# Compressive Reconstruction Based on Sparse Autoencoder Network Prior for Single-Pixel Imaging

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## Abstract

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## 1. Introduction

- We proposed a novel compressed sensing reconstruction method with a sparse autoencoder network prior that can be directly applied in photon-counting compressed imaging systems. Compared with the traditional one-norm prior, this sparse autoencoder network prior has significant advantages in terms of reconstruction quality;
- We proposed the concept of multi-channel prior information, and our experiments demonstrated that reconstructing images under the constraints of multi-channel network priors could effectively solve the problems encountered with single-channel network priors. Good reconstruction results can be achieved regardless of the low or high measurement rate.

## 2. Compressive Reconstruction System for the Single-Pixel Imaging

## 3. Sparse Autoencoder Network Prior-Based Reconstruction Method

#### 3.1. Compressed Sensing

#### 3.2. Sparse Autoencoder Network

#### 3.3. Single-Pixel Imaging Based on the Sparse Autoencoder Network Prior

- The superiority of the network prior. It has been demonstrated in earlier works that neural networks themselves are a form of prior knowledge, and their different structures limit their ability to learn information. The image restoration ability constrained by the prior of the network is superior to many state-of-the-art, non-local, patch-based priors, such as the BM3D prior [17]. We replace one-norm prior with SAE prior to change the role of the prior from a sparsity constraint to a contour similarity constraint, making this process more interpretable;
- The powerful feature extraction ability of the sparse autoencoder network allows it to capture the most significant features in the training data. Using these features as prior information for image restoration is advantageous in obtaining more accurate results.

#### 3.4. The Sparse Autoencoder Network Training and Reconstruction Method

#### 3.4.1. The Sparse Autoencoder Network Training

#### 3.4.2. Reconstruction Method

Algorithm 1. The algorithm steps for solving with gradient method |

1: Initialization: ${x}^{0}$ |

2: for $k=\mathrm{1,2},...,K$ do |

3: $\nabla f({x}^{k})\approx \frac{f({x}^{k}+h)-f({x}^{k}-h)}{2h}$ |

4: while the following equation is satisfied to stop iterating: |

5: $\nabla f\left({x}^{k}\right)<loss\_max$ |

6: end(while) |

7: update ${x}^{k+1}={x}^{k}-\nabla f\left(x\right)\times leaning\_rate$ |

8: end(for) |

9: Output: ${x}^{k+1}$ |

## 4. Result and Discussion

#### 4.1. Implementation Details

#### 4.2. Experiment Result

#### 4.2.1. Comparison between Sparse Autoencoder (SAE) Network Prior and Other Methods

#### 4.2.2. Select the Optimal SAE Network Parameters and Prior Reconstruction Parameter λ

- Parameters of SAE Network

- 2.
- The prior balance parameter λ in the prior reconstruction

^{10}and gradually decreases to 10

^{3}with increasing training iterations. To enable the fidelity term and the prior to achieve a more synchronized gradient impact, we should select the λ value that allows for the prior constraint result2 to fit into this range. When λ = 0.5, result2 is at the scale of 10

^{6}, meeting our expectations.

#### 4.2.3. The Effects of Sampling Rate SR and Reconstruction Measurement Rate MR of SAE Prior on Image Reconstruction of Different Datasets

#### 4.2.4. The Effect of Multi-Channel Prior Experiment on Prior Reconstruction

#### 4.2.5. Application of Network Prior in Single-Photon Image Reconstruction Results in Laboratory Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The compressive reconstruction system we used for single-pixel imaging. (

**a**) The structure of the system. The object is imaged on the DMD after the illumination by the parallel light and then collected by PMT. The signal processed by FPGA is the measurement value Y, which is used to reconstruct the picture with the prior information. (

**b**) The relationship between photon pulse and light intensity.

**Figure 2.**The principle and structure of sparse autoencoder. The input image is reshaped to a column vector; xn is the input data, representing n-dimensional features of the input. h1, h2, and h3 are the hidden layers, which means the extraction and reconstruction of image features. The SAE_LOSS is the loss function we use in this experiment, consisting of two parts, MSELoss and sparsity penalty.

**Figure 3.**The feature extraction diagram of SAE network. We use two down-sampling layers to obtain the texture features and visualize the features through shape transformation.

**Figure 5.**The calculation structure of single-channel and multi-channel network priors. (

**a**) The structure of single-channel. We use the pre-trained SAE network to obtain the prior information D(x) and then initialize the image to be reconstructed x (img_h height and img_w width) as an all-zero vector (img_h* img_w height and 1 width). A is the measurement matrix, and Y is the measurement value obtained via this experiment. x is calculated using gradient descent method as function 5. (

**b**) The structure of multi-channel. We use two different SAE sampling rates to obtain the prior information.

**Figure 6.**The structure of the simulations. We use the calculated y = $A{x}_{0}$ as the measurement value, and the size of $A$ is changed along with the MR (4096*MR height and 4096 width). Then, we use the procedure mentioned in Section 3.4 to reconstruct the image.

**Figure 7.**Reconstruction results with different methods (MNIST). We use the image with the Chinese characters of our affiliation on it and reconstruct it using TVAL3 and norm-1 prior as comparison and then test the SAE prior under different SR.

**Figure 8.**Reconstruction results with different parameters using different datasets. We used MNIST and FACE datasets to train the SAE network and test the best performance parameters Rho_t and beta separately.

