# Adaptive Single Photon Compressed Imaging Based on Constructing a Smart Threshold Matrix

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principle and Realization of Single-Photon Compressed Imaging

## 3. The Construction of an Adaptive Measurement Matrix

## 4. Experimental Results and Discussion

#### 4.1. Effect of Measurement Times on Imaging Quality

#### 4.2. Effect of Reconstruction Algorithm on Imaging Quality

#### 4.3. Anti-Noise Ability of Adaptive Measurement Matrix

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Studer, V.; Bobin, J.; Chahid, M.; Mousavi, H.S.; Candes, E.; Dahan, M. Compressive fluorescence microscopy for biological and hyperspectral imaging. Proc. Natl. Acad. Sci. USA
**2012**, 109, 10136–10137. [Google Scholar] [CrossRef] [PubMed] - Becker, W.; Bergmann, A.; Hink, M.A.; König, K.; Benndorf, K.; Biskup, C. Fluorescence lifetime imaging by time-correlated single-photon counting. Microsc. Res. Tech.
**2004**, 63, 58–66. [Google Scholar] [CrossRef] [PubMed] - Pourmorteza, A.; Symons, R.; Sandfort, V.; Mallek, M.; Fuld, M.K.; Henderson, G.; Jones, E.C.; Malayeri, A.A.; Folio, L.R.; Bluemke, D.A. Abdominal Imaging with Contrast-enhanced Photon-counting CT: First Human Experience. Radiology
**2016**, 279, 239–245. [Google Scholar] [CrossRef] [PubMed] - Taguchi, K.; Iwanczyk, J.S. Vision 20/20: Single photon counting x-ray detectors in medical imaging. Med. Phys.
**2013**, 40, 100901. [Google Scholar] [CrossRef] [PubMed] - Michel, J.; Liu, J.; Kimerling, L.C. High-performance Ge-on-Si photodetectors. Nat. Photonics
**2010**, 4, 527–534. [Google Scholar] [CrossRef] - Wang, Z.; Bovik, A.C.; Sheikh, H.R.; Simoncelli, E.P. Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process.
**2004**, 13, 600–612. [Google Scholar] [CrossRef] [PubMed] - Namekata, N.; Sasamori, S.; Inoue, S. 800 MHz single-photon detection at 1550-nm using an InGaAs/InP avalanche photodiode operated with a sine wave gating. Opt. Express
**2006**, 14, 10043–10049. [Google Scholar] [CrossRef] [PubMed] - He, W.; Sima, B.; Cheng, Y.; Chen, W.; Chen, Q. Photon counting imaging based on GM-APD. Opt. Precis. Eng.
**2012**, 20, 1831–1837. [Google Scholar] [CrossRef] - Roth, J.M.; Murphy, T.E.; Xu, C. Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube. Opt. Lett.
**2002**, 27, 2076. [Google Scholar] [CrossRef] [PubMed] - Yu, W.K.; Liu, X.F.; Yao, X.R.; Wang, C.; Zhai, G.J.; Zhao, Q. Single-photon compressive imaging with some performance benefits over raster scanning. Phys. Lett. A
**2014**, 378, 3406–3411. [Google Scholar] [CrossRef] - Ji, S.; Xue, Y.; Carin, L. Bayesian Compressive Sensing. IEEE Trans. Signal Process.
**2008**, 56, 2346–2356. [Google Scholar] [CrossRef] [Green Version] - Averbuch, A.; Dekel, S.; Deutsch, S. Adaptive compressed image sensing using dictionaries. SIAM J. Imaging Sci.
**2012**, 5, 57–89. [Google Scholar] [CrossRef] - Aβmann, M.; Bayer, M. Compressive adaptive computational ghost imaging. Sci. Rep.
**2013**, 3, 1545. [Google Scholar] [CrossRef] - Donoho, D.L. Compressed sensing. IEEE Trans. Inf. Theory
**2006**, 52, 1289–1306. [Google Scholar] [CrossRef] - Hou, J.M.; Ning, H.E.; Ke, L.V. Image Fast Reconstruction Method Based on Compressive Sensing. Comput. Eng.
**2011**, 37, 215–216. [Google Scholar] [CrossRef] - Chen, S.S.; Donoho, D.L.; Saunders, M.A. Atomic decomposition by basis pursuit. SIAM Rev.
**2001**, 43, 129–159. [Google Scholar] [CrossRef] - Tropp, J.A.; Gilbert, A.C. