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Abstract
Fourier ptychographic microscopy (FPM) is a recently developed computational microscopy approach that can produce both wide field-of-view and high-resolution intensity and phase image of the sample. Traditionally, a periodic LED array is used as the illumination source. The frequency spectrum is a grid structure that is sparse in the low-middle frequency region and dense in the high-frequency region. However, for most biological samples, the useful information is mostly concentrated in the low-middle frequency region. So the traditional sampling method may lead to distortion in the reconstructed images because of insufficient sampling. Therefore, we propose a convenient and feasible method to continuously improve the sampling conditions without increasing the LED units and changing the pitch, termed double-height illumination FPM. The LED array is moved from one plane to another to achieve the sample illuminations at two different heights and the low-resolution intensity images are totally utilized to reconstruct the high-resolution complex amplitude image. Both simulation results and experimental results illustrate that the proposed method not only improves the reconstruction accuracy and efficiency, but also has strong robustness under noisy conditions. Furthermore, the rule of choosing the two illumination heights is discussed, and the physical insight of the proposed method is analyzed from the view of the data redundancy. In addition, a wave vector correction method is employed to reduce the artifacts of the reconstructed image at the edge of the large full field of view.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fourier ptychographic microscopy (FPM) [1–3] is a recently developed computational technology, which overcomes the physical spatial bandwidth product limitation of low numerical aperture (NA) imaging systems. Similar to the traditional ptychography approach [4–8], FPM shares the roots of synthetic aperture concept [9,10] and other phase retrieval methods [11]. While FPM configuration does not involve mechanical scanning module, a programmable LED array is adopted to realize oblique illumination of the object. After acquiring multiple low-resolution (LR) intensity images, the spectrum is computationally synthesized in the Fourier domain, and finally a complex amplitude image with large field of view (FOV) and high-resolution is reconstructed. Since FPM technology was put forward in 2013, it has made great progress in many biomedical applications such as hematology [12] and pathology [13,14].
In the original FPM system, a periodic LED array is used for sample illumination. Although this kind of planar LED panel is easy to design and process, it causes the grid artifact problem in the reconstructed result. Much research work has been reported to further improve the performance of FPM, which can be classified into two types. The first type is to optimize the illumination structure. Zheng et al. designed and built a circular LED illuminator [15], corresponding to a non-uniform sampling pattern in the Fourier domain, which solved the raster grid artifact problem, and improved quality of reconstructed images. Subsequently, hemispherical digital condensers were used in FPM [16,17] to acquire a non-uniform sub-spectrum distribution and avoid the problem of illumination intensity fluctuation. It can also realize larger NA of the system, and further improve the imaging resolution. However, the above methods increase the difficulty of manufacture, and the LED units are fixedly arranged, being difficult to adjust. In addition, there is more difficulty to determine and calibrate the position of each sub-spectrum. The second type is to improve the reconstruction algorithm. In response to the limitation of traditional GS algorithms [18], some scholars have modified the phase retrieval algorithms by proposing EPRY [19], WFP [20], TPWFP [21], Adaptive step [22], Apodized CTF constraint [23], etc. In view of the existence of noise in experiments, some scholars have proposed a series of noise-resistant and denoising algorithms [24–28]. For the LED position misalignment problem, some scholars proposed the algorithms for position error correction [29–34] based on simulated annealing algorithm, quasi-Newton method, nonlinear regression analysis, etc. In addition, some people applied deep learning [35–40] to FPM to improve the quality of reconstructed image.
At the same time, large full FOV is one of the advantages of FPM. While directly reconstructing the large full FOV will cause serious artifacts, especially at the edge of the FOV. That is because the spherical wave emitted by each LED element is approximately treated as a plane wave on the object plane. Zhu et al. [41] combined LED array with telecentric lenses to provide plane wave with different angles and uniform intensity, thus achieving accurate large full FOV reconstruction. In most of the current FPM research, a small segment in the center of the FOV is intercepted for the reconstruction to demonstrate such as the resolution improvement, the phase retrieval and various parameter corrections. In practice, the reconstruction of the non-central area of FOV will have various aberrations [19], vignetting [42], etc. For the reconstruction of large full FOV, the inconsistency of illumination angle in different FOV is also a problem that cannot be ignored. Zheng pointed out the issue of an illumination angle correction for different segments of the large full FOV when proposing FPM [1]. However, this issue has not been studied and discussed in detail.
