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Abstract
Optical aberrations introduced by sample or system elements usually degrade the image quality of a microscopic imaging system. Computational adaptive optics has unique advantages for 3D biological imaging since neither bulky wavefront sensors nor complicated indirect wavefront sensing procedures are required. In this paper, a stochastic parallel gradient descent computational adaptive optics method is proposed for high-efficiency aberration correction in the fluorescent incoherent digital holographic microscope. The proposed algorithm possesses the advantage of parallelly estimating various aberrations with fast convergence during the iteration; thus, the wavefront aberration is corrected quickly, and the original object image is retrieved accurately. Owing to its high-efficiency adaptive optimization, the proposed method exhibits better performances for a 3D sample with complex and anisotropic optical aberration. The proposed method can be a powerful tool for the visualization of dynamic events that happen inside cells or thick tissues.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fluorescence microscopy provides a valuable means to directly access information about the 3D structure of biological samples in the specific life process. However, the in-vivo visualization of the dynamic events that happened on fine structures inside the cells or tissues is still a challenge. The main problem is that the spatial resolution and signal noise ratio (SNR) of imaging system will be dramatically degraded because of the presence of the aberration introduced by sample or system. In the past decades, the above-mentioned problem has been solved by the development and application of different adaptive optics (AO) methods in microscopy. Either wavefront sensor, wavefront modulator, or even both of them are required in conventional AO microscopy [1,2]. However, the performances of the systems are severely limited by the parameters of the wavefront sensors and correctors [3]. In the sensorless AO methods [4], the systems also suffer photobleaching caused by the relatively time-consuming wavefront detection and correction procedure [5].
On the other hand, the emergence of computational adaptive optics (CAO) has brought a new candidate for adaptive microscopy [6]. Computational adaptive optics (CAO) exhibits unique advantages such as neither wavefront sensor nor wavefront modulator is necessary for the aberration detection and compensation of optical aberration [7,8,23,24]. Alternatively, the basic principle of CAO is trying to optimize the quality of reconstructed images of the sample in a numerical iterative procedure. The reconstructions are obtained by applying a compensation phase mask in the spatial frequency domain of the measurements, and the quality metric of reconstructions is used as the feedback to obtain an optimized reconstructed image [9,10]. The key point that makes it possible to implement the CAO in a specific imaging modality lies in whether the imaging system possess the capability of recording the amplitude and the phase of the sample-originated wavefront. The incoherent 3D imaging [11–13] with CAO system will be an ideal tool for biological imaging because only one extended-object hologram is sufficient to reconstruct the aberration-compensated 3D structure of the sample [14,15]. In our previous work, we demonstrated the approach to non-scanning 3D imaging of fluorescence samples in a CAO self-interference holographic microscopic system [16]. In the conventional CAO and our previous work, the estimation and correction of wavefront aberration are accomplished by optimizing Zernike aberration order-by-order sequentially and separately. The effect and speed of this sequential optimization are not always satisfactory, especially for a complex aberration mode, the result can be disturbed by the local optimized values [17,18].
In this work, we develop more efficient computational adaptive optics in self-interference fluorescence microscopy by employing a stochastic parallel gradient descent (SPGD) algorithm in the CAO optimization. The proposal is named SPGD-CAO in this work. Experiment results demonstrate that the SPGD-CAO has more advantageous than stepwise optimization in the case of complex aberration and serious aberration. Compared with the sequential optimization procedure, the SPGD-CAO algorithm significantly improves the aberration correcting speed, while the accuracy of the CAO aberration correction is also improved owing to the global optimization. As the result, reconstructed images with better spatial resolution and higher SNR can be obtained in a system with complex and anisotropic optical aberration.
2. Methodology
Fresnel incoherent correlation holography (FINCH) is a unique imaging method in which the wavefront of the light that originated from each point light source can be recorded as holograms of point objects quantitatively and separately. This characteristic makes it possible to implement the CAO in FINCH. The wavefront information is encoded in the hologram of each point source, which is the interference pattern of two mutually coherent waves obtained by dividing the light from each point light source into two different beams. The optical system as shown in Fig. 1 is used to briefly demonstrate the basic principle of the FINCH.
