Fast and stable Fourier ptychographic microscopy based on improved phase recovery strategy

Jiaxiong Luo; Haishu Tan; Hanbao Chen; Sicong Zhu; Jiancong Li; Ruofei Wu; Yanxiong Wu; Yanxiong Wu

doi:10.1364/OE.454615

1. Introduction

The optical microscope is an important tool for observing and exploring the microscopic world. It plays an important role in life sciences and clinical medical research [1,2]. Due to the diffraction limit and the limitations of lens manufacturing technology, there is a mutual restriction between a wide field-of-view (FOV) and high-resolution (HR) imaging in the traditional microscopic imaging system. This is a limitation for achieving large space-bandwidth product (SBP) imaging in current systems. To overcome this limitation, a common solution is to use a high numerical aperture (NA) objective lens in conjunction with a mechanical scanning device to achieve HR imaging with a wide FOV [3,4]. However, improvements in the SBP of this technology come at the expense of time resolution imaging and the introduction of image stitching errors that are difficult to eliminate [5,6]. Further, biological samples are generally transparent or semi-transparent, and the difference in its internal structure causes uneven refractive index distribution; thus, its main structural information is studied through phase imaging. However, limited by the sampling rate of current detectors, the internal structure and quantitative characteristics of the cells reflected in a phase image cannot be directly obtained by traditional bright-field microscopy imaging systems [7]. For this reason, Fourier ptychographic microscopy (FPM) imaging technology, which expands the traditional microscopic imaging system, was proposed [1]. FPM combines the advantages of phase recovery [8,9] and synthetic aperture [10–12], bypassing the inherent diffraction limit of the objective lens. The resolution of the final reconstructed image depends on the sum of the objective lens NA_obj and the maximum angle illumination NA_ill. In a typical FPM imaging platform, the microscope’s light source is replaced by a programmable light-emitting diode (LED) matrix. The LED matrix has the characteristics of low price, high degree of freedom, and can be used to provide plane-wave illumination with varying angles, which greatly reduces the hardware requirements of the system for HR imaging. A microscope system with a low NA is used to collect low-resolution (LR) images corresponding to different illumination angles. The LR images describe the different subspectral regions of the sample. Subsequently, the LR image spectrum is stitched in the frequency domain through a phase recovery algorithm and, finally, HR imaging with a wide FOV is realized [1,8].

Since the introduction of FPM imaging technology, various methods that improve the imaging performance of FPM have been proposed. These methods mainly focus on improving imaging systems and reconstruction algorithms [13,14]. The conventional FPM framework uses the alternating projection (AP) algorithm to reconstruct HR images. This widely used algorithm iterates alternately in the spatial domain and the Fourier domain, while imposing corresponding constraints on the obtained complex amplitude function to reconstruct the sample’s HR complex amplitude image. However, this method lacks robustness (often converging to a local optimum), and is sensitive to noise. In addition, the starting image is noisy and has a low signal-to-noise ratio because the LR image is obtained under high-angle illumination [15,16], which drowns or even distorts the high-frequency information of the image. This seriously affects the stability of the FPM algorithm and reduces the reconstructed image quality.

To improve the robustness of FPM image reconstruction and minimize the negative impact of noise, a number of studies have been conducted. These studies can be divided into two schemes: preprocessing of the image, and improving the convergence of the reconstruction algorithm. In practical applications, it is generally considered necessary to preprocess images. The threshold method is a common method for the preprocessing of dark-field (DF) images. It calculates the average intensity value by selecting several small areas of the DF image, which it uses as a fixed threshold to denoise the image [17]. However, the threshold calculated by this method is random and cannot reflect the noise level; therefore, it cannot effectively distinguish the background noise from the signal. To solve this problem, Hou et al. proposed an improved threshold noise reduction method that can provide a reliable background noise threshold for noise reduction and is applicable to any sampled data [15]. However, because the conventional FPM algorithm framework is used in the subsequent iterative recovery process, the method has a slow convergence speed and low reconstruction efficiency. Regarding schemes that eliminate the influence of noise by optimizing the global convergence of the algorithm, Bian et al. proposed the Wirtinger–Fourier ptychography scheme [16], which uses an iterative optimization framework that combines phase recovery and noise relaxation. Similarly, Zuo et al. proposed an adaptive step-size (AS) algorithm [18]. The algorithm adopts a reconstruction strategy of decreasing the step size when optimizing the search step size, which improves the accuracy of image reconstruction. This method effectively improves the robustness of the algorithm toward noise, and achieves better image reconstruction results. However, the long recovery time for some experimental data requires powerful processing units for general usage. Therefore, the development of an imaging method with a faster convergence speed and higher robustness that is built on the current FPM imaging platform is of great importance.