In this paper, we reconstruct the large FOV, high-fidelity and high-resolution image by optimizing the illumination mode and developing the reconstruction algorithm. Firstly, we propose a novel idea to improve the sampling rate of the central low-middle frequency region termed double-height illumination (DHI)-FPM, which needs no more additional design and processing of the light source. A corresponding double-height reconstruction algorithm is developed for the phase retrieval. Secondly, we propose a method to correct the illumination wave vector and reduce the artifacts of the reconstructed images in the edge region. Simulation and experimental results show that the proposed method not only has a good performance in noisy environment, improves the quality of the reconstructed image, but also improves the convergence efficiency.
2. Principle
2.1 Frequency coverage and reconstruction algorithm of DHI-FPM
The DHI-FPM platform is shown in Fig. 1(a), which consists of a LED array and a conventional microscope with a low NA microscopic objective (MO). The core idea of DHI-FPM is to move the LED array from one plane to another to achieve the sample illuminations at two different heights. Then two groups of LR images with different frequency information are obtained, as shown in Fig. 1(b). After, the sub-frequency spectrums are synthesized in Fourier domain as Fig. 1(c) by double-height reconstruction algorithm, so as to restore the high-resolution image of the sample with high fidelity.
Fig. 1. Schematic diagram of DHI-FPM. (a) Basic schematic for the whole system, (b) different illumination height projections, (c) schematic diagram of spectrum sampling pattern.
According to the theory of optical coherent imaging, when the ith LED unit is turned on, the illumination wave vector can be expressed as:
From the above equations, in order to adjust the position of the sub-frequency spectrum and the overlapping ratio, the normal way is to change ${x_{i0}}$ and ${y_{i0}}$ parameters, which is realized by lighting the different LED units. Here, we take the third variable d into the consideration. By altering the illumination height d, the sampling rate and bandwidth in the Fourier frequency domain can be adjusted flexibly.
The double-height reconstruction algorithm is accordingly developed, and it can realize the functions of both the sub-frequency spectrum registration and the phase retrieval. For faster convergence, the LR images from the higher height(d2) are firstly employed in the reconstruction algorithm. Because the related sub-spectrums of these LR images have higher overlapping ratio, the reconstructed image can be obtained with high speed and good quality. Then the LR images from the other height(d1) are reconstructed correspondingly. The steps in detail are as follows:
Step 1: define the LR images recorded at the two heights as ${I_{l,m,i,{d_1}}}$ and ${I_{l,m,i,{d_2}}}$, Subscripts l and m denote low-resolution and measurement, respectively. Subscript i defines the ith illumination. The high-resolution spectrum of the initial estimation can be expressed as $\tilde{O}({k_x},{k_y}) = \Im \{{B[{{O_{l,1,{d_2}}}(x,y)} ]exp [j\varphi (x,y)]} \},$ where B[…] is the interpolation function, the multiple of the interpolation is equal to the magnification of MO, ${O_{l,1,{d_2}}}(x,y)$ is the square root of ${I_{l,m,i,{d_2}}}$, and $\varphi (x,y)$ is a zero matrix.
Step 2: for the ith illumination angle, calculate the spectrum distribution ${\tilde{\psi }_{j,i,d}}({k_x},{k_y})$ as Eq. (2). Then an inverse Fourier transform is performed to obtain a simulated low-resolution complex amplitude ${\psi _{j,i,d}}(x,y)$, where subscript j is the number of iterations, and subscript d is the distance from object to the LED array plane.
Step 3: the amplitude is replaced by the square root of the measured intensity images, and the phase value is retained to achieve the updated complex amplitude as $\psi ^{\prime}_{j,i,d}{(x,y)} = \sqrt {{I_{l,i,d}}(x,y)} \frac{{{\psi _{j,i,d}}(x,y)}}{{|{{\psi_{j,i,d}}(x,y)} |}}.$
Step 4: return to the spectrum plane through the Fourier transform, $\tilde{\psi }{^{\prime}_{j,i,d}}({k_x},{k_y}) = \Im \{{\psi {^{\prime}_{j,i,d}}(x,y)} \}.$ And then update the sample spectrum and the pupil function:
Step 5: repeat the process from step 2 to step 4 for all illumination plane waves of d2. Then, start again from the central spectrum, repeat the above steps until the whole spectrum related with d1 is updated.