Assuming an on-axis point-like fluorescent object is located at a distance of ${z_s}$ to the objective lens L (with a focal length of $f$). The emitted light is collected by the lens L and then passes through the polarizer P1 and the spatial light modulator (SLM) sequentially. The distance between the SLM and L is ${d_s}$. The light is divided into two beams after passing through the SLM, which converge toward two different axial positions ${d_1}$ and ${d_2}$ after the SLM. On the CCD plane (located at a distance of ${z_h}$ after the SLM), the interference pattern (point spread hologram) has the intensity distribution of Eq. (1). The point spread hologram (PSH) is usually regarded as the point spread function (PSF) of FINCH recording system.
In combination with the phase-shift technique, multiple phase-shifted holograms are recorded using different ${\theta _i}$ and superimposed to obtain a complex-valued hologram without the twin images and zero-level term. The complex-valued hologram of object ${H_a}$ can be represented as:
${z_r}$ is the reproduction distance, which can be obtained by the autofocus algorithm.${H_a}$ is the aberrated complex-valued hologram caused by optical aberration, it can be represented as an ideal hologram $Q(1/{z_r})$ multiplied by Ψ which included the phase aberration ${\varphi _{ab}}$. The hologram can be reconstructed using the numerical propagation algorithm and the reproduced image can be obtained as follows:Using the backpropagation algorithms, the 3D reconstruction of object can be obtained from the hologram. The reconstructed images obtained directly from the OH are, however, blurred because of the presence of optical aberrations. The aberration can be compensated using the CAO adaptive optimization by finding out and then applying a properly compensated phase mask in the spatial frequency domain of the reconstructed images. The CAO correction procedure on a reconstructed complex-valued image can be expressed as:
The flowchart of the SPGD-CAO algorithm is shown as Fig. 2 and the basic procedure of the optimization is also described in detail.
- (1) Evaluate the reconstructed aberrated image and define it as the initial value ${M_0}(OH)$. Set the iteration numbers i. Set the control variables $U = {U_0}$ according to the orders of the aberration to be solved, usually ${U_0} = (0,0, \cdot{\cdot} \cdot ,0)$.
- (2) Generate random perturbations obeying Bernoulli distribution $\delta u$. Update control variables to $U + \delta u$, calculate the phase mask ${\varphi _ + }(AO)$, load it into the diffraction reconstruction image of the object complex-valued hologram, and calculate the performance evaluation function after forwarding perturbation ${M_ + }(OH)$. Update control variables to $U\textrm{ - }\delta u$, calculate the phase mask ${\varphi _ - }(AO)$, load it into the diffraction reconstruction image of the object complex-valued hologram, calculate the performance evaluation function after reverse perturbation ${M_ - }(OH)$.
- (3) Calculate $\gamma (i) = {\gamma _0} \cdot (c + M/{M_0})$, where c and ${\gamma _0}$ are basic parameters, ${\gamma _i}$ is the update gain according to the procedure of maximization and minimization respectively. Calculate $\delta M = \frac{{{M_ + }(OH) + {M_ - }(OH)}}{2}$, calculate the control variables for the next iteration $U^{\prime} = U - {\gamma _i}\delta M\delta u$, update the control variables $U = U^{\prime}$. Return to (2) and continue iterating until the iteration condition or the number of iterations is satisfied. The whole optimization is computed well can be seen from the final convergence.
There are many functions for evaluating image sharpness, which is generally chosen according to the image recovery effect. For targets with more detailed texture features, the Tenengrad evaluation function is a more commonly used evaluation function [21].
${G_x}(x,y)$ and ${G_y}(x,y)$ denote the convolution of the image $I(x,y)$ with the horizontal and vertical templates of the Sobel operator, respectively.The horizontal and vertical templates of the Sobel operator can be expressed as:
In traditional FINCH, the backward propagation algorithm was used to construct the complex-valued hologram. The reconstructed images usually became aberrated due to the existence of aberrations. We proposed a FINCH-based computational adaptive method to construct a virtual phase board to compensate for the phase distribution of reconstructed images. Tenengrad evaluation function was used as the evaluation index of image quality, and the SPGD algorithm was used for iterative optimization. Finally, corrected the aberration, optimized image quality, and improved resolution.