In this study, we propose an FPM reconstruction method based on the improved phase recovery strategy. The first step is to use the improved threshold denoising method in the image preprocessing stage, which does not need the prior knowledge of noise characteristics, and can be applied to the processing of any sample. The second step is to add adaptive adjustment factors in the subsequent iterative update process to optimize the update process of the spectrum function, so as to achieve rapid convergence and better reconstruction results. Since the iterative updating part of the proposed method concerns the development of the traditional alternating iterative algorithm, it maintains the advantages of a simple algorithmic framework and good scalability, and can be easily combined with other algorithms such as error correction algorithms to achieve better imaging performance. The effectiveness of the proposed method is verified by simulation and experiments, and it is demonstrated that the method has faster convergence speed and stronger robustness.

The remainder of this paper is organized as follows. In Section II, we introduce the conventional FPM algorithm framework, and the improvement of the algorithm is proposed. In Section III, we compare the simulation experiment results and actual experimental data to verify the effectiveness of the proposed method. Finally, we summarize and discuss the study in Section IV.

2. Materials and methods

The FPM imaging process comprises image acquisition and reconstruction. Fig. 1(a) shows the FPM imaging system model, which is contains a programmable LED matrix and microscope system with a low NA objective lens. When the center LED is illuminated vertically (as shown by the red arrow), it will be limited by the coherent transfer function of the microscope system. When the nth off-axis LED illuminates the sample with an oblique plane wave U_n = sinφ_n/λ (as shown by the blue arrow), the illumination numerical aperture is NA_illu= U_n/λ= sinφ_n, where, φ_n represents the lighting angle of the nth LED. Therefore, the illumination aperture provided by FPM is determined by the maximum incident angle of the LED array [19].

In addition, In Fig. 1, the part labeled “a1” shows the overlap of sub-apertures due to angular varying illumination in the Fourier domain, and part “a2” shows circles of different colors that correspond to the positions of the LED units. Fig. 1(b) depicts the image reconstruction result of FPM imaging. A series of acquired LR images are reconstructed to obtain the HR intensity and phase images of the sample. Before introducing the details of our method, it is necessary to briefly review the image processing of the FPM system.

Fig. 1. System setup#of FPM and reconstruction results. (a) FPM imaging system model. (a1) Overlap of sub-apertures in the Fourier domain. (a2) Shows the LED in different positions. (b) Image reconstruction result of FPM imaging.

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2.1 Principle of the conventional FPM

Traditional large-field HR imaging technology is based on mechanical scanning combined with FOV splicing technology. In contrast, the imaging process of FPM is to light up each independent LED unit in turn, irradiate the samples with plane waves with different incident angles emitted by LED units at different positions, obtain the corresponding LR images through the camera, and use them for subsequent image reconstruction. Owing to the irradiation of plane waves with different incident angles, it is equivalent to shifting the spectrum of the sample to the corresponding position on the back focal plane of the objective lens, and then shifting the spectral information from beyond the NA to within the NA of the objective lens. In addition, to avoid confusion in the final reconstruction result, it is necessary to ensure that the image acquisition sequence is consistent with the update sequence.