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit. IEEE Trans. Inf. Theory
**2007**, 53, 4655–4666. [Google Scholar] [CrossRef] [Green Version] - Needell, D.; Vershynin, R. Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Found. Comput. Math.
**2009**, 9, 317–334. [Google Scholar] [CrossRef] - Blumensath, T.; Davies, M.E. Iterative hard thresholding for compressed sensing. Appl. Comput. Harmonic Anal.
**2009**, 27, 265–274. [Google Scholar] [CrossRef] [Green Version] - Thomas, B.; Davies, M.E. Normalized Iterative Hard Thresholding: Guaranteed Stability and Performance. IEEE J. Sel. Top. Signal Process.
**2010**, 4, 298–309. [Google Scholar] [CrossRef] [Green Version] - Blumensath, T. Accelerated iterative hard thresholding. Signal Process.
**2012**, 92, 752–756. [Google Scholar] [CrossRef] [Green Version] - Li, C.; Yin, W.; Jiang, H.; Zhang, Y. An efficient augmented Lagrangian method with applications to total variation minimization. Comput. Optim. Appl.
**2013**, 56, 507–530. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.; Liu, S.; Xiong, R.; Ma, S.; Zhao, D. Improved total variation based image compressive sensing recovery by nonlocal regularization. In Proceedings of the 2013 IEEE International Symposium on Circuits and Systems (ISCAS2013), Beijing, China, 19–23 May 2013; pp. 2836–2839. [Google Scholar] [CrossRef]
- Candes, E.J.; Tao, T. Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? IEEE Trans. Inf. Theory
**2006**, 52, 5406–5425. [Google Scholar] [CrossRef] [Green Version] - Candes, E.J.; Romberg, J.; Tao, T. Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information. IEEE Trans. Inf. Theory
**2006**, 52, 489–509. [Google Scholar] [CrossRef] - Candès, E.J. Compressive Sampling. In Proceedings of the International Congress of Mathematiciansm, Madrid, Spain, 22–30 August 2006; pp. 1433–1452. [Google Scholar] [CrossRef]
- Chen, T.; Li, Z.-W.; Wang, J.L.; Wang, B.; Guo, S. Imaging system of single pixel camera based on compressed sensing. Opt. Precis. Eng.
**2012**, 20, 2523–2530. [Google Scholar] [CrossRef] - Candes, E.J.; Tao, T. Decoding by linear programming. IIEEE Trans. Inf. Theory
**2005**, 51, 4203–4215. [Google Scholar] [CrossRef] [Green Version] - Candès, E.J.; Romberg, J.K.; Tao, T. Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math.
**2010**, 59, 1207–1223. [Google Scholar] [CrossRef] - Li, S.-T.; Wei, D. A Survey on Compressive Sensing. Acta Autom. Sin.
**2009**, 35, 1369–1377. [Google Scholar] [CrossRef] [Green Version] - Liu, X.F.; Yao, X.R.; Wang, C.; Guo, X.Y.; Zhai, G.J. Quantum limit of photon-counting imaging based on compressed sensing. Opt. Express
**2017**, 25, 3286–3296. [Google Scholar] [CrossRef] [PubMed]

**Figure 3.**Reconstitution of images measured with the Gaussian random matrix under compression ratio at (

**a**) 0.1, (

**b**) 0.2, (

**c**) 0.3, (

**d**) 0.4, (

**e**) 0.5, and reconstitution of images measured with adaptive measurement matrix under compression ratio at (

**f**) 0.1, (

**g**) 0.2, (

**h**) 0.3, (

**i**) 0.4, (

**j**) 0.5, the reconstruction algorithm is TVAL3 and the resolution of the image is $64\times 64$ for all.

**Figure 4.**PSNR of image reconstructed with number of measurement time from 0 to 1800 with a step of 20, which measured with the Gaussian random matrix and the adaptive measurement matrix. When the PSNR of both image is 39.6 dB, the image measured with the Gaussian random matrix requires 779 measurements, while the image measured with the adaptive measurement matrix requires only 184 measurements.