3. Experiments and results
To validate the performance of the proposed method, a fluorescent incoherent digital holographic microscope was built as shown in Fig. 3. A 457 nm solid-state laser (85-BLT-605, Melles Griot) was used as the excitation light. The output light from the fiber was collimated and then magnified by a 4f telescope system (composed of two doublet lenses L1 and L2, AC254-150A and AC254-400A, Thorlabs) to ensure a sufficient illumination area on the sample plane. After being reflected by the dichroic mirror (Q495lp, Olympus), the light was focused onto the back focal plane of the microscopic objective lens (TU Plan Fluor 20×/N. A. 0.45, Nikon) to wide-field illuminate the sample. The fluorescent samples was translated by nano-z scanning translation stage (P-736. ZR2S, PI). The emitted fluorescent was collected by the same objective lens. After passing through the Dichroic Mirror (DM), the fluorescent was guided to the SLM (1920 × 1080 pixels, 6.4 µm pixel pitch, LETO 3.0, Holoeye) using lens L4 and finally to the CCD camera (2048 × 2048 pixels, 7.4 µm pixel pitch, ML4022, FLI). Two 45° oriented polarizers were used for the beam dividing.
3.1 Demonstration of SPGD-CAO aberration-corrected effectiveness
The performances of the proposed method were firstly demonstrated in a system with artificial aberration. In the system, as shown in Fig. 3, optical aberration was deliberately introduced by adding a phase mask to the SLM. In the experiments, single Zernike-mode aberration (astigmatism, RMS = 0.634λ, PV = 3.505λ) as shown in Fig. 4(a) was introduced. A fluorescence resolution test chart (1951 USAF resolution target, negative, Edmund Optics, with a yellow-green fluorescent backplate) was used as the sample. Holograms with four different phase shifting values θi (θ1 = 0, θ2=π/2, θ3=π, θ4 = 3π/2) were recorded to construct the complex-valued hologram. The amplitude and phase of the complex-valued hologram of the object are shown in Fig. 4 (b) and Fig. 4 (c), respectively. The conventional holographic reconstructed image obtained using the Fresnel backpropagation algorithm (Fig. 4 (d), intensity image) is severely degraded under the influence of optical aberration. The first line pair of the ninth group of the resolution chart (line width of 980 nm) is blurred and indistinguishable. The corrected image obtained by applying the proposed SPGD-CAO algorithm to the complex-valued reconstruction is shown in Fig. 4(g). The resolution and signal-to-noise ratio of the reconstructed image is significantly improved, while the third line pair of the ninth group (line width of 780 nm) is visible. The improvement in the image quality can be quantitatively observed from the line profile as shown in Fig. 4(h). The convergence curve of the SPGD optimization process is shown in Fig. 4 (e). It can be seen that the relative image quality metric quickly converged after 30 times of iterations. The whole optimization process takes 13s on a Windows desktop (Intel Core i5 3.2 GHz and 8 GB RAM, and the size of the target image is 290 × 290 pixels). Figure 4(f) gives the phase distribution of the phase mask that corresponds to the optimal corrected image (Fig. 4(g)). For comparison, the reconstructed image without any aberration is shown in Fig. 4(i). It should be emphasized that these optimal correction phase mask does not necessarily have the same distribution as the system aberration (astigmatism as shown in Fig. 4(a) here). The reason for this phenomenon is that in incoherent digital holography, both the two beams involved in the interference contain optical aberration and the phase information of the optical system. The hologram records the wavefront by recording the phase difference between the two parts of information. Therefore, the magnitude of the reconstructed image is different from that of the loaded image.
Fig. 4. Experimental results: (a) Loaded 45° astigmatism, Wavefront RMS = 0.634λ, PV = 3.505λ, (b) Intensity distribution of complex-valued hologram, (c) Phase distribution of complex-valued hologram, (d) Aberrated image reconstructed by diffraction, (e) Graph of Convergence, (f) Phase distribution calculated by the algorithm, (g) Corrected image by SPGD, (h) Intensity distribution of the underlined part, (i) Unaberrated image reconstructed by diffraction.
The performances of the proposed methods in a system with more complex Zernike-mode aberration were further investigated. The first 20 orders (except tilt, tip, and defocus) of the Zernike polynomials with random magnitudes were used to simulate a complex aberration mode. The correction process is similar to other types of aberration. The phase distribution of the generated complex aberration is shown in Fig. 5(a) (aberration wavefront RMS = 0.42λ, PV = 4.725λ). Since the aberration is more complex, the optimization process takes more time (40s) than the case for the single mode aberration, and the algorithm converges after 100 iterations (Fig. 5(e)). Overall, compared with the conventional result (Fig. 5(d)), the quality of the reconstructed image was greatly improved after applying the SPGD-CAO optimization (Fig. 5(g) and Fig. 5(h)), and the resolution and signal to noise ratio of the reconstructed image are significantly improved. For comparison, the reconstructed image without any aberration is shown in Fig. 5(i).