The conventional FPM reconstruction process is shown in Fig. 2. The specific recovery process can be described as follows [20]. 1) The HR spectrum of the sample is initialized. Often, the LR image obtained under a normal incident plane wave is upsampled to obtain the initial HR amplitude image. The phase image is initialized to zero to generate an initial estimate of the HR complex amplitude distribution, u₀(r), where r = (x, y) denotes the coordinates in the focal plane. 2) For a certain incident angle, the spectral information in the corresponding sub-aperture in the initial HR spectrum of the sample is intercepted by the pupil function, and the complex amplitude distribution of the LR target is generated by the inverse Fourier transform. 3) The LR image obtained under the corresponding LED illumination angle is used to replace the amplitude component of the target image, while the phase is unchanged to update the target complex amplitude image. 4) The spectrum of the updated target complex amplitude image is obtained by Fourier transform, and the spectral information in the corresponding sub-aperture in the object HR spectrum is then updated by the spectrum distribution difference of the target light field before and after the update. 5) Steps 2–4 are repeated for different lighting angles until the spectral information in all the sub-apertures has been updated, and the obtained spectrum is used as the initial solution for the next iteration. Finally, the entire process is repeated to perform further iterations until the termination conditions are met.

Fig. 2. Flowchart of the reconstruction algorithm.

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It is worth noting that because FPM is a series of intensity images collected in the spatial domain for phase recovery and splicing in the frequency domain, the sufficient overlap rate of adjacent sub-regions in the Fourier domain is very important to ensure data redundancy and convergence. According to Dong et al., the FPM requires an overlap rate of at least 35% to obtain a better reconstruction effect [21].

2.2 Improved thresholding method

Before performing the FPM iterative reconstruction, it is necessary to preprocess the raw image data to minimize the impact of image noise on the subsequent image reconstruction. The threshold method is a background noise reduction method commonly used in FPM data preprocessing; however, this method cannot obtain an accurate noise reduction threshold, and the noise reduction effect is greatly affected by the selected area. The noise reduction method is based on the improved threshold proposed by Hou et al. A weighting factor $\alpha $ is introduced by calculating the arithmetic mean intensity difference between the target image and the actual measured image to balance the noise reduction process. This denoising method can obtain a more accurate noise threshold, and can be easily applied to any sample data set. In this method, to maintain good noise reduction performance, it is recommended to use a weighting factor value in the range of 1–1.05 (we set the value of the weighting factor α to 1). For the nth DF image, the threshold is calculated as follows:

(1)$$Threshol{d^n} = \left\langle {I_m^n({\mathbf r})} \right\rangle - \left\langle {I_t^n({\mathbf r})} \right\rangle$$

where $I_m^n({\boldsymbol r} )$ and $I_t^n({\boldsymbol r} )$ are the average intensity values of the nth measured and target image, respectively. Subsequent noise reduction processing is performed by subtracting the noise threshold from the LR image:

(2)$$I_d^n(r) = I_m^n({\mathbf r}) - \alpha \cdot Threshol{d^n}$$

where $I_d^n({\boldsymbol r} )$ represents the nth LR image after noise reduction. This is then used to update the amplitude component of the target image in step 3 above, as shown in process ① in Fig. 2.

2.3 Proposed improvement scheme for FPM

The method proposed in this study is based on an improvement of the AP algorithm. By introducing an adaptive regulation factor β in the update process of the spectrum function, the update of the sample spectrum function can be balanced, maintaining good convergence performance and high efficiency. The specific reconstruction algorithm is described as follows.

In the FPM experiments, each LED can be used to provide plane wave illumination with different wave vectors [22]. The nth LED illumination angle is equivalent to a sample that is illuminated by the wave vector U_n = (U_xn, U_yn), denoting the coordinates in Fourier domain. The corresponding frequency spectrum sub-region of the sample can be extracted using the pupil function [i.e., ${\phi _\textrm{n}}({\mathbf u}) = {P_n}({\mathbf u}){S_n}({\mathbf u} - {{\mathbf U}_n})$]; the corresponding target light field function on the image surface is generated by applying the inverse Fourier transform (i.e., ${\Phi _\textrm{n}}({\mathbf r}) = {{\cal F}^{ - 1}}\{{{\phi_n}({\mathbf u})} \}$).