Step 6: the updated spectrum is used as the new initial estimation of spectrum, and the above update process is repeated until convergence, which is judged by the error matric indicated as below
It is set the iteration stops when the value reaches to 0.01. The flow chart of the algorithm is shown in Fig. 2.
2.2 Wave vector correction of large full FOV
For the full FOV, the illumination wave vectors of different areas are inconsistent as shown in Fig. 3. For the middle area of the FOV, the wave vector from the red LED unit can be expressed by Eq. (1), the illumination angle is $\theta$. But for the edge area of the FOV, such as a sub-region centered at $({x_c},{y_c})$, the illumination angle at this point is ${\theta ^{\prime}}$, corresponding to the angle in Eq. (1) needing to be calibrated. Otherwise, there will be a shift of the Fourier spectrum, resulting in the artifacts in the reconstrued images. For simplicity, it can be treated as if the relative position of the LED array plane and the object plane has been altered, which results in a phase distortion in reconstruction, similar to that caused by oblique illumination. According to the geometric relationship, the illumination wave vector can be modified as follows:
Fig. 3. Schematic diagram of the wave vector error in the small region at the edge generated by spherical wave illumination.
The above calibrated wave vector is then used in the reconstructed algorithm to get rid of the artifacts.
3. Simulation results
The simulation parameters of DHI-FPM are set according to the practical experimental conditions. The LED array consists of 11×11 units with a spacing of 4 mm, and the central wavelength of the illumination light is 625 nm. The magnification and NA of the objective lens are 4× and 0.1. A CCD with the pixel size of 6.5 µm×6.5 µm is used for recording the raw images. In the actual acquisition process of FPM, the images acquired by CCD may be affected by various noises, which would degrade the quality of the reconstructed image. To further verify the robustness to noise of the proposed method, we add 30% Gaussian noises to each LR image.
Figure 4 shows the simulated results. Figures 4(a1)-(a2) are the ground truth of the simulated amplitude and the phase image, respectively. And Fig. 4(a3) is the LR intensity image when the central LED unit is turned on, and it is now seriously disturbed by noises. Figures 4(b1)-(b4) represent the reconstructed results when the illumination height is 40 mm. It can be seen that, the smaller illumination height increases the bandwidth of the spectrum, but at the same time, it reduces the sampling rate of the spectrum. Especially the low-middle frequency region containing main information of the sample is sampled more sparsely, resulting in blurring artifacts and uneven of the background. Figures 4(c1)-(c4) show the reconstructed results when the LED array is 90 mm away from the object surface, and it can be seen that the higher illumination distance increases the sampling rate of the low-middle frequency region, but limits the bandwidth of the spectrum at the expense of imaging resolution. Figures 4(d1)-(d4) denote the results of DHI-FPM reconstruction, which increases the sampling rate of the central low-middle frequency region while ensuring the spectrum bandwidth to cover high Fourier frequency spectrum. The result shows that the DHI-FPM method has clearer background and higher fidelity compared with the single-height illumination method.
Fig. 4. Reconstruction results of different illumination heights under noisy conditions. (a1)-(a2) The ground truth HR amplitude and phase images, (a3) LR image acquired with central LED unit illumination, (b1)-(b4)&(c1)-(c4)&(d1)-(d4) the reconstructed intensity, phase, sampling pattern and frequency spectrum when the illumination height is 40 mm, 90 mm and double heights, respectively.
To quantitatively evaluate the performance of the DHI-FPM system, the root mean square error (RMSE) is used here to evaluate the reconstructed quality of the images. The RMSE is defined as
Fig. 5. Comparison of reconstruction quality performance with illumination height is 40 mm, 90 mm and double heights, respectively. (a) The RMSE of intensity images versus the iteration number, (b) the RMSE of phase images versus the iteration number.
It is worth to note that the single-height method with more LEDs can also improve image quality. However, assuming the same spectral bandwidth (imaging resolution) and sufficient overlapping ratio, if the comparable enhancement effect is expected to be achieved, the single-height method requires more LEDs to be lighted than the double-height method through the simulations with noise.
