# Fourier Ptychographic Microscopy via Alternating Direction Method of Multipliers

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principles

#### 2.1. Problem Formulation

^{n}. The operation diag(x)y represents the element-by-element multiplication between two vectors, x and y. Matrix

**Q**

_{j}∈R

^{m×n}denotes the downsampling of the object in the Fourier domain corresponding to the j-th illumination angle. The pupil function p∈C

^{m}can be considered as a circular aperture that imposes constraint because of its finite size. To recover the HR object function from a stack of LR intensity images, we describe the reconstruction process as the following optimization problem:

^{n}denotes the object in the Fourier domain. Under regular circumstances, as the lens aberration is almost not completely known, the problem can be further transformed to:

#### 2.2. ADMM Solution

_{j}= diag(p)

**Q**

_{j}s and solve:

_{j}is the Lagrange multiplier corresponding to the intermediate variables q

_{j}. If we denote x

^{k}as the k-th estimated value of x, the update of variables is given according to the following order:

_{j}= λ

_{j}

_{/}α and the scaled-form ADMM iterations can be further expressed as:

^{tol}> 0 is the stopping tolerance and the normalized error metrics are defined as:

Algorithm 1: ADMM Solution for FPM (ADMM-FPM) |

Input:Q_{j}, I_{j}Output:s, p |

Initializes^{0}, p^{0}, ω^{0}_{j}, q^{0}_{j}fork = 1: Niter (iterations)for j = 1: N (different incident angles)update q ^{k}_{j} according to Equation (16)update s ^{k} according to Equation (13)update p ^{k} according to Equation (17)update ω ^{k}_{j} according to Equation (15)break when (21) and (22) are satisfiedendend |

## 3. Simulations

_{x}and σ

_{x}denote the mean intensity and standard deviation (the square root of variance), respectively, of the image vector x, and σ

_{xy}represents the correlation coefficient between two image vectors, x and y. Constants C

_{1}and C

_{2}are included to avoid instability when these statistics are very close to zero. The SSIM index ranges from 0 to 1 and higher value indicates that two images are of more similar structural information.

#### 3.1. Performance Comparison under Noiseless Conditions

^{0}= 1, β

^{0}= 1. It is noteworthy that the values of step size are not fixed but vary for each iteration. We adopt the concept of self-adaptive step size and update the step size according to the following rule [37]:

^{k}denotes the normalized error metric at the k-th iteration and ε should be a constant much less than 1. It is found that fixed ε = 0.01 usually works well to produce desired reconstruction results [37]. The introduction of this strategy allows the algorithm to approach a solution promptly at the early iterations and then gradually converge to a stable level as the step size decreases. Hence, the Gauss–Newton method we use as contrast is actually a modification of the original algorithm intended to improve robustness. Empirical evidence in terms of ptychography reveals that appropriately delaying the update time of reliable variables helps to achieve more stable reconstruction in mPIE algorithm. In our simulations, we are confident about the initial pupil, which is designed to update at the 15th illumination position while the object function is updated at the very beginning. Simultaneously, the momentum parameter for pupil update should accordingly be tuned smaller. Referring to [39], the parameters of mPIE algorithm are properly chosen as: α = 0.1, β = 0.8, γ = 1, η

_{obj}= 0.9, η

_{pupil}= 0.3, T

_{pupil}= 15.

#### 3.2. Performance Comparison under Noisy Conditions

^{−5}to σ = 0.01 in Figure 4(a1,a2). Here, the horizontal axis of coordinate system takes the form of logarithm in order to display the trend of curves more reasonably. Similar behavior can be observed in the graphs as the case of Gaussian noise that the curves extend downward with Poisson noise increasing. The curve of mPIE algorithm is always located at the bottom, implying its equally poor performance under Poisson noise. From somewhere between σ = 10

^{−4}and σ = 10

^{−3}, the Gauss–Newton method starts to outperform the ADMM method especially in phase reconstruction, which benefits from the self-adaptive step size strategy adopted in Gauss–Newton. At this range, the iteration curve of the Gauss–Newton method goes through a stair-type increase when the step size updates for each iteration [Figure 4(b1,b2)]. Generally speaking, we can safely conclude that our proposed ADMM method holds an advantage when σ < 10

