# Noise Suppression in Compressive Single-Pixel Imaging

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## Abstract

**:**

## 1. Introduction

## 2. Compressive Imaging Theory

## 3. Multiplicative Noise

## 4. Additive Noise

#### 4.1. The Influence of Additive Noise

#### 4.2. Theoretical Explanations

#### 4.3. Experimental Verifications and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**). The schematic diagram for normalized ghost imaging (NGI). The light source (LS) is expanded through a beam expander (BE). A structured light is generated through a spatial light modulator (SLM) and projected on the object trough a project lens (PL). Then, the structured light is divided into two arms by a beam splitter (BS). The object beam is detected by a single-pixel detector (SPD${}_{1}$) after the modulation of the object (Obj). The intensity in reference arm is directly detected by another SPD (SPD${}_{2}$). L${}_{1}$–L${}_{2}$: lens; Pr: prism. (

**b**). Practical experiment arrangements for multiplicative noise (EPSON projector) and additive noise (DMD projector) experiments. The rough white paper in reference arm is employed to homogenize the reference light.

**Figure 2.**Experimental comparison between compressive single-pixel imaging (CSPI) and normalized CSPI (NCSPI) with 1.7% fluctuation and 40% artificial noise, respectively. Two classical reconstruction solvers, ${L}_{1}$-magic and TVAL3, are adopted to retrieve final results. The spatial resolution is set at $128\times 128$ pixels and 8000 times differential detections (48.8% sampling ratio) are realized to reconstruct every image.

**Figure 3.**(

**a**) The qualitative comparison for NCSPI and CSPI under different noise ratios using ${L}_{1}$-magic solver. The reconstruction resolution is set at $64\times 64$ pixels. In simulation, 2000 times measurements are carried out, which means the compression ratio is about 48.8%. (

**b**) The detailed reconstruction images and their SSIM indices under 1% and 3% noise ratios. NCSPI performs better in offsetting multiplicative noise and produces clearer images in both noise ratios.

**Figure 4.**Experiment results for ${L}_{1}$-magic and TVAL3 algorithms using either direct or differential detection. Different sampling ratios (25%, 50%, 75%, and 100%) are displayed in column. The experiment resolution is set as $128\times 128$ pixels.

**Figure 5.**Normalized mean square error (MSE) curves calculated for one-step iteration in (

**a**) Newton’s method and (

**b**) steepest descent method with different sampling ratio, respectively. It is noted that as MSE in 100% sampling ratio in (

**a**) is extremely large, MSE in 95% sampling ratio under direct detection is thus adopted to normalize the data.

**Figure 6.**(

**a**) The structural similarity index (SSIM) curves for reconstructed images at different sampling ratio under different noise ratios, the parameter $\u03f5$ for these results is set as 10. (

**b**) The detailed images reconstructed from different sampling ratios under 2% noise ratio. (

**c**) The SSIM curves reconstructed using different $\u03f5$ values under 1% additive noise ratio. (

**d**) The detailed reconstructed images at 50% sampling ratio. The reconstruction resolution is set at $64\times 64$ pixels.

**Figure 7.**Reconstruction results from two designed matrices. (

**a**) The comparison of reconstruction results from the designed zero-mean matrix ${\mathsf{\Phi}}_{1}$ and a random matrix; 1% additive noise is simulated and ${L}_{1}$-magic solver is employed. (

**b**) The comparison of reconstruction results from another designed matrix ${\mathsf{\Phi}}_{2}$ and a random matrix; 1% additive noise is simulated and TVAL3 solver is employed. The reconstruction resolution is set at $64\times 64$ pixels.

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**MDPI and ACS Style**

Li, X.; Qi, N.; Jiang, S.; Wang, Y.; Li, X.; Sun, B.
Noise Suppression in Compressive Single-Pixel Imaging. *Sensors* **2020**, *20*, 5341.
https://doi.org/10.3390/s20185341

**AMA Style**

Li X, Qi N, Jiang S, Wang Y, Li X, Sun B.
Noise Suppression in Compressive Single-Pixel Imaging. *Sensors*. 2020; 20(18):5341.
https://doi.org/10.3390/s20185341

**Chicago/Turabian Style**

Li, Xianye, Nan Qi, Shan Jiang, Yurong Wang, Xun Li, and Baoqing Sun.
2020. "Noise Suppression in Compressive Single-Pixel Imaging" *Sensors* 20, no. 18: 5341.
https://doi.org/10.3390/s20185341