In order to further analyze the physical insight of DHI-FPM to enhance image reconstruction, the overlapping times and the overlapping ratio of the sub-spectrums are introduced to illustrate the comparison between the single-height method and our proposed method. The results are shown in Fig. 6. Figures 6(a1)-(c1) show the sampling pattern when the illumination height is 40 mm(red circle), 90 mm(blue dot) and double heights, respectively. Figures 6(a2)-(c2) indicate the overlap of the sub-spectrums correspondingly. The value represents the overlapping times, which is shown in Figs. 6(a3)-(c3) along the diagonal yellow line of Figs. 6(a2)-(c2). It is obvious that the DHI method offers more redundant information than the single-height method. After that, Figs. 6(a4)-(c4) show the magnified sub-apertures along the white line of Figs. 6(a2)-(c2). The curves represent the variation tendency of the overlapping ratio with the Fourier frequency. The overlapping ratio is calculated as the normalized overlapped area of the adjacent sub-spectrums. For the DHI data, the two sequences of the sub-spectrums related with two heights are combined and treated as a series of data to be dealt with. It can be seen that the overlapping ratio is higher in the proposed method. The comparison reveals that the DHI-FPM method improves the redundancy of the data especially in the low-medium frequency regions, making it obtain better reconstruction and faster convergence.
Fig. 6. Comparison of the data redundancy between single-height and double-height method. (a1)-(c1)&(a2)-(c2) The sampling pattern and overlap of sub-spectrums when the illumination height is 40 mm, 90 mm and double heights, respectively, (a3)-(c3) the overlapping times along the diagonal yellow line of (a2)-(c2), (a4)-(c4) the variation tendency of the overlapping ratio along the white line of (a2)-(c2).
We further explore the influence of the choice of two illumination heights on the image quality in the DHI-FPM method. The general rule to choose the two height parameters is to further improve the sampling conditions in the low-middle frequency region with the larger-distance illumination. So one height is firstly fixed according to the requirement of the highest imaging resolution, and then another larger height is adjustable and optimized. In the actual experiment, the adjustable range of illumination height is limited by the space of the commercial optical microscopy between the sample stage and the bottom frame. Besides, it is worth to note that when d2/d1 is an integer, in the Fourier frequency domain some sub-spectrum produced by the LED at the first height would be overlapped with other sub-spectrum produced by the LED at another height totally. These data cannot offer more significant information.
We have performed simulations to illustrate the optimization of the second adjustable height. One illumination height denoted as d1 is set to be 40 mm to ensure a larger spectral bandwidth and higher imaging resolution. Another one denoted as d2 varies from 50 mm to 90 mm with the step of 1 mm. The simulation results are shown in Fig. 7, from which we can see that the best reconstruction results are obtained when the second illumination height is around 79 mm. Therefore, in the actual experiment, we should also preferably choose this height for illumination.
Fig. 7. RMSE of reconstructed images as a function of illumination height. (a) RMSE of intensity images versus d2, (b) RMSE of phase images versus d2.
4. Experimental results
Our system is based on a modified commercial microscope, consisting of a 4×, NA=0.1 microscopic objective, a top-mounted camera and a programmable LED array as the illumination source, the photo of which is shown in Fig. 8. A scientific-grade sCMOS camera with resolution of 2560×2160 and pixel size of 6.5 µm×6.5 µm is used for recording the raw images. The LED array can emit the quasi-monochromatic light, whose central wavelength is 625 nm. We use 11×11 units to provide multi-angle illumination in the practical experiment.