**Figure 5.**In the case of the sampling rate is 0.15, the image measured by Gaussian random matrix reconstructed by OMP (

**a**), IHT (

**b**), and TVAL3 (

**c**) algorithms. The resolution of the image is all $64\times 64$.

**Figure 6.**In the case of the sampling rate is 0.15, the image measured by adaptive measurement matrix reconstructed by OMP (

**a**), IHT (

**b**), and TVAL3 (

**c**) algorithms. The resolution of the image is all $64\times 64$.

**Figure 7.**In the case of the sampling rate is 0.4, the image measured by Gaussian random matrix reconstructed by OMP (

**a**), IHT (

**b**), and TVAL3 (

**c**) algorithms. The resolution of the image is all $64\times 64$.

**Figure 8.**In the case of the sampling rate is 0.4, the image measured by adaptive measurement matrix reconstructed by OMP (

**a**), IHT (

**b**), and TVAL3 (

**c**) algorithms. The resolution of the image is all $64\times 64$.

**Figure 9.**Image added Gaussian noise with a mean of 0 and variance of (

**a**) 0, (

**b**) 0.02, (

**c**) 0.04, (

**d**) 0.06, (

**e**) 0.08, and (

**f**) 0.1. The resolution of the images are all $64\times 64$.

**Figure 10.**When the sampling rate is 0.1, the image reconstructed from the original image measured by Gaussian random matrix added with a mean of 0 and variance of (

**a**) 0, (

**b**) 0.02, (

**c**) 0.04, (

**d**) 0.06, (

**e**) 0.08, and (

**f**) 0.1. The reconstructed algorithm is TVAL3. The resolution of the image is $64\times 64$ for all.

**Figure 11.**When the sampling rate is 0.1, the image reconstructed from the original image measured by adaptive measurement matrix added with a mean of 0 and variance of (

**a**) 0, (

**b**) 0.02, (

**c**) 0.04, (

**d**) 0.06, (

**e**) 0.08, and (

**f**) 0.1. The reconstructed algorithm is TVAL3. The resolution of the image is $64\times 64$ for all.

**Figure 12.**PSNR of image reconstructed from the original image added with Gaussian noise, which with a mean of 0 and different variance of 0.0–0.1 by using the adaptive matrix measure with the Gaussian random matrix and the adaptive matrix, the resolution of the image is all $64\times 64$, and the reconstructed algorithm is TVAL3.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shangguan, W.; Yan, Q.; Wang, H.; Yuan, C.; Li, B.; Wang, Y.
Adaptive Single Photon Compressed Imaging Based on Constructing a Smart Threshold Matrix. *Sensors* **2018**, *18*, 3449.
https://doi.org/10.3390/s18103449

**AMA Style**

Shangguan W, Yan Q, Wang H, Yuan C, Li B, Wang Y.
Adaptive Single Photon Compressed Imaging Based on Constructing a Smart Threshold Matrix. *Sensors*. 2018; 18(10):3449.
https://doi.org/10.3390/s18103449

**Chicago/Turabian Style**

Shangguan, Wentao, Qiurong Yan, Hui Wang, Chenglong Yuan, Bing Li, and Yuhao Wang.
2018. "Adaptive Single Photon Compressed Imaging Based on Constructing a Smart Threshold Matrix" *Sensors* 18, no. 10: 3449.
https://doi.org/10.3390/s18103449