# Comparison of Common Algorithms for Single-Pixel Imaging via Compressed Sensing

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Compressed Sensing SPI

_{0}-norm minimization [39,59]:

## 3. Selection of Measurement Matrix

#### 3.1. Random Measurement Matrix

#### 3.1.1. Gaussian Random Measurement Matrix

#### 3.1.2. Bernoulli Random Measurement Matrix

#### 3.2. Partial Orthogonal Measurement Matrix

#### 3.2.1. Partial Hadamard Matrix

_{N}is the $N\times N$ identity matrix and H

^{T}is the transpose of the matrix H. This measurement matrix is incoherent for most sparse signals and sparse dictionaries. However, since the order of the Hadamard matrix must satisfy 2n, $n=1,2,3,\cdots $, there are strict requirements for the dimension of the target signal, which limits its use in practice.

#### 3.2.2. Partial Fourier Matrix

#### 3.3. Semi-Deterministic Random Measurement Matrix

#### 3.3.1. Toeplitz and Circulant Matrix

_{i,j}= T

_{i+1,j+1}). A circulant matrix is a special form of the Toeplitz matrix. The elements in the first row of the matrix obey the same random distribution as the random measurement matrix.

#### 3.3.2. Sparse Random Matrix

## 4. Selection of the Reconstruction Algorithm

#### 4.1. Convex Optimization Algorithms

#### 4.1.1. Basis Pursuit

#### 4.1.2. Basis Pursuit Denoising/Least Absolute Shrinkage and Selection Operator

#### 4.1.3. Decoding by Linear Programming

_{1}-minimization problem:

#### 4.1.4. Dantzig Selector

#### 4.2. Greedy Algorithms

#### 4.2.1. Orthogonal Matching Pursuit

#### 4.2.2. Compressive Sampling Matching Pursuit/Subspace Pursuit

#### 4.2.3. Iterative Hard Thresholding

_{k}is the hard thresholding function and $\lambda $ denotes the step size.

#### 4.3. Non-Convex Optimization Algorithms

_{1}-norm with l

_{p}-norm where $\mathrm{p}\le 1$ [125,126,127].

#### 4.3.1. Iterative Reweighted Least Square Algorithm

_{p}objective function in (16) with a weighted l

_{2}-norm:

^{(n−1)}so that the objective in (17) is a first-order approximation ${\omega}_{i}={\left|{s}_{i}^{(n-1)}\right|}^{p-2}$. The solution of (17) can be given explicitly, giving the next iteration s

^{(n)}:

_{n}is the diagonal matrix with entries $1/{\omega}_{i}={\left|{s}_{i}^{(n-1)}\right|}^{2-p}$.

#### 4.3.2. Bayesian Compressed Sensing Algorithm

#### 4.4. Bregman Distance Minimization Algorithms

^{T}s can be calculated by fast transformation, which makes it possible to solve the unconstrained problem:

#### 4.5. Total Variation Minimization Algorithms

_{ij}denote the pixel in the i-th row and j column of an $n\times n$ image, and define the operators

_{ij}x can be interpreted as a kind of discrete gradient of the digital image $x$. The total variation of $x$ is simply the sum of the magnitudes of this discrete gradient at every point:

#### 4.5.1. Min-TV with Equality Constraints

_{ij}x is nonzero for only a small number of indices $ij$ for the image $x$, the image signal can be restored by solving the following equality constraint problems, which are called min-TV with equality constraints (TV-EQ):

#### 4.5.2. Min-TV with Quadratic Constraints

_{ij}x is non-zero) and the measured value of the single pixel is polluted by noise, the equality constraint problem of Equation (24) can be transformed into the following min-TV with quadratic constraints (TV-QC) problem:

#### 4.5.3. TV Dantzig Selector

#### 4.5.4. Total Variation Augmented Lagrangian Alternating Direction Algorithm

#### 4.6. Other Algorithms

## 5. Simulation

#### 5.1. Comparison of Measurement Matrix Performance

#### 5.2. Comparison of Reconstruction Algorithm Performance

## 6. Experiment

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Quantitative comparison of various reconstruction algorithms at different sampling rates. (

**a**–

**d**) are the quantitative comparison results of the convex optimization algorithm, greedy algorithm, TV minimization algorithm, and non-convex optimization algorithm.