4.1 Reconstructed results of the edge area of the large full FOV under single-height illumination with wave vector correction
The first experiment is to reconstruct the central and edge area of the large full FOV under single-height illumination. The ovarian tissue sample is adopted, and the illumination height is 73 mm. We used a 4×, 0.1 NA objective lens to acquire the raw image sequence, the maximum illumination angle of LED unit produces the effective NA about 0.36, and the final synthetic NA is about 0.46. Figure 9 shows the reconstructed results of different segments. The solid areas indicate the three different regions selected, and the dashed areas indicate their reconstructed results. Figure 9(a) shows the full-field low resolution images with central LED unit illumination, and the corresponding central positions of the three sub-region images denoted as Figs. 9(a1)- 9(a3). Figures 9(b1)–9(b6) show the reconstructed results of Fig. 9(a1). Among them, Figs. 9(b1-b3) show the directly reconstructed intensity image, phase image and spectrum, respectively, and Figs. 9(b4)–9(b6) show the reconstructed results optimized by the wave vector correction method proposed in this paper. It can be seen that the image quality is improved by removing the irregular black lines in the intensity image and the wrapped phase distortion in the phase image. Comparing Figs. 9(b3) and 9(b6), it can be seen there is a shift of the directly reconstructed spectrum to the upper left direction, which corresponds to the relative position of Fig. 9(a1) in the whole FOV. Thus, the phase distortion in Fig. 9(b2) appears also in the tilted direction, and this error is corrected by using the proposed wave vector correction method. Similarly, Figs. 9(c1)–9(c6) show the reconstructed results of Fig. 9(a2), where the phase aberration extended in the horizontal direction is eliminated. The reconstructed results of the center region of the FOV are shown in Fig. 9(d), and there are no artifacts and aberrations in this result. The experimental results verify the effectiveness of the proposed method, which improves the reconstruction quality of the edge area. It opens the possibility of better large full FOV reconstruction.
Fig. 9. Comparison of reconstructed results before and after wave vector correction of different regions with full FOV of 4.2 mm×3.5 mm. (a) The raw image acquired with central LED unit illumination, and three sub-region images denoted as (a1-a3), (bc1-b6) & (c1-c6) & (d1-d3) reconstructed results of (a1)-(a3).
4.2 DHI-FPM experimental results for the central part of the full FOV
The second experiment is DHI-FPM results for the central part of the full FOV with the ovarian tissue sample, and the raw images are acquired at two illumination heights of 40 mm and 79 mm, respectively. The effective illumination NA is 0.58 and 0.34, and so the synthetic NA is 0.68 and 0.44. The single-height algorithm and double-height algorithm are used for reconstruction, and the experimental results are shown in Fig. 10. Figure 10(a) shows the LR image under the normal illumination captured by the camera and the selected small region of interest of 128 × 128 pixels. Figures. 10(b)–10(d) show the reconstructed results of the chosen region using the different methods. The yellow solid line frame area is the target region selected for comparison, and the dotted line frame is the enlargement of the target region. Comparing groups (b) and (c) of Fig. 10, it can be seen that when the illumination height is 40 mm, the dark-field images are increased, which are susceptible to the noise. So the reconstructed image is more affected by the noise, and the distortion of the luteal cells occurs due to the insufficient sampling rate in the low-middle frequency region. When the illumination height is 79 mm, the reconstructed image has a cleaner background because the LR images have relatively more bright field images [42]. But at the same time, the image resolution is sacrificed due to the narrow spectral bandwidth, resulting in blurred details of the luteal cells of the ovarian tissue. The group in Fig. 10(d) shows that the double-height results are significantly better, with clearer cell outlines, abundant cell details, and significant reduction of the number of iterations. From the intensity outline of the two cells, indicated by the two arrows, the DHI method present sharper border. The calculation time required for reconstruction is 17.564 s, 17.253 s and 16.981 s, respectively, when the convergence condition is fulfilled as descried in section 2.1. The reconstruction is performed by a computer with 8GB RAM, Intel i5-8400 CPU. In terms of data acquisition time, the DHI method presently needs more time to collect the raw images compared with one height method, meanwhile for the reconstruction process, the time is almost the same due to the improvement of the iterative efficiency. In addition, our proposed method produces the better quality of the reconstructed image without increasing the computation time.
Fig. 10. Comparison of single-height and double-height reconstructed results of ovarian tissue. (a) Full-field image acquired with central LED unit illumination, (b1-b2)&(c1-c2) the reconstructed results with single-height approach, (d1-d2) the reconstructed results with double-height approach.