Next, the intensity constraint is applied to keep the phase information of the target light field unchanged. In addition, the amplitude component of the target light field is updated with the square root of the corresponding LR image after threshold noise reduction:

(3)$${\mathop \Phi \limits^ {\wedge}}_{n} ({\mathbf r}) = \sqrt {I_d^n({\mathbf r})} \frac{{{\Phi _n}({\mathbf r})}}{{|{{\Phi _n}({\mathbf r})} |}}$$

The spectrum of the updated target complex amplitude image is obtained through Fourier transform (i.e., $\phi _n^{\prime}({\boldsymbol u} )= F\{{{{\mathrm{\hat{\Phi }}}_n}({\boldsymbol r} )} \}$), where the spectral information in the corresponding sub-aperture in the HR sample spectrum is updated by updating the spectrum distribution difference $\mathrm{\Delta }{\phi _n}({\boldsymbol u} )= \phi _n^{\prime}({\boldsymbol u} )- {\phi _n}({\boldsymbol u} )$. Subsequently, the adaptive control factor β is introduced to balance the update process of the sample spectrum function, which can effectively improve the reconstructed image quality, as shown in process ② in Fig. 2. In this method, the best β value can be determined according to different image data. The β value is determined using the following formulas:

(4)$$\varepsilon = \left\langle {\sum\limits_{n = 1}^N {Threshol{d^n}} } \right\rangle$$

(5)$$\beta = 2 \cdot ({{{|{{P_n}({\mathbf u} + {{\mathbf U}_{\mathbf n}})} |}_{\max }}} )- \varepsilon$$

where $\varepsilon $ is the arithmetic mean of the thresholds of N images calculated by (4), and the best regulatory factor β value is calculated using (5). N denotes the number of LR images collected. Further, to minimize the impact of sudden changes in pupil function value on the spectrum update while considering the interaction between the spectrum and the pupil function, the current and maximum values of the pupil function are added during the spectrum update. The ratio is defined as follows:

(6)$${W_1} = \frac{{|{{P_n}({\mathbf u} + {{\mathbf U}_n})} |}}{{{{|{{P_n}({\mathbf u} + {{\mathbf U}_n})} |}_{\max }}}}$$

We offer a new and better spectral update function, which is expressed as follows:

(7)$${\textrm{S}_{n + 1}}({\mathbf u}) = {S_n}({\mathbf u}) + \gamma$$

(8)$$\gamma = (\beta - {W_1})\frac{{|{{P_n}^\ast ({\mathbf u} + {{\mathbf U}_n})} |}}{{{{|{{P_n}({\mathbf u} + {{\mathbf U}_n})} |}^2} + {\delta _1}}}\Delta {\phi _n}({\mathbf u})$$

where ${\delta _1}$ is a regularization parameter used to ensure numerical stability. We set ${\delta _1} = 1$ to represent the HR sample spectrum function obtained in the nth iteration, and ‘*’ indicates the complex conjugation operation. This process continues until all images in the sequence are used for updating; it represents the completion of a single iteration. The entire iterative process is repeated many times to obtain the optimal solution of the sample HR complex amplitude image. The pupil function remains unchanged throughout the iteration process (it is assumed that the pupil function is precisely known). In addition, to evaluate the convergence of the proposed method, we define the error metric as:

(9)$$E(k) = \sum\limits_n {\sum\limits_r {{{|{I_d^n({\mathbf r}) - {{|{\Phi _n^k({\mathbf r})} |}^2}} |}^2}} }$$

where $I_d^n(\mathbf{r})$ is the intensity distribution after noise reduction, and ${|{\Phi _\textrm{n}^k({\mathbf r})} |^2}$ indicates the target intensity distribution after the kth iteration. Finally, the sample spectrum is returned to the spatial domain by inverse Fourier transform to obtain the HR amplitude and phase distribution of the sample.

3. Experiment and results

3.1 Simulation

Before applying the proposed method to actual experimental data, first we validated the effectiveness of this method through simulation experiments; the setting of the simulation parameters is consistent with the existing FPM experimental platform. For illumination, we simulated a 21 × 21 LED matrix with a 2.5-mm pitch, placed 87.5 mm beneath the sample. The incident wavelength of LEDs is 626 nm, the NA of the 4× objective lens is 0.1, and the pixel size of the camera is 6.5 µm. A photograph of a camera operator, and an orthophoto of the business district of West Concord, MA, US with the same pixel size were used as the HR input intensity image and phase image, respectively, as shown in Figs. 3(a1) and (a2). We simulated and generated 441 LR images; each image corresponds to a plane-wave illumination at different angles. We then compared the imaging performance of the proposed method with two other common reconstruction algorithms, namely, the AP and AS algorithms. All simulation and experimental calculations in this study were carried out using MATLAB R2018a on a computer with an Intel i5-8500 CPU, 16 GB of memory, and a 64-bit Windows 10 operating system.

