of the histogram. Simulation and experimental results suggested that the proposed method can obtain higher accuracy in phase aberration removal, as compared with the traditional least-squares algorithm. By extending the fitting model [29] to include other aberration coefficients by extending Eq. (2), high-order aberrations could be removed using our proposed technique. The use of the proposed method also can be extended to numerical reconstruction of dual-wavelength digital holograms. We plan to look into this aspect further in future.
Funding
National Natural Science Foundation of China (61975112); National Key Research and Development Program of China (2020YFE024600).
Acknowledgments
This project was supported by the National Natural Science Foundation of China (No.61975112) and the National Key Research and Development Program of China (2020YFE024600).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291 (1999). [CrossRef]
2. W. Zhou, Y. Yu, and A. Asundi, “Structure measurement of phase grating on post-magnification digital micro-holography,” Proc. SPIE 7155, 71550A (2008). [CrossRef]
3. W. Zhou, S. Liu, C. Wang, H. Zhang, Y. Yu, and T.-C. Poon, “Elimination of Quadratic Phase Aberration in Digital Holographic Microscopy by Using Transport of Intensity,” Front. Photonics 3, 848453 (2022). [CrossRef]
4. C. J. Mann, L. Yu, C.-M. Lo, and M. K. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13(22), 8693 (2005). [CrossRef]
5. S. Li, D. Yin, S. Feng, J. Ma, Q. Ma, and C. Yuan, “Phase Aberration Compensation for Resolution Enhancement in Digital Holographic Microscopy under Structured Illumination,” in Biophotonics Congress: Optics in the Life Sciences Congress (OSA, 2019), paper JT4A.28.
6. E. Sánchez-Ortiga, P. Ferraro, M. Martínez-Corral, G. Saavedra, and A. Doblas, “Digital holographic microscopy with pure-optical spherical phase compensation,” J. Opt. Soc. Am. A 28(7), 1410 (2011). [CrossRef]
7. E. Sánchez-Ortiga, A. Doblas, M. Martínez-Corral, G. Saavedra, and J. Garcia-Sucerquia, “Aberration compensation for objective phase curvature in phase holographic microscopy: comment,” Opt. Lett. 39(3), 417 (2014). [CrossRef]
8. V. A. Matkivsky, A. A. Moiseev, P. A. Shilyagin, D. V. Shabanov, G. V. Gelikonov, and V. M. Gelikonov, “A New Method for Finding Optical Aberrations on the Basis of Analysis of the Object Hologram Without Additional Measurements,” Radiophys Quantum El 59(5), 393–404 (2016). [CrossRef]
9. A. Doblas, D. Hincapie-Zuluaga, G. Saavedra, M. Martínez-Corral, and J. Garcia-Sucerquia, “Physical compensation of phase curvature in digital holographic microscopy by use of programmable liquid lens,” Appl. Opt. 54(16), 5229 (2015). [CrossRef]
10. D. Deng, J. Peng, W. Qu, Y. Wu, X. Liu, W. He, and X. Peng, “Simple and flexible phase compensation for digital holographic microscopy with electrically tunable lens,” Appl. Opt. 56(21), 6007 (2017). [CrossRef]
11. S. Liu, Q. Lian, Y. Qing, and Z. Xu, “Automatic phase aberration compensation for digital holographic microscopy based on phase variation minimization,” Opt. Lett. 43(8), 1870 (2018). [CrossRef]
12. T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14(10), 4300 (2006). [CrossRef]
13. P. Ferraro, S. D. Nicola, A. Finizio, G. Coppola, S. Grilli, C. Magro, and G. Pierattini, “Compensation of the inherent wave front curvature in digital holographic coherent microscopy for quantitative phase-contrast imaging,” Appl. Opt. 42(11), 1938–1946 (2003). [CrossRef]
14. D. Deng, W. Qu, W. He, Y. Wu, X. Liu, and X. Peng, “Off-axis tilt compensation in common-path digital holographic microscopy based on hologram rotation,” Opt. Lett. 42(24), 5282 (2017). [CrossRef]
15. P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31(10), 1405 (2006). [CrossRef]
16. C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation,” Opt. Commun. 309, 221–226 (2013). [CrossRef]