4.3 Reconstructed results by DHI-FPM for the whole area of the large FOV with wave vector correction
To further demonstrate the validity of the proposed method, the onion apical tissue is selected to experimentally check the reconstruction for the central and edge region of the large full FOV under the double-height illumination. The experimental results are shown in Fig. 11. The yellow solid line areas are the target regions selected for comparison, and the dotted line areas are the enlargement of the target regions. Figures 11(b1)–11(b6) shows the reconstructed results of the central region of the FOV. When the illumination height is 40 mm, the spectrum bandwidth increases, which improves the imaging resolution. However, due to the insufficient overlapping ratio in the low-middle frequency region, the cell contour is distorted, accompanied by water-ripple artifacts in the intensity image, and the background of the phase image is uneven due to the influence of the noise. When the illumination height is 79 mm, the central low-middle frequency region is densely sampled making the outline of the cells clear, and the background of the reconstructed image is clean due to more bright-field images with less noise influence. But at the same time, the imaging resolution is also reduced due to the narrow spectrum bandwidth, resulting in indistinguishable details such as the cytoderm. The reconstructed results using the double-height algorithm proposed in this paper is less affected by noise, with clear contours, abundant details and clean background. Meanwhile the reconstruction accuracy and efficiency are greatly improved. Figures 11(c1)–11(c6) shows the reconstructed results of the edge region of the large full FOV. It can be seen clearly that the reconstructed results obtained by DHI-FPM are significantly better than those of the single-height illumination. It reduces the water ripple-like artifacts and background inhomogeneity without sacrificing resolution. To quantitatively evaluate the performance of the DHI-FPM, we choose a relatively flat area (blue rectangle area) to calculate its standard deviation (SD). The standard deviation under the double-height illumination is smaller than that under the single-height illumination, which shows that the background of the DHI-FPM is smoother and has better ability to adapt to noise.
Fig. 11. Comparison of reconstruction results of onion root tip tissue by single-height and double-height methods, with full FOV of 4.2 mm×3.5 mm. (a1-a2) Raw image with central LED unit illumination, (b1-b6) reconstruction results of the center region of the FOV, (c1-c6) reconstruction results of the edge region of the FOV.
5. Conclusions
In this paper, we firstly analyze the influence of the traditional LED array on the reconstruction results corresponding to the spectrum distribution with different illumination heights, and propose a convenient and feasible method to improve the performance, termed DHI-FPM. The LED array is moved from one plane to another to achieve the sample illuminations at two different heights and the low-resolution intensity images are totally utilized to reconstruct the high-resolution complex amplitude image. The method has three advantages: Firstly, it uses a traditional LED array with fixed number and spacing as the illumination light source without additional design and processing, reduce the difficulty of illumination source manufacturing and usage. Secondly, the illumination at two different heights optimizes the sampling pattern in the frequency domain, which can better capture the signal distribution of the samples in the Fourier domain and suppress the raster grid artifact problem. Thirdly, it increases the overlapping times and the overlapping ratio in the low-middle frequency region while maintaining the spectrum bandwidth, and the sampling rate can be changed by continuously adjusting the height, which is more flexible and convenient. Moreover, the double-height illumination method can be extended to triple-height or even multi-height to improve the Fourier domain sampling conditions, thus further improving the reconstruction image quality. In addition, we propose a wave vector correction method for removing the reconstruction artifacts in the edge region. Simulations and experiments have verified that the proposed method can not only effectively improve the reconstruction accuracy and reconstruction efficiency but also has strong robustness to the noise, having prospective application in the biomedical microscopic imaging. In order to reduce the data acquisition time, it is hopeful that the number of the lighted LED units would be decreased further by using some advanced computational imaging techniques in the next work, such as the deep learning or compressed sensing methods.
Funding
National Natural Science Foundation of China (62075001, 62175004); Science Foundation of Education Commission of Beijing (KZ202010005008).
Acknowledgement
The authors would like to thank Ms. Honghong Wang for her fundamental research work.