Fig. 3. Fourier ptychographic reconstructions using different approaches. (a1) and (a2) Input HR intensity and phase profiles of the simulated complex sample. (b1)–(d2) FPM reconstructions with AP algorithm, AS algorithm, and the proposed method (β = 1.95). (e) and (f) RMSE curves of the recovered HR intensity image and phase image with the three methods under different iterations.

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Figures 3(b1) – (d2) show the HR intensity and phase images recovered using the three methods. In addition to an intuitive comparison of the results, we used the root-mean-square error (RMSE) to evaluate the reconstructed image quality. Figs. 3(e) and (f) show the RMSE curves of the intensity and phase image reconstructed under different iterations. A comparison of the results shows that the HR image recovered by the AP algorithm has the worst quality in addition to serious artifacts in the background. In contrast, the method proposed in this study and the AS algorithm yielded better reconstruction results. More specifically, the RMSE values of the method proposed in this study are smaller than those of the AS algorithm (i.e., faster convergence conditions). It should be noted that in the simulation data, the calculated optimal control factor β value was 1.95. This evidence verifies the effectiveness of the proposed method for reconstructing HR complex amplitude images.

3.2 Experiment on real captured data

To verify the performance of the proposed method in practical experiments, we used the method on a public FPM data set (USAF 1951 resolution target) by Zuo et al. [18]. Furthermore, we compared the reconstruction results with existing recovery algorithms. The experimental setup follows the simulation parameters in Section 3.1, and it was calculated that the frequency overlap rate is 85.72%, meeting the requirements of the minimum overlap rate. In the experiment, the LED matrix was lit to illuminate the samples sequentially. A total of 441 corresponding LR images were collected during the experiment. The LR image corresponding to the center LED is upsampled as the initial spectrum amplitude, as shown in Fig. 4(a1). Fig. 4 shows the results obtained using the three different FPM reconstruction methods. The HR image reconstructed by the AP algorithm has the worst definition and the most turbid background, as shown in Fig. 4(b1). Fig. 4(b2) shows the HR image recovered by the AS algorithm. The image clarity has been greatly improved, and the line edge is sharper than that of the AP algorithm, but there are still some ring artifacts of local details. The AS algorithm stops automatically after 21 iterations, because it adopts an adaptive convergence strategy. Inspired by the simulation results, we set the regulatory factor calculated from (5) for the proposed method as β=1.9. This value is substituted into the subsequent iterative process to recover and obtain the best result, as shown in Fig. 4(d1). Fig. 4(a2) - (d2) display an enlarged view of the area shown in the colored squares in (a1) - (d1). The comparison results show that the proposed method not only recovers clear image details, but also improves the uniformity of the image background. Moreover, for the experimental data, the proposed method only requires 8 iterations to achieve the convergence condition, which is significantly more efficient compared to existing algorithms.

Fig. 4. Comparison of reconstruction results using the USAF data set in Ref. 18. (a1)–(d1) Reconstruction results with AP algorithm (converges in 15 iterations), AS algorithm (converges in 21 iterations), and the proposed approach (β = 1.9, converges in 8 iterations). (a2)–(d2) Corresponding zoom-in of the smallest features.

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To further evaluate the imaging performance of the proposed method, we compared the peak signal-to-noise ratio (PSNR) and the final error of the reconstruction results for the three methods. Table 1 shows that, compared with the other two methods, the proposed method has a faster convergence speed, higher PSNR, and smallest final reconstruction error.

Table 1. Comparison of PSNR, final error, and running time of AP algorithm, AS algorithm, and the proposed method.