**Figure A2.**The “Cameraman” images were reconstructed by various reconstruction algorithms at different sampling rates.

**Figure A3.**Quantitative comparison of various reconstruction algorithms with different noises. (

**a**–

**d**) are the quantitative comparison results of the convex optimization algorithm, greedy algorithm, TV minimization algorithm, and non-convex optimization algorithm.

**Figure A4.**The “Cameraman” image was reconstructed by various reconstruction algorithms with Gaussian noise.

## References

- Duarte, M.F.; Davenport, M.A.; Takhar, D.; Laska, J.N.; Sun, T.; Kelly, K.F.; Baraniuk, R.G. Single-pixel imaging via compressive sampling. IEEE Signal Process. Mag.
**2008**, 25, 83–91. [Google Scholar] [CrossRef] - Edgar, M.P.; Gibson, G.M.; Padgett, M.J. Principles and prospects for single-pixel imaging. Nat. Photonics
**2019**, 13, 13–20. [Google Scholar] [CrossRef] - Sen, P.; Chen, B.; Garg, G.; Marschner, S.R.; Horowitz, M.; Levoy, M.; Lensch, H.P. Dual photography. In ACM SIGGRAPH 2005 Papers; Association for Computing Machinery: New York, NY, USA, 2005; pp. 745–755. [Google Scholar]
- Bian, L.; Suo, J.; Situ, G.; Li, Z.; Fan, J.; Chen, F.; Dai, Q. Multispectral imaging using a single bucket detector. Sci. Rep.
**2016**, 6, 24752. [Google Scholar] [CrossRef] - Rousset, F.; Ducros, N.; Peyrin, F.; Valentini, G.; D’andrea, C.; Farina, A. Time-resolved multispectral imaging based on an adaptive single-pixel camera. Opt. Express
**2018**, 26, 10550–10558. [Google Scholar] [CrossRef] - Zhang, Z.; Liu, S.; Peng, J.; Yao, M.; Zheng, G.; Zhong, J. Simultaneous spatial, spectral, and 3D compressive imaging via efficient Fourier single-pixel measurements. Optica
**2018**, 5, 315–319. [Google Scholar] [CrossRef] - Qi, H.; Zhang, S.; Zhao, Z.; Han, J.; Bai, L. A super-resolution fusion video imaging spectrometer based on single-pixel camera. Opt. Commun.
**2022**, 520, 128464. [Google Scholar] [CrossRef] - Tao, C.; Zhu, H.; Wang, X.; Zheng, S.; Xie, Q.; Wang, C.; Wu, R.; Zheng, Z. Compressive single-pixel hyperspectral imaging using RGB sensors. Opt. Express
**2021**, 29, 11207–11220. [Google Scholar] [CrossRef] - Liu, A.; Gao, L.; Zou, W.; Huang, J.; Wu, Q.; Cao, Y.; Chang, Z.; Peng, C.; Zhu, T. High speed surface defects detection of mirrors based on ultrafast single-pixel imaging. Opt. Express
**2022**, 30, 15037–15048. [Google Scholar] [CrossRef] - Wang, Y.; Huang, K.; Fang, J.; Yan, M.; Wu, E.; Zeng, H. Mid-infrared single-pixel imaging at the single-photon level. Nat. Commun.
**2023**, 14, 1073. [Google Scholar] [CrossRef] - Watts, C.M.; Shrekenhamer, D.; Montoya, J.; Lipworth, G.; Hunt, J.; Sleasman, T.; Krishna, S.; Smith, D.R.; Padilla, W.J. Terahertz compressive imaging with metamaterial spatial light modulators. Nat. Photonics
**2014**, 8, 605–609. [Google Scholar] [CrossRef] - Lu, Y.; Wang, X.K.; Sun, W.F.; Feng, S.F.; Ye, J.S.; Han, P.; Zhang, Y. Reflective single-pixel terahertz imaging based on compressed sensing. IEEE Trans. Terahertz Sci. Technol.