^{−4}. Figure 4d gives one typical group of visual results when Poisson noise is valued σ = 10

^{−4}. For the mPIE algorithm, high-frequency details seem to be oversmoothed and many small-scale features are so blurred that they cannot be distinguished easily. The reconstruction results are also corrupted with strong artifacts, which leads to the uneven distribution of background. The Gauss–Newton method and ADMM method perform similarly from the perspective of visual perception, both with crosstalk between amplitude and phase information as well as artifact corruption, but the ADMM method works better in terms of numerical results. Overall, the removal of Poisson noise seems not so efficient as in Gaussian noise simulations, which we guess might stem from different characteristics of the two kinds of noise. To our knowledge, Poisson distribution can be approximated to Gaussian distribution when the arrival rate of photons is high enough, thus the performance of reconstruction methods under high-intensity Poisson noise should generally resemble that under Gaussian noise. Actually, Gaussian noise and Poisson noise respectively correspond to readout noise and photon shot noise in practical imaging systems. In our simulations, the former is “added to” the darkfield images, while the latter is “generated” based on the pixel intensity according to certain statistical property. Since the simulated Poisson noise added by the above-mentioned method does not certainly conform to strict Poisson contribution, the relationship of approximation cannot be established. Consequently, the performance of the ADMM method is quite different or can be even worse.

#### 3.3. Comparison and Analysis of Algorithm Efficiency

## 4. Experiments

## 5. Conclusions and Discussions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagram of FPM reconstruction process. The Fourier transform of the object is firstly downsampled by the illumination matrix, then confined by the pupil function and finally inverse Fourier transformed back to form LR intensity images in the spatial domain. The object is to be reconstructed by solving a blind phase retrieval problem.

**Figure 2.**Comparison of simulated reconstruction results under noiseless conditions. (

**a1**,

**a2**) Amplitude and phase SSIM curves for 100 iterations. (

**b1**–

**b3**) Simulated ground truth of amplitude, phase and pupil for FPM reconstruction. Corresponding visual reconstruction results: (

**c1**–

**c3**) Gauss–Newton method. (

**d1**–

**d3**) mPIE algorithm. (

**e1**–

**e3**) ADMM method.

**Figure 3.**Comparison of simulated reconstruction results under Gaussian noise. (

**a1**,

**a2**) Amplitude and phase SSIM curves as a function of Gaussian noise level. (

**b**) Visual reconstruction results of three algorithms under 70% Gaussian noise with SSIM values marked below the images. Magnified images for the religion of interest in phase reconstruction results are shown in (

**b1**–

**b4**).

**Figure 4.**Comparison of simulated reconstruction results under Poisson noise. (

**a1**,

**a2**) Amplitude and phase SSIM curves as a function of Poisson noise level. (

**b1**,

**b2**) Amplitude and phase SSIM curves as a function of iteration numbers under Poisson noise (σ = 10

^{−3}). (

**c1**,

**c2**) Amplitude and phase SSIM curves as a function of iteration numbers under Poisson noise (σ = 10

^{−4}). (

**d**) Visual reconstruction results of three algorithms under Poisson noise (σ = 10

^{−4}) with SSIM values marked below the images.

**Figure 5.**The curves of normalized error metric for three algorithms under (

**a**) 50% Gaussian noise and (

**b**) Poisson noise (σ = 10

^{−4}). All algorithms are designed to run for 100 times of iteration.