17. S. Liu, W. Xiao, and F. Pan, “Automatic compensation of phase aberrations in digital holographic microscopy for living cells investigation by using spectral energy analysis,” Opt. Laser Technol. 57, 169–174 (2014). [CrossRef]
18. G. Coppola, G. Di Caprio, M. Gioffré, R. Puglisi, D. Balduzzi, A. Galli, L. Miccio, M. Paturzo, S. Grilli, A. Finizio, and P. Ferraro, “Digital self-referencing quantitative phase microscopy by wavefront folding in holographic image reconstruction,” Opt. Lett. 35(20), 3390 (2010). [CrossRef]
19. T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, “Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation,” Appl. Opt. 45(5), 851 (2006). [CrossRef]
20. J. Min, B. Yfao, S. Ketelhut, C. Engwer, B. Greve, and B. Kemper, “Simple and fast spectral domain algorithm for quantitative phase imaging of living cells with digital holographic microscopy,” Opt. Lett. 42(2), 227 (2017). [CrossRef]
21. C. Liu, X. Yu, and M. K. Kim, “Phase aberration correction by correlation in digital holographic adaptive optics,” Appl. Opt. 52(12), 2940 (2013). [CrossRef]
22. L. Miccio, D. Alfieri, S. Grilli, P. Ferraro, A. Finizio, L. De Petrocellis, and S. D. Nicola, “Direct full compensation of the aberrations in quantitative phase microscopy of thin objects by a single digital hologram,” Appl. Phys. Lett. 90(4), 041104 (2007). [CrossRef]
23. Y. X. Wang, D. Y. Wang, Y. Z. Zhang, L. Rong, and J. Zhao, “Pure-optical quadratic phase compensation in image-plane digital holographic microscopy,” J. Opt. 43(2), 130–136 (2014). [CrossRef]
24. C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Phase aberration compensation in digital holographic microscopy based on principal component analysis,” Opt. Lett. 38(10), 1724 (2013). [CrossRef]
25. J. Di, J. Zhao, W. Sun, H. Jiang, and X. Yan, “Phase aberration compensation of digital holographic microscopy based on least squares surface fitting,” Opt. Commun. 282(19), 3873–3877 (2009). [CrossRef]
26. T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23(12), 3177 (2006). [CrossRef]
27. E. van den Berg and M. P. Friedlander, “Sparse Optimization with Least-Squares Constraints,” SIAM J. Optim. 21(4), 1201–1229 (2011). [CrossRef]
28. Z. Ren, Z. Xu, and E. Y. Lam, “Phase aberration compensation in digital holographic microscopy using regression analysis,” in Imaging and Applied Optics (OSA, 2018), paper JTh3B.5.
29. Z. Ren, J. Zhao, and E. Y. Lam, “Automatic compensation of phase aberrations in digital holographic microscopy based on sparse optimization,” APL Photonics 4(11), 110808 (2019). [CrossRef]
30. J. Espinosa, J. Pérez, D. Mas, and C. Illueca, “Weighted Zernike polynomial fitting in steep corneas sampled in Cartesian grid,” J. Mod. Opt. 58(19-20), 1710–1715 (2011). [CrossRef]
31. J. Nam and J. Rubinstein, “Weighted Zernike expansion with applications to the optical aberration of the human eye,” J. Opt. Soc. Am. A 22(9), 1709 (2005). [CrossRef]
32. T. Nguyen, V. Bui, V. Lam, C. B. Raub, L.-C. Chang, and G. Nehmetallah, “Automatic phase aberration compensation for digital holographic microscopy based on deep learning background detection,” Opt. Express 25(13), 15043 (2017). [CrossRef]
33. H. Guo, “A Simple Algorithm for Fitting a Gaussian Function [DSP Tips and Tricks],” IEEE Signal Process. Mag. 28(5), 134–137 (2011). [CrossRef]
34. H. Wang, Z. Dong, X. Wang, Y. Lou, and S. Xi, “Phase compensation in digital holographic microscopy using a quantitative evaluation metric,” Opt. Commun. 430, 262–267 (2019). [CrossRef]
35. S. Liu, Q. Lian, and Z. Xu, “Phase aberration compensation for digital holographic microscopy based on double fitting and background segmentation,” Opt. Lasers Eng. 115, 238–242 (2019). [CrossRef]
36. Y. Liu, Z. Wang, and J. Huang, “Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography,” Appl. Sci. 8(3), 444 (2018). [CrossRef]
37. S. Liu, W. Zhu, Z. Xu, and M. Gao, “Automatic and robust phase aberration compensation for digital holographic microscopy based on minimizing total standard deviation,” Opt. Lasers Eng. 134, 106276 (2020). [CrossRef]