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In addition, the above experiments used the best regulatory factor, which greatly improves the clarity and background uniformity of the final reconstructed HR image. To explore the impact of the regulatory factor on the proposed method, we repeated the experiment under different β values. The reconstructed images are presented in Figs. 5(a1) – (e1). All reconstruction processes were subject to eight iterations. Fig. 5(a2) – (e2) show an enlargement of the corresponding region of interest. By comparing the reconstruction results, we observe that different β values affect the clarity of some segments and the uniformity of the image background. For example, when the introduced regulatory factor is less than the optimal value (1.9 in this experiment), the image background artifacts increase gradually as $\beta $ decreases. In contrast, when the regulation factor is greater than the optimal value, the image artifacts disappeared, but the clarity of the reconstructed image was diminished. The optimal regulatory factors calculated by the proposed method have a good balance on the clarity and background uniformity of the reconstructed images.

Fig. 5. Reconstruction results of the proposed method with the corresponding regulatory factor ranging from 1.0 to 2.0. (a1)–(e1) are the reconstruction results with different regulatory factors (all iterations are 8 times). (a2)–(e2) shows the corresponding zoom-in of interest.

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Next, we conducted experiments on the USAF target public data set provided by Zheng et al. [14] to further verify the stability of the method when used to reconstruct experimental data. In the acquisition of the experimental data, the FPM experimental setup included a conventional microscope with a 2× objective lens with an NA of 0.1, and an illuminating light source consisting of a 32 × 32 color-programmable LED matrix with a pitch of 4 mm, which was placed 90.88 mm away from the sample. The red channel was selected for the experiment; the center wavelength of the red is 630 nm. In the experiment, the LED matrix with a center size of 15×15 was sequentially lit, and a total of 225 LR images, each corresponding to one LED, were captured. The frequency overlap rate is 78.01%. The Fourier transform of the upsampled image captured under the center LED illumination was selected as the initial estimate of the HR spectrum. The data set reconstruction results using different processing methods are shown in Fig. 6, where 6(a) shows the full-field LR image illuminated by the center LED, and 6(b) – (d) show the reconstruction results using AP, AS, and the proposed method, respectively. A small portion (150 × 150 pixels) is extracted from the corresponding full FOV image, marked with colored squares, as shown in Figs. 6(a1) – (d1). Compared with the raw image, the FPM reconstruction algorithms improve the resolution, but the quality of each reconstruction is different. For AP, the background of the HR image is uneven, and the edges are blurred in some sub-regions. For AS, the clarity of the HR image is further improved, but the image background has artifacts, and the features are hollow. In contrast, the proposed method eliminates the influence of noise, retains more image details, and obtains better reconstruction results. To further quantify and analyze the resolution characteristics of the reconstruction results, Figs. 6(a2) – (d2) show the intensity distribution along the dotted line in (a1) – (d1). The red curve indicates that the line pair in group 9 element 1 of the raw image cannot be distinguished. Other color curves show a certain line resolution. More specifically, the curve results in Fig. 6(d2) show that the image line reconstructed by the proposed method has a better edge and higher contrast. In conclusion, compared with the conventional FPM reconstruction algorithm, the HR image reconstructed by the proposed method has clearer details, a more uniform background, maintains good stability, and has good resolution when applied to different experimental datasets.

Fig. 6. Comparison of reconstruction results using the USAF data set. (a) FOV of the USAF target image [(a1) presents a zoom-in image of a sub-region of (a)]. Groups (b), (c) and (d) demonstrate the reconstructed results by AP, AS, and the proposed method (β = 1.95), respectively. (a2)–(d2) Pixel contrast curves of the intensity line traces corresponding to (a1)–(d1).