**Figure 6.**Comparison of experimental reconstruction results for a USAF resolution target under (

**a**) 20 dB gain and (

**b**) 30 dB gain. (

**c1**,

**c2**) Plot of the grayscale value of pixels along a certain horizontal section (marked in (

**a**,

**b**)) for amplitude reconstruction images under 20 dB and 30 dB gain, respectively.

**Figure 7.**Comparison results of experimental reconstruction for a biological sample (a pathological section of onion scale leaf epidermal cells). (

**a**,

**a1**) The full FOV image under 20 dB gain and magnified image for the region of interest. (

**b**) Reconstruction results under 20 dB gain. (

**c**) Reconstruction results under 30 dB gain.

**Table 1.**Classification of typical reconstruction algorithms for FPM (Adapted from Ref. [36]).

First-Order | Second-Order | |
---|---|---|

Sequential | G-S algorithm ${O}^{\left(i,l+1\right)}={O}^{\left(i,l\right)}-\frac{1}{{\left|P\right|}_{\mathrm{max}}^{2}}{\nabla}_{O}{f}_{A,l+1}\left({O}^{\left(i.l\right)}\right)$ | Sequential Gauss–Newton $\left[\begin{array}{c}{O}^{\left(i,l+1\right)}\\ {\overline{O}}^{\left(i,l+1\right)}\end{array}\right]=\left[\begin{array}{c}{O}^{\left(i,l\right)}\\ {\overline{O}}^{\left(i,l\right)}\end{array}\right]-\left[\begin{array}{cc}{Q}_{l}^{*}diag\left(\frac{P}{{\left|P\right|}_{\mathrm{max}}}\right){Q}_{l}& 0\\ 0& {Q}_{l}^{T}diag\left(\frac{P}{{\left|P\right|}_{\mathrm{max}}}\right){\overline{Q}}_{l}\end{array}\right]{\left({H}_{cc,l}^{A}\right)}^{-1}\left[\begin{array}{c}{\nabla}_{O}{f}_{A,l+1}\left({O}^{\left(i,l\right)}\right)\\ {\nabla}_{\overline{O}}{f}_{A,l+1}\left({O}^{\left(i,l\right)}\right)\end{array}\right]$ |

Global | Wirtinger flow ${O}^{\left(i+1\right)}={O}^{\left(i\right)}-{\alpha}^{\left(i\right)}{\nabla}_{O}{f}_{A,l}\left({O}^{\left(i\right)}\right)$ | Global Gauss–Newton $\left[\begin{array}{c}{O}^{\left(i+1\right)}\\ {\overline{O}}^{\left(i+1\right)}\end{array}\right]=\left[\begin{array}{c}{O}^{\left(i\right)}\\ {\overline{O}}^{\left(i\right)}\end{array}\right]-{\alpha}^{\left(i\right)}{\left({H}_{cc,l}^{A}\right)}^{-1}\left[\begin{array}{c}{\nabla}_{O}{f}_{A,l+1}\left({O}^{\left(i\right)}\right)\\ {\nabla}_{\overline{O}}{f}_{A,l+1}\left({O}^{\left(i\right)}\right)\end{array}\right]$ |

Parameter | α | β | γ | η |

Range | 0.5~1 | 1000 | 0.1~0.5 | 1 |

Approach | No Noise | Gaussian Noise | Poisson Noise | |||
---|---|---|---|---|---|---|

Iterations | Time | Iterations | Time | Iterations | Time | |

Gauss–Newton | 72 | 21.84 | 35 | 10.33 | 46 | 15.13 |

mPIE | 17 | 8.80 | 6 | 2.73 | 5 | 2.28 |

ADMM | 80 | 35.54 | 42 | 15.81 | 50 | 20.82 |

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**MDPI and ACS Style**

Wang, A.; Zhang, Z.; Wang, S.; Pan, A.; Ma, C.; Yao, B.
Fourier Ptychographic Microscopy via Alternating Direction Method of Multipliers. *Cells* **2022**, *11*, 1512.
https://doi.org/10.3390/cells11091512

**AMA Style**

Wang A, Zhang Z, Wang S, Pan A, Ma C, Yao B.
Fourier Ptychographic Microscopy via Alternating Direction Method of Multipliers. *Cells*. 2022; 11(9):1512.
https://doi.org/10.3390/cells11091512

**Chicago/Turabian Style**

Wang, Aiye, Zhuoqun Zhang, Siqi Wang, An Pan, Caiwen Ma, and Baoli Yao.
2022. "Fourier Ptychographic Microscopy via Alternating Direction Method of Multipliers" *Cells* 11, no. 9: 1512.
https://doi.org/10.3390/cells11091512