Fig. 5. Experimental results: (a) Load 20 orders hybrid aberration, Wavefront RMS = 0.42λ, PV = 4.725λ, (b) Intensity distribution of complex-valued hologram, (c) Phase distribution of complex-valued hologram, (d) Aberrated image reconstructed by diffraction, (e) Graph of Convergence, (f) Phase distribution calculated by the algorithm, (g) Corrected image by SPGD, (h) Intensity distribution of the underlined part, (i) Unaberrated image reconstructed by diffraction.
In bio-imaging applications, there is fundamental diversity in both the type and magnitude of the optical aberration introduced by different types of cells or tissues. As demonstrated above, by applying a numerical aberration correction and image optimization process, the proposed SPGD-CAO method can significantly improve the imaging performances of a fluorescence holographic microscope with simple-Zernike mode and complex-Zernike mode optical aberration. We then performed the analysis on the efficiency of the proposed method in correcting the optical aberration with different levels of magnitudes. The combinations of the first 20 orders (except tilt, tip, and defocus) of the Zernike polynomials, with different levels of magnitudes (aberration wavefront RMS = 0.34λ, 0.42λ, 0.53λ and 0.62λ, respectively), were chosen as the system aberration. As expected, the SPGD-CAO optimization correction can always provide results (Fig. 6(c)) with improved quality than the uncorrected images (Fig. 6(a)). For the system aberration with a relatively low level of magnitudes (RMS = 0.34λ, 0.42λ, and 0.53λ), the image quality of the proposed method is comparable with the results obtained using our previous order-by-order sequential CAO method (Fig. 6(b)), more details about the optical system can be found in [16]. The whole optimization process takes 196s when using the traditional method and only takes the 40s when the image quality reached optimum through the proposed method. However, for a serious system aberration (RMS = 0.62λ), the proposed method is superior since it provides better reconstruction (comparing Fig. 6(c) to Fig. 6(b)).
Fig. 6. Experimental results: (a) Imaging results of loaded 20 order hybrid aberration at different degrees, (b) Corrected images by the method of stepwise optimization, (c) Corrected images by the method of SPGD, (d) Intensity distribution of the underlined part, Wavefront RMS = 0.53λ, PV = 5.73λ, (e) Intensity distribution of the underlined part, Wavefront RMS = 0.62λ, PV = 6.81λ.
The comparison of the intensity distribution of different color delineations (Fig. 6(d) to Fig. 6(e)) shows that the image corrected by our proposed method has significantly improved both contrast and resolution compared with an uncorrected image, and has the advantage over the method of stepwise optimization in the case of complex aberration forms and large aberration. This can be explained as the parallel SPGD optimization being more accurate in finding the optimal correction phase mask because it is more robust to the local optimized values with fast convergence speed [22].
3.2 Non-scanning 3D CAO imaging of the fluorescent sample
We prepared a 3D sample by embedding 500 nm diameter yellow-green fluorescent microspheres into the gel. The 1:5000 dilution of the microspheres were mixed with the Polymethyl methacrylate (PMMA) powder. The mixture was then heated and stirred to a colloidal state and sandwiched between the microscope slide and coverslip. The axial range of the resulting 3D sample was measured to be about 8 µm by using a nano-z stage.
The amplitude and phase distribution of the complex-valued hologram of 3D sample are shown in Fig. 7(a) and (b) respectively. The FZP-like holograms as shown in From Fig. 7(b) quantitatively encode the 3D positions of the microspheres. 3D images of the sample can be reconstructed directly from a single frame of the complex-valued hologram, without any mechanical scanning of the optical elements. However, because of the presence of optical aberrations (introduced by the inhomogeneous refractive index of PMMA gel), the holograms are slightly distorted. As a result, the imaging performances of the system are degraded. Figure 8(a1)-(b1) gives the focal plane intensity distributions of two selected microspheres 1 and 2. The different in-focused reconstruction distances for the two microspheres (z1 = 8.72 µm, and z2 = 9.21 µm, respectively. Measured by using self-focusing algorithm) clearly indicate that they are located at different depth planes in the object space. With the presence of optical aberration, elliptical and uneven distributed images are obtained. The reconstructed images in XZ plane (Fig. 8 (a2)-(b2)) are also distorted. As shown in Fig. 8(a3)-(b3) and Fig. 8(a4)-(b4), without any additional measurements, this image degradation problems in non-scanning 3D reconstructions can be effectively corrected after the application of the proposed method. It is worth to be noticed that the CAO corrections are anisotropic since the correction phase masks (inserted) have different distributions for the two microspheres that are located at different spatial coordinates. The axial distribution of PSF is more uniform and concentrated as illustrated in Fig. 8(a4)-(b4). As shown in Fig. 8(a5)-(b5), The intensity profile of the XY images of microsphere after optimization shows an increased signal to noise ratio, suggesting that the optimization corrected system and sample aberrations. Figure 8(c) shows a 3D view of the clipped CAO corrected 3D reconstruction that obtained from the hologram as shown in Fig. 7. The whole optimization process takes 45s (on our desktop computer with Intel Core i5 3.2 GHz and 8 GB RAM) when the first 20 orders of the Zernike polynomials are chosen as the aimed aberration of the sample. In these experiments, the sum-squared of intensities of the reconstructed image was used as a metric.