**2020**, 10, 495–501. [Google Scholar] [CrossRef] - Li, W.; Hu, X.; Wu, J.; Fan, K.; Chen, B.; Zhang, C.; Hu, W.; Cao, X.; Jin, B.; Lu, Y. Dual-color terahertz spatial light modulator for single-pixel imaging. Light Sci. Appl.
**2022**, 11, 191. [Google Scholar] [CrossRef] [PubMed] - Gibson, G.M.; Sun, B.; Edgar, M.P.; Phillips, D.B.; Hempler, N.; Maker, G.T.; Malcolm, G.P.A.; Padgett, M.J. Real-time imaging of methane gas leaks using a single-pixel camera. Opt. Express
**2017**, 25, 2998–3005. [Google Scholar] [CrossRef] [PubMed] - 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, E1679–E1687. [Google Scholar] [CrossRef] [PubMed] - Radwell, N.; Mitchell, K.J.; Gibson, G.M.; Edgar, M.P.; Bowman, R.; Padgett, M.J. Single-pixel infrared and visible microscope. Optica
**2014**, 1, 285–289. [Google Scholar] [CrossRef] - Mostafavi, S.M.; Amjadian, M.; Kavehvash, Z.; Shabany, M. Fourier photoacoustic microscope improved resolution on single-pixel imaging. Appl. Opt.
**2022**, 61, 1219–1228. [Google Scholar] [CrossRef] - Durán, V.; Soldevila, F.; Irles, E.; Clemente, P.; Tajahuerce, E.; Andrés, P.; Lancis, J. Compressive imaging in scattering media. Opt. Express
**2015**, 23, 14424–14433. [Google Scholar] [CrossRef] - Deng, H.; Wang, G.; Li, Q.; Sun, Q.; Ma, M.; Zhong, X. Transmissive single-pixel microscopic imaging through scattering media. Sensors
**2021**, 21, 2721. [Google Scholar] [CrossRef] - Guo, Y.; Li, B.; Yin, X. Dual-compressed photoacoustic single-pixel imaging. Natl. Sci. Rev.
**2023**, 10, nwac058. [Google Scholar] [CrossRef] - Radwell, N.; Johnson, S.D.; Edgar, M.P.; Higham, C.F.; Murray-Smith, R.; Padgett, M.J. Deep learning optimized single-pixel LiDAR. Appl. Phys. Lett.
**2019**, 115, 231101. [Google Scholar] [CrossRef] - Huang, J.; Li, Z.; Shi, D.; Chen, Y.; Yuan, K.; Hu, S.; Wang, Y. Scanning single-pixel imaging lidar. Opt. Express
**2022**, 30, 37484–37492. [Google Scholar] [CrossRef] [PubMed] - Sefi, O.; Klein, Y.; Strizhevsky, E.; Dolbnya, I.P.; Shwartz, S. X-ray imaging of fast dynamics with single-pixel detector. Opt. Express
**2020**, 28, 24568–24576. [Google Scholar] [CrossRef] [PubMed] - He, Y.H.; Zhang, A.X.; Li, M.F.; Huang, Y.Y.; Quan, B.G.; Li, D.Z.; Wu, L.A.; Chen, L.M. High-resolution sub-sampling incoherent x-ray imaging with a single-pixel detector. APL Photonics
**2020**, 5, 056102. [Google Scholar] [CrossRef] - Salvador-Balaguer, E.; Latorre-Carmona, P.; Chabert, C.; Pla, F.; Lancis, J.; Tajahuerce, E. Low-cost single-pixel 3D imaging by using an LED array. Opt. Express
**2018**, 26, 15623–15631. [Google Scholar] [CrossRef] - Gao, L.; Zhao, W.; Zhai, A.; Wang, D. OAM-basis wavefront single-pixel imaging via compressed sensing. J. Light. Technol.
**2023**, 41, 2131–2137. [Google Scholar] - Gong, W. Performance comparison of computational ghost imaging versus single-pixel camera in light disturbance environment. Opt. Laser Technol.
**2022**, 152, 108140. [Google Scholar] [CrossRef] - Padgett, M.J.; Boyd, R.W. An introduction to ghost imaging: Quantum and classical. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2017**, 375, 20160233. [Google Scholar] [CrossRef] - Zhang, Z.; Ma, X.; Zhong, J. Single-pixel imaging by means of Fourier spectrum acquisition. Nat. Commun.