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Lastly, we compared the proposed method with the other two algorithms applied on a biological sample (i.e., a blood smear) provided by Zheng et al. A full FOV image of the sample is shown in Fig. 7(a). The FPM experimental setup is consistent with the above description, except that in the acquisition of experimental data, a 15 × 15 blue LED (with a central wavelength of 475 nm) was used for illuminating samples placed on a glass slide with a thickness of 1 mm and a refractive index of 1.52; a total of 225 LR images, each corresponding to one illumination angle, were captured. In the process of FPM image reconstruction, we used three different algorithms for all 225 LR images, until the iteration mets the convergence condition. The selected area is indicated by the red square in the FOV image (120 × 120 pixels), as shown in Fig. 7(b). Fig. 7(c1) and (d1) show the amplitude and phase distribution reconstructed by the AP algorithm; the reconstruction results have obvious distortion and poor image quality. Figs. 7(c2) and (d2) show the results obtained using the AS algorithm. Although this method improves the clarity of the reconstructed image and obtains more detail than the AP algorithm, artifacts still exist in the image background. In contrast, using the proposed method, clear cell contours and richer details are recovered, artifacts are eliminated more effectively, and a cleaner image background is obtained, as shown in Figs. 7(c3) and (d3). The imaging performance of this method was further evaluated using the mean square error (MSE) and PSNR, and the results are listed in Table 2. It can be seen that compared with the other two methods, the HR intensity image and phase image recovered by the proposed method exhibit better performance in terms of the MSE and PSNR. In addition, a lower number of iterations is required, indicating that the method has a faster convergence speed. This further proves the ability of the proposed method to realize quantitative phase imaging.

Fig. 7. Experimental results of a blood smear sample. (a) LR FOV image of the specimen recorded by the camera. (b) Corresponding region of interest. Panel groups (c) and (d) show amplitude and phase images reconstructed using three methods, respectively.

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Table 2. Comparison of reconstruction performance of three different methods.

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4. Conclusions

In summary, this paper proposed an FPM reconstruction method based on an improved phase recovery strategy. This method is combined with the existing improved threshold noise reduction framework to eliminate noise more effectively and retain more useful information. The experimental results show that, compared with the conventional FPM reconstruction algorithm, the proposed method has a faster convergence speed and greater robustness. It not only recovers clear image details, but also ensures the uniformity of the image background. In addition, although the proposed method improves the efficiency of image reconstruction, the problem of unknown errors in the FPM imaging system is not discussed. For example, the uncertainty of LED array illumination brightness fluctuation and microscopic image differences can affect the stability of the FPM image reconstruction. This will be the direction of our future work.

Funding

National Natural Science Foundation of China (62075042, X190311UZ190); Foshan University (2021 graduate free exploration).

Acknowledgments

We sincerely acknowledge the open source data set of the blood smear and USAF data set provided by Zheng et al., and the USAF data set provided by Zuo et al.

Disclosures

The authors declare no conflicts of interest.

Data availability

The USAF and blood smear data set provided by Zheng et al. are available in Ref. [14]. The USAF data set provided by Zuo et al. are available in Ref. [18].

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	AP algorithm	AS algorithm	Proposed method
Iteration	15	21	8
PSNR	23.91	33.67	39.32
Final error	1.9147×10³	1.2351×10³	1.0264×10³
Running time (s)	56.86	70.65	33.24

	Recovery with AP algorithm		Recovery with AS algorithm		Recovery with proposed method
Reconstructed image	Amplitude	Phase	Amplitude	Phase	Amplitude	Phase
MSE (×10⁻²)	7.31	5.92	3.58	2.62	1.75	1.06
PSNR (dB)	24.69	25.01	33.68	29.24	49.01	46.77
Iteration (times)	15		10		8
Running time (s)	20.79		17.64		12.09

	AP algorithm	AS algorithm	Proposed method
Iteration	15	21	8
PSNR	23.91	33.67	39.32
Final error	1.9147×10³	1.2351×10³	1.0264×10³
Running time (s)	56.86	70.65	33.24

	Recovery with AP algorithm		Recovery with AS algorithm		Recovery with proposed method
Reconstructed image	Amplitude	Phase	Amplitude	Phase	Amplitude	Phase
MSE (×10⁻²)	7.31	5.92	3.58	2.62	1.75	1.06
PSNR (dB)	24.69	25.01	33.68	29.24	49.01	46.77
Iteration (times)	15		10		8
Running time (s)	20.79		17.64		12.09

Fast and stable Fourier ptychographic microscopy based on improved phase recovery strategy

Abstract

1. Introduction

2. Materials and methods

2.1 Principle of the conventional FPM

2.2 Improved thresholding method

2.3 Proposed improvement scheme for FPM

3. Experiment and results

3.1 Simulation

3.2 Experiment on real captured data

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (2)

Equations (9)

Optics Express