Fig. 8. Experimental 3D imaging results. Uncorrected XY (a1,b1) and XZ (a2, b2) reconstructed images of microsphere 1 and 2 (indicated with blue and green boxes in Fig. 7, respectively). Corrected XY (a1,b1) and XZ (a2, b2) reconstructed images of microsphere 1 and 2 obtained after applying the proposed method. (a5,b5) Intensity distribution of the scribed parts. (c) A 3D view of the 3D reconstruction of the microspheres after CAO correction (clipped from the 3D reconstruction of the hologram as shown in Fig. 7). White scale bars: 0.72 µm. Yellow scale bars: 1.73 µm.
4. Conclusions
In this work, the speed and precision of CAO optical aberration correction in a fluorescence holographic microscopic system were improved by using SPGD parallel optimization method. The theoretical discussion on the basic principle of phase aberration encoding in fluorescence holography and SPGD-CAO optimization procedure were presented. SPGD algorithm is a numerical method for solving multi-dimensional unconstrained optimization problems. It estimates the gradient of the performance evaluation function with each disturbance, optimizes control variables along the gradient direction, and searches for the optimal solution for performance evaluation value. To investigate the performance of our approach, we perform experiments on a fluorescence resolution test chart, with deliberately introduced system aberration with different types and magnitudes. Significant improvements in image resolution and signal-to-noise ratio were demonstrated by the results. Our approach exhibits superior performances and better reconstruction quality under a high level of magnitudes of aberration. The experiments on the 3D sub-resolution microspheres sample have demonstrated the validity of our method in correcting the anisotropic aberration. Compared with the previous sequential optimization method, our parallel SPGD-CAO optimization method provides a 3D reconstructed image of the sample with better quality, especially in a system with strong and complex optical aberration (for example thick tissues). In addition, the whole processing speed of the proposed method is faster. The AO corrected 3D imaging results of nanoparticles demonstrates the non-scanning 3D imaging ability of the proposed method. The proposed approach can be applied in a certain imaging mechanism, if the system can detect both the amplitude and phase of the light signal from the sample (for example in OCT, FINCH and digital holography). We noticed that in purpose of improving the axial imaging resolution, FINCH-based methods have been combined with confocal scanning microscopy [15]. We believe that our method can be applied to variety of different microscopic imaging methods such as multiphoton microscopy and super-resolution microscopy, and can then potentially benefit the biological imaging of cells and tissues.
Funding
National Natural Science Foundation of China (61575009); Beijing Municipal Natural Science Foundation (4182016); Tianjin Key Laboratory of Micro-scale Optical Information Science and Technology.
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. K Tyson R, Principles of Adaptive Optics, Fourth Edition, (CRC Press, 2016).