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. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013). [CrossRef]
2. P. Konda, L. Loetgering, and R. Horstmeyer, “Fourier ptychography: current applications and future promises,” Opt. Express 28(7), 9603–9629 (2020). [CrossRef]
3. A. Pan, C. Zuo, and B. Yao, “High-resolution and large field-of-view Fourier ptychographic microscopy and its applications in biomedicine,” Rep. Prog. Phys. 83(9), 096101 (2020). [CrossRef]
4. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004). [CrossRef]
5. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009). [CrossRef]
6. L. Yi, C. Zhen, and Y. Han, “Electron ptychography of 2D materials to deep sub-angstrom resolution,” Nature 559(7714), 343–349 (2018). [CrossRef]
7. D. Shapiro, Y. Yu, and T. Tyliszczak, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8(10), 765–769 (2014). [CrossRef]
8. L. Rong, F. Tan, and D. Wang, “High-resolution terahertz ptychography using divergent illumination and extrapolation algorithm,” Opt. Lasers Eng. 147, 106729 (2021). [CrossRef]
9. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture fourier holographic optical microscopy,” Phys. Rev. Lett. 97(16), 168102 (2006). [CrossRef]
10. V. Mico, Z. Zalevsky, P. Garcia-Martinez, and J. Garcia, “Synthetic aperture super resolution with multiple off axis holograms,” J. Opt. Soc. Am. A 23(12), 3162–3170 (2006). [CrossRef]
11. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32(10), 1737–1746 (1993). [CrossRef]
12. J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting white blood cells from a blood smear using Fourier ptychographic microscopy,” PLoS ONE 10(7), e0133489 (2015). [CrossRef]
13. A. Williams, J. Chung, X. Ou, G. Zheng, S. Rawal, Z. Ao, R. Datar, C. Yang, and R. Cote, “Fourier` ptychographic microscopy for filtration-based circulating tumor cell enumeration and analysis,” J. Biomed. Opt. 19(6), 066007 (2014). [CrossRef]
14. R. Horstmeyer, X. Ou, G. Zheng, P. Willems, and C. Yang, “Digital pathology with Fourier ptychography,” Comput. Med. Imaging Graph. 42, 38–43 (2015). [CrossRef]
15. K. Guo, S. Dong, and G. Zheng, “Optimization of sampling pattern and the design of Fourier ptychographic illuminator,” Opt. Express 23(5), 6171–6180 (2015). [CrossRef]
16. S. Sen, D. Desai, and M. Alsubaie, “Imaging photonic crystals using Fourier plane imaging and Fourier ptychographic microscopy techniques implemented with a computer controlled hemispherical digital condenser,” Opt. Commun. 383, 500–507 (2017). [CrossRef]
17. A. Pan, Y. Zhang, and K. Wen, “Subwavelength resolution Fourier ptychography with hemispherical digital condensers,” Opt. Express 26(18), 23119–23131 (2018). [CrossRef]
18. K. Guo, S. Dong, and G. Zheng, “Fourier ptychography for brightfield, phase, darkfield, reflective, multi-slice, and fluorescence imaging,” IEEE J. Sel. Top. Quantum Electron. 22(4), 77–88 (2016). [CrossRef]
19. X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22(5), 4960–4972 (2014). [CrossRef]
20. L. Bian, J. Suo, and G. Zheng, “Fourier Ptychographic Reconstruction Using Wirtinger Flow Optimization,” Opt. Express 23(4), 4856–4866 (2015). [CrossRef]
21. L. Bian, J. Suo, and J. Chung, “Fourier Ptychographic Reconstruction Using Poisson Maximum Likelihood and Truncated Wirtinger Gradient,” Sci. Rep. 6(1), 27384 (2016). [CrossRef]
22. C. Zuo, J. Sun, and C. Qian, “Adaptive Step-size Strategy for Noise-robust Fourier Ptychographic Microscopy,” Opt. Express 24(18), 20724–20744 (2016). [CrossRef]
23. X. Chen, Y. Zhu, M. Sun, D. Li, Q. Mu, and L. Xuan, “Apodized coherent transfer function constraint for partially coherent Fourier ptychographic microscopy,” Opt. Express 27(10), 14099–14111 (2019). [CrossRef]