**2015**, 6, 6225. [Google Scholar] [CrossRef] - Chen, Y.; Yao, X.R.; Zhao, Q.; Liu, S.; Liu, X.F.; Wang, C.; Zhai, G.J. Single-pixel compressive imaging based on the transformation of discrete orthogonal Krawtchouk moments. Opt. Express
**2019**, 27, 29838–29853. [Google Scholar] [CrossRef] - Su, J.; Zhai, A.; Zhao, W.; Han, Q.; Wang, D. Hadamard Single-pixel Imaging Using Adaptive Oblique Zigzag Sampling. Acta Photonica Sin.
**2021**, 50, 311003. [Google Scholar] - Wang, Z.; Zhao, W.; Zhai, A.; He, P.; Wang, D. DQN based single-pixel imaging. Opt. Express
**2021**, 29, 15463–15477. [Google Scholar] [CrossRef] - Xu, C.; Zhai, A.; Zhao, W.; He, P.; Wang, D. Orthogonal single-pixel imaging using an adaptive under-Nyquist sampling method. Opt. Commun.
**2021**, 500, 127326. [Google Scholar] [CrossRef] - Kallepalli, A.; Innes, J.; Padgett, M.J. Compressed sensing in the far-field of the spatial light modulator in high noise conditions. Sci. Rep.
**2021**, 11, 17460. [Google Scholar] [CrossRef] [PubMed] - Shin, Z.; Chai, T.Y.; Pua, C.H.; Wang, X.; Chua, S.Y. Efficient spatially-variant single-pixel imaging using block-based compressed sensing. J. Signal Process. Syst.
**2021**, 93, 1323–1337. [Google Scholar] [CrossRef] - Sun, M.J.; Meng, L.T.; Edgar, M.P.; Padgett, M.J.; Radwell, N. A Russian Dolls ordering of the Hadamard basis for compressive single-pixel imaging. Sci. Rep.
**2017**, 7, 3464. [Google Scholar] [CrossRef] - Shin, J.; Bosworth, B.T.; Foster, M.A. Single-pixel imaging using compressed sensing and wavelength-dependent scattering. Opt. Lett.
**2016**, 41, 886–889. [Google Scholar] [CrossRef] [PubMed] - Donoho, D.L. Compressed sensing. IEEE Trans. Inf. Theory
**2006**, 52, 1289–1306. [Google Scholar] [CrossRef] - Candès, 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] - Wang, L.H.; Zhang, W.; Guan, M.H.; Jiang, S.Y.; Fan, M.H.; Abu, P.A.R.; Chen, C.A.; Chen, S.L. A Low-Power High-Data-Transmission Multi-Lead ECG Acquisition Sensor System. Sensors
**2019**, 19, 4996. [Google Scholar] [CrossRef] - Gibson, G.M.; Johnson, S.D.; Padgett, M.J. Single-pixel imaging 12 years on: A review. Optics Express
**2020**, 28, 28190. [Google Scholar] [CrossRef] - Qaisar, S.; Bilal, R.M.; Iqbal, W.; Naureen, M.; Lee, S. Compressive sensing: From theory to applications, a survey. J. Commun. Netw.
**2013**, 15, 443–456. [Google Scholar] [CrossRef] - Rani, M.; Dhok, S.B.; Deshmukh, R.B. A systematic review of compressive sensing: Concepts, implementations and applications. IEEE Access
**2018**, 6, 4875–4894. [Google Scholar] [CrossRef] - Arjoune, Y.; Kaabouch, N.; El Ghazi, H.; Tamtaoui, A. A performance comparison of measurement matrices in compressive sensing. Int. J. Commun. Syst.
**2018**, 31, e3576. [Google Scholar] [CrossRef] - Abo-Zahhad, M.M.; Hussein, A.I.; Mohamed, A.M. Compressive sensing algorithms for signal processing applications: A survey. Int. J. Commun. Netw. Syst. Sci.
**2015**, 8, 197–216. [Google Scholar] - Gunasheela, S.K.; Prasantha, H.S. Compressed Sensing for Image Compression: Survey of Algorithms. In Emerging Research in Computing, Information, Communication and Applications; Springer: Singapore, 2019; pp. 507–517. [Google Scholar]
- Marques, E.C.; Maciel, N.; Naviner, L.; Cai, H.; Yang, J. A review of sparse recovery algorithms. IEEE Access
