# Optimization of Sampling Mode in Macro Fourier Ptychography Imaging Based on Energy Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Principle

#### 2.1. Image Formation Model

**Figure 1.**Macroscopic Fourier ptychography technique physical model. From left to right: The laser emits quasi-monochromatic light that passes through the spatial filter to the converging lens. The light field interacts with the object and converges at the camera lens plane. The lens shortens the distance conditions required for Fraunhofer diffraction, and the intensity image is finally captured by the camera.

#### 2.2. Recovery Algorithm

- Take the average of a series of captured images as the initial estimated image. The estimated spectrum ${\widehat{U}}^{k}$ is obtained from its Fourier transform, and k represents the number of iterations.
- Set the aperture function as $O\left({x}^{\prime}-{a}_{i},{y}^{\prime}-{b}_{i}\right)$, $({a}_{i,}{b}_{i})$ represents the position of the i-th aperture. Multiply it with the estimated object spectrum ${\widehat{U}}^{k}$ to get the spectrum intercepted by the aperture, and inverse Fourier transform to get the spatial domain image ${\Phi}_{i}{}^{k}$ containing the phase information.

- 3.
- Replace the magnitudes of ${\Phi}_{i}{}^{k}$ with the magnitude of the corresponding observed images ${I}_{i}$ while preserving the phase:

- 4.
- Update the estimate of ${\widehat{U}}^{k}$ by solving the following regularized, least-squares problem:

- 5.
- Repeat steps 2–4 iterative operations until the preset number of iterations k is reached.
- 6.
- Calculate the final updated spectrum and restore the final high-resolution image by inverse Fourier transform.

## 3. Simulation Results and Analysis

#### 3.1. Irregular Sampling Mode

#### 3.2. Non-Uniform Sampling Mode

## 4. Experimental Results and Analysis

#### 4.1. Experimental Design

#### 4.2. Pre-Sampling

#### 4.3. Irregular Sampling Mode

#### 4.4. Non-Uniform Sampling Mode

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Different sampling modes, reconstruction images and analysis. (

**a1**–

**d1**) For different spectral sampling modes, the overlap ratio of adjacent apertures in the horizontal and vertical directions is 60%. (

**a2**–

**d2**) Reconstructed images and partial enlarged images corresponding to different sampling modes. (

**e**) Pixel intensity distribution map of the horizontal stripes at group numbers 2 and 3 intercepted at the drawn line. (

**f**) The structural similarity values K corresponding to different reconstructed images.

**Figure 3.**Different sampling modes, reconstructed images, and analysis. (

**a1**,

**c1**) Non-uniform sampling, where (

**a1**) has a low overlap rate in the center and cross-hair area, and a high overlap rate away from the center and cross-hair area. (

**c1**) Contrary to (

**a1**), (

**b1**,

**d1**) are uniform sampling modes. (

**a2**−

**d2**) are the corresponding image reconstruction results and partial enlarged images. (

**e**) Pixel intensity distribution map of the horizontal stripes at group numbers 2 and 3 intercepted at the drawn line. (

**f**) The structural similarity values K corresponding to different reconstructed images.

**Figure 4.**Different sampling modes, reconstructed images and analysis. (

**a1**,

**c1**) are non-uniform sampling, (

**a1**) has a low overlap rate in the central area and a high overlap rate away from the center. (

**c1**) is the opposite of (

**a1**). (

**b1**,

**d1**) are uniform sampling modes. (

**a2**−

**d2**) are the corresponding image reconstruction results and partial enlarged images. (

**e**) The structural similarity values K corresponding to different reconstructed images.

**Figure 5.**Schematic diagram of the experiment. From left to right: The 532 nm wavelength laser is passed through a spatial filter as a coherent light source. The focusing lens will make the transmitted light field of the object converge on the aperture plane and Fourier transform. The light field passes through an aperture of a certain size, and the lens inverse Fourier transforms the signal and focuses it onto the camera sensor. The electronically controlled high-precision displacement stage drives the camera to capture image information at different positions.

**Figure 6.**(

**a**) Images were taken at different positions. (

**b**) Image intensity sum corresponding to (

**a**) position, where the square brightness is proportional to the corresponding image intensity. (

**c**) The image was taken in the central position. (

**d**) The images captured by the large aperture are used as a reference to evaluate the quality of the reconstructed image.

**Figure 7.**Experimental sampling mode, imaging results and analysis. (

**a1**–

**d1**) Four different experimental sampling areas. The overlap of adjacent apertures is 70% in both the horizontal and vertical directions. (

**a2**–

**d2**) Imaging results and enlarged images corresponding to four different sampling modes. (

**e**) The pixel intensity distribution map of the horizontal stripes at group number 2 intercepted at the drawn line. (

**f**) The structural similarity values K corresponding to different reconstructed images.

**Figure 8.**Experimental sampling mode, imaging results, and analysis. (

**a1**) Uniform sampling with 70% overlap of adjacent apertures. (

**b1**,

**c1**) Non-uniform sampling. (

**b1**) The overlap rate is low in the central area, and the overlap rate gradually increases toward the surrounding area. (

**c1**) The overlap rate in the center area is high, and the overlap rate gradually decreases toward the surrounding area. (

**a2**−

**c2**) The corresponding reconstructed image results and enlarged images. (

**d**) The pixel intensity distribution map of the horizontal stripe at group number 2 intercepted at the drawn line. (

**e**) The structural similarity values K corresponding to different reconstructed images.

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**MDPI and ACS Style**

Jiang, R.; Shi, D.; Wang, Y.
Optimization of Sampling Mode in Macro Fourier Ptychography Imaging Based on Energy Distribution. *Photonics* **2023**, *10*, 321.
https://doi.org/10.3390/photonics10030321

**AMA Style**

Jiang R, Shi D, Wang Y.
Optimization of Sampling Mode in Macro Fourier Ptychography Imaging Based on Energy Distribution. *Photonics*. 2023; 10(3):321.
https://doi.org/10.3390/photonics10030321

**Chicago/Turabian Style**

Jiang, Runbo, Dongfeng Shi, and Yingjian Wang.
2023. "Optimization of Sampling Mode in Macro Fourier Ptychography Imaging Based on Energy Distribution" *Photonics* 10, no. 3: 321.
https://doi.org/10.3390/photonics10030321