2. N Ji, “Adaptive optical fluorescence microscopy,” Nat. Methods 14(4), 374–380 (2017). [CrossRef]
3. J Booth M, “Adaptive optics in microscopy,” The Royal Society 365(1861), 2829–2843 (2008).
4. Acner Camino, Pengxiao Zang, Arman Athwal, Shuibin Ni, Yali Jia, David Huang, and Yifan Jian, “Sensorless adaptive-optics optical coherence tomographic angiography,” Biomed. Opt. Express 11(7), 3952–3967 (2020). [CrossRef]
5. C Jang, J Kim, DC Clark, S Lee, B Lee, and MK Kim, “Holographic fluorescence microscopy with incoherent digital holographic adaptive optics,” J. Biomed. Opt. 20(11), 111204 (2015). [CrossRef]
6. Steven G. Adie, Benedikt W. Graf, Adeel Ahmad, P. Scott Carney, and Stephen A. Boppart, “Computational adaptive optics for broadband optical interferometric tomography of biological tissue,” Proc. Natl. Acad. Sci. U. S. A. 109(19), 7175–7180 (2012). [CrossRef]
7. D Hillmann, H Spahr, C Pfäffle, H Sudkamp, G Franke, and G Hüttmann, “In vivo optical imaging of physiological responses to photostimulation in human photoreceptors,” Proc. Natl. Acad. Sci. U.S.A. 113(46), 13138–13143 (2016). [CrossRef]
8. Dierck Hillmann, Hendrik Spahr, Carola Hain, Helge Sudkamp, Gesa Franke, Clara Pfäffle, Christian Winter, and Gereon Hüttmann, “Aberration-free volumetric high-speed imaging of in vivo retina,” Sci. Rep. 6(1), 35209 (2016). [CrossRef]
9. Paritosh Pande, Yuan Zhi Liu, Fredrick A. South, and Stephen A. Boppart, “Automated computational aberration correction method for broadband interferometric imaging techniques,” Opt. Lett. 41(14), 3324–3327 (2016). [CrossRef]
10. Dan Zhu, Ruoyan Wang, Mantas Žurauskas, Paritosh Pande, Jinci Bi, Qun Yuan, Lingjie Wang, Zhishan Gao, and Stephen A. Boppart, “Automated fast computational adaptive optics for optical coherence tomography based on a stochastic parallel gradient descent algorithm,” Opt. Express 28(16), 23306–23319 (2020). [CrossRef]
11. Joseph Rosen and Gary Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007). [CrossRef]
12. Joseph Rosen, Nisan Siegel, and Gary Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011). [CrossRef]
13. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008). [CrossRef]
14. Gary Brooker, Nisan Siegel, Victor Wang, and Joseph Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011). [CrossRef]
15. Nisan Siegel, Vladimir Lupashin, Brian Storrie, and Gary Brooker, “High-magnification super-resolution FINCH microscopy using birefringent crystal lens interferometers,” Nat. Photonics 10(12), 802–808 (2016). [CrossRef]
16. Tianlong Man, Yuhong Wan, Wujuan Yan, Xiuhong Wang, Erwin J. G. Peterman, and Dayong Wang, “Adaptive optics via self-interference digital holography for non-scanning three-dimensional imaging in biological samples,” Biomed. Opt. Express 9(6), 2614–2626 (2018). [CrossRef]
17. Karen M. Hampson, Raphaël Turcotte, Donald T. Miller, Kazuhiro Kurokawa, Jared R. Males, Na Ji, and Martin J. Booth, “Adaptive optics for high-resolution imaging,” Nat. Rev. Methods Primers 1(1), 68 (2021). [CrossRef]
18. S. Bonora and R. J. Zawadzki, “Wavefront sensorless modal deformable mirror correction in adaptive optics: optical coherence tomography,” Opt. Lett. 38(22), 4801–4804 (2013). [CrossRef]
19. Zongliang Xie, Haotong Ma, Xiaojun He, Bo Qi, Ge Ren, Li Dong, and Yufeng Tan, “Adaptive piston correction of sparse aperture systems with stochastic parallel gradient descent algorithm,” Opt. Express 26(8), 9541–9551 (2018). [CrossRef]
20. M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22(12), 907–909 (1997). [CrossRef]
21. J. L. Pech-Pacheco, G. Cristobal, J. Chamorro-Martinez, and J. Fernandez-Valdivia, “Diatom autofocusing in brightfield microscopy: A comparative study,” Proceedings 15th International Conference on Pattern Recognition3, 314–317 (2000).
22. Huizhen Yang and Xinyang Li, “Comparison of several stochastic parallel optimization algorithms for adaptive optics system without a wavefront sensor,” Opt. Laser Technol. 43(3), 630–635 (2011). [CrossRef]
23. Rishyashring R. Iyer, Yuan-Zhi Liu, and Stephen A. Boppart, “Automated sensorless single-shot closed-loop adaptive optics microscopy with feedback from computational adaptive optics,” Opt. Express 27(9), 12998–13014 (2019). [CrossRef]
24. Siyang Liu, Fei Xia, Xusan Yang, Meiqi Wu, Laurie A. Bizimana, Chris Xu, and Steven G. Adie, “Closed-loop wavefront sensing and correction in the mouse brain with computed optical coherence microscopy,” Biomed. Opt. Express 12(8), 4934–4954 (2021). [CrossRef]