24. Y. Zhang, Z. Cui, and J. Zhang, “Group-based Sparse Representation for Fourier Ptychography Microscopy,” Opt. Commun. 404(23), 55–61 (2017). [CrossRef]
25. Y. Zhang, P. Song, and Q. Dai, “Fourier Ptychographic Microscopy Using a Generalized Anscombe Transform Approximation of the Mixed Poisson-Gaussian Likelihood,” Opt. Express 25(1), 168–179 (2017). [CrossRef]
26. L. Hou, H. Wang, and M. Sticker, “Adaptive Background Interference Removal for Fourier Ptychographic Microscopy,” Appl. Opt. 57(7), 1575–1580 (2018). [CrossRef]
27. X. Chen, H. Cheng, Y. Wen, and Y. Feng, “Fourier ptychographic reconstruction based on augmented Lagrangian method and sparse approximations for phase and magnitude,” Appl. Opt. 60(9), 2471–2482 (2021). [CrossRef]
28. R. Claveau, P. Manescu, D. Fernandez-Reyes, and M. Shaw, “Structure-dependent amplification for denoising and background correction in Fourier ptychographic microscopy,” Opt. Express 28(24), 35438–35453 (2020). [CrossRef]
29. J. Sun, Q. Chen, Y. Zhang, and C. Zuo, “Efficient positional misalignment correction method for Fourier ptychographic microscopy,” Biomed. Opt. Express 7(4), 1336–1350 (2016). [CrossRef]
30. A. Pan, Y. Zhang, T. Zhao, Z. Wang, D. Dan, M. Lei, and B. Yao, “System calibration method for Fourier ptychographic microscopy,” J. Biomed. Opt. 22(09), 1–11 (2017). [CrossRef]
31. J. Liu, Y. Li, W. Wang, H. Zhang, Y. Wang, J. Tan, and C. Liu, “Stable and robust frequency domain position compensation strategy for Fourier ptychographic microscopy,” Opt. Express 25(23), 28053–28067 (2017). [CrossRef]
32. R. Eckert, Z. F. Phillips, and L. Waller, “Efficient illumination angle self-calibration in Fourier ptychography,” Appl. Opt. 57(19), 5434–5442 (2018). [CrossRef]
33. J. L. Zhang, X. Tao, and P. Sun, “A positional misalignment correction method for Fourier ptychographic microscopy based on the quasi-Newton method with a global optimization module,” Opt. Commu. 452, 296–305 (2019). [CrossRef]
34. A. Zhou, W. Wang, and N. Chen, “Fast and robust misalignment correction of Fourier ptychographic microscopy for full field of view reconstruction,” Opt. Express 26(18), 23661–23674 (2018). [CrossRef]
35. T. Nguyen, Y. Xue, Y. Li, L. Tian, and G. Nehmetallah, “Deep learning approach for Fourier ptychography microscopy,” Opt. Express 26(20), 26470–26484 (2018). [CrossRef]
36. S. Jiang, K. Guo, J. Liao, and G. Zheng, “Solving Fourier ptychographic imaging problems via neural network modeling and TensorFlow,” Biomed. Opt. Express 9(7), 3306–3319 (2018). [CrossRef]
37. J. Zhang, T. Xu, Z. Shen, Y. Qiao, and Y. Zhang, “Fourier ptychographic microscopy reconstruction with multiscale deep residual network,” Opt. Express 27(6), 8612–8625 (2019). [CrossRef]
38. Y. Cheng, M. Strachan, Z. Weiss, M. Deb, D. Carone, and V. Ganapati, “Illumination pattern design with deep learning for single-shot Fourier ptychographic microscopy,” Opt. Express 27(2), 644–656 (2019). [CrossRef]
39. J. Zhang, X. Tao, L. Yang, R. Wu, P. Sun, C. Wang, and Z. Zheng, “Forward imaging neural network with correction of positional misalignment for Fourier ptychographic microscopy,” Opt. Express 28(16), 23164–23175 (2020). [CrossRef]
40. R. Wang, P. Song, S. Jiang, C. Yan, J. Zhu, C. Guo, Z. Bian, T. Wang, and G. Zheng, “Virtual brightfield and fluorescence staining for Fourier ptychography via unsupervised deep learning,” Opt. Lett. 45(19), 5405–5408 (2020). [CrossRef]
41. Y. Zhu, M. Sun, X. Chen, H. Li, Q. Mu, D. Li, and L. Xuan, “Single full-FOV reconstruction Fourier ptychographic microscopy,” Biomed. Opt. Express 11(12), 7175–7182 (2020). [CrossRef]
42. A. Pan, C. Zuo, Y. Xie, and B. Yao, “Vignetting effect in Fourier Ptychographic Microscopy,” Opt. Lasers Eng. 120, 40–48 (2019). [CrossRef]