**Figure 9.**Reconstruction results with different λ using different datasets. We used MNIST and FACE datasets to train the SAE network and test the best performance parameter λ.

**Figure 15.**The gradient results of different MRs in laboratory experiments reconstruction process. We recorded the gradient data under different MRs and plotted them every 5 steps. (

**a**) The result of MR = 0.05. (

**b**) The result of MR = 0.1. (

**c**) The result of MR = 0.3. (

**d**) The result of MR = 0.4.

Prior | MR = 0.05 | MR = 0.1 | MR = 0.2 | MR = 0.3 | MR = 0.6 |
---|---|---|---|---|---|

TVAL3 | 10.82683 | 9.84631 | 15.41595 | 22.62640 | 44.17010 |

one-norm | 15.60288 | 15.83556 | 16.57922 | 17.34535 | 20.72902 |

SAE (0.05) | 15.63417 | 15.93219 | 16.71110 | 17.35355 | 19.37229 |

SAE (0.6) | 15.59581 | 15.85697 | 16.78925 | 17.55059 | 20.41493 |

Computational Complexity Estimation | Compute Complexity | Parameter Count | Training Epochs | Training Time (h) |
---|---|---|---|---|

SAE | 16.778 M | 16.788 M | 1000 | 0.025 |

$\underset{x}{Min}||Ax-y|{|}^{2}+\lambda \times ||D(x)-x|{|}^{2}$ | Around 50,000 iterations | 60 | 1.5 | |

$\underset{x}{Min}||Ax-y|{|}^{2}+\lambda \times ||x|{|}_{1}$ | Around 50,000 iterations | 80 | 2 |

Rho_t | 0.01 | 0.05 | 0.055 | 0.056 | 0.1 |
---|---|---|---|---|---|

PSNR | 20.27770 | 20.43110 | 20.52952 | 20.50590 | 20.41738 |

Beta | 0.1 | 0.15 | 0.2 | 0.23 | 0.25 |
---|---|---|---|---|---|

PSNR | 20.43130 | 20.44929 | 20.48450 | 20.44459 | 20.43981 |

Rho_t | 0.005 | 0.01 | 0.015 | 0.02 | 0.05 |
---|---|---|---|---|---|

PSNR | 18.67160 | 19.23764 | 18.67760 | 19.15409 | 18.95618 |

Beta | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
---|---|---|---|---|---|

PSNR | 19.08152 | 19.36060 | 19.37492 | 19.27794 | 19.15430 |

λ | 0.01 | 0.1 | 0.5 | 1 | 10 |
---|---|---|---|---|---|

PSNR | 17.32141 | 17.32242 | 17.32894 | 17.32025 | 17.31337 |

λ | 25 | 50 | 100 | 110 | 150 |
---|---|---|---|---|---|

PSNR | 17.11100 | 17.12348 | 17.14735 | 17.40971 | 17.24071 |

SR | MR = 0.05 | MR = 0.1 | MR = 0.3 | MR = 0.6 | MR = 0.9 |
---|---|---|---|---|---|

0.05 | 15.35845 | 15.74198 | 17.19696 | 20.58298 | 26.98725 |

0.1 | 15.38565 | 15.73773 | 17.19660 | 20.59198 | 27.01659 |

0.3 | 15.39108 | 15.73896 | 17.20087 | 20.61011 | 27.01370 |

0.6 | 15.38773 | 15.73296 | 17.20256 | 20.60886 | 27.01445 |

0.9 | 15.38594 | 15.73431 | 17.20620 | 20.60880 | 27.03663 |

SR | MR = 0.05 | MR = 0.1 | MR = 0.3 | MR = 0.6 | MR = 0.9 |
---|---|---|---|---|---|

0.05 | 15.39450 | 15.69398 | 17.17027 | 20.30837 | 25.02564 |

0.1 | 15.37207 | 15.75328 | 17.13487 | 20.13807 | 24.98679 |

0.3 | 15.38561 | 15.66410 | 17.24386 | 20.11817 | 24.93752 |

0.6 | 15.43811 | 15.70000 | 17.10307 | 20.18926 | 25.35408 |

0.9 | 15.40044 | 15.68401 | 17.31690 | 20.19478 | 25.22858 |

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**MDPI and ACS Style**

Zeng, H.; Dong, J.; Li, Q.; Chen, W.; Dong, S.; Guo, H.; Wang, H.
Compressive Reconstruction Based on Sparse Autoencoder Network Prior for Single-Pixel Imaging. *Photonics* **2023**, *10*, 1109.
https://doi.org/10.3390/photonics10101109

**AMA Style**

Zeng H, Dong J, Li Q, Chen W, Dong S, Guo H, Wang H.
Compressive Reconstruction Based on Sparse Autoencoder Network Prior for Single-Pixel Imaging. *Photonics*. 2023; 10(10):1109.
https://doi.org/10.3390/photonics10101109

**Chicago/Turabian Style**

Zeng, Hong, Jiawei Dong, Qianxi Li, Weining Chen, Sen Dong, Huinan Guo, and Hao Wang.
2023. "Compressive Reconstruction Based on Sparse Autoencoder Network Prior for Single-Pixel Imaging" *Photonics* 10, no. 10: 1109.
https://doi.org/10.3390/photonics10101109