**2018**, 7, 1300–1322. [Google Scholar] [CrossRef] - Bian, L.; Suo, J.; Dai, Q.; Chen, F. Experimental comparison of single-pixel imaging algorithms. J. Opt. Soc. Am. A
**2018**, 35, 78–87. [Google Scholar] [CrossRef] - Qiu, Z.; Zhang, Z.; Zhong, J. Comprehensive comparison of single-pixel imaging methods. Opt. Lasers Eng.
**2020**, 134, 106301. [Google Scholar] - Arjoune, Y.; Kaabouch, N.; El Ghazi, H.; Tamtaoui, A. Compressive sensing: Performance comparison of sparse recovery algorithms. In Proceedings of the 2017 IEEE 7th Annual Computing and Communication Workshop and Conference (CCWC), Las Vegas, NV, USA, 9–11 January 2017; IEEE: Piscataway, NJ, USA, 2017; pp. 1–7. [Google Scholar]
- Candès, E.J. Compressive sampling. In Proceedings of the International Congress of Mathematicians, Madrid, Spain, 22–30 August 2006; Volume 3, pp. 1433–1452. [Google Scholar]
- Baraniuk, R.; Davenport, M.A.; Duarte, M.F.; Hegde, C. An introduction to compressive sensing. Connex. e-Textb.
**2011**. [Google Scholar] - Donoho, D.; Tanner, J. Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Am. Math. Soc.
**2009**, 22, 1–53. [Google Scholar] [CrossRef] - Mallat, S. A Wavelet Tour of Signal Processing; Elsevier: Amsterdam, The Netherlands, 1999. [Google Scholar]
- Kovacevic, J.; Chebira, A. Life beyond bases: The advent of frames (Part I). IEEE Signal Process. Mag.
**2007**, 24, 86–104. [Google Scholar] [CrossRef] - Kovacevic, J.; Chebira, A. Life beyond bases: The advent of frames (Part II). IEEE Signal Process. Mag.
**2007**, 24, 115–125. [Google Scholar] [CrossRef] - Rubinstein, R.; Bruckstein, A.M.; Elad, M. Dictionaries for sparse representation modeling. Proc. IEEE
**2010**, 98, 1045–1057. [Google Scholar] [CrossRef] - Gribonval, R.; Nielsen, M. Sparse representations in unions of bases. IEEE Trans. Inf. Theory
**2003**, 49, 3320–3325. [Google Scholar] [CrossRef] - Candes, E.J. The restricted isometry property and its implications for compressed sensing. Comptes Rendus Math.
**2008**, 346, 589–592. [Google Scholar] [CrossRef] - Donoho, D.L.; Tsaig, Y. Fast Solution of ℓ0-Norm Minimization Problems When the Solution May Be Sparse. IEEE Trans. Inf. Theory
**2008**, 54, 4789–4812. [Google Scholar] [CrossRef] - Candes, E.; Romberg, J. Sparsity and incoherence in compressive sampling. Inverse Probl.
**2007**, 23, 969–985. [Google Scholar] [CrossRef] - Chen, Z.; Dongarra, J.J. Condition numbers of Gaussian random matrices. SIAM J. Matrix Anal. Appl.
**2005**, 27, 603–620. [Google Scholar] [CrossRef] - Zhang, G.; Jiao, S.; Xu, X.; Wang, L. Compressed sensing and reconstruction with bernoulli matrices. In Proceedings of the The 2010 IEEE International Conference on Information and Automation, Harbin, China, 20–23 June 2010; IEEE: Piscataway, NJ, USA, 2010; pp. 455–460. [Google Scholar]
- Duarte, M.F.; Eldar, Y.C. Structured compressed sensing: From theory to applications. IEEE Trans. Signal Process.
**2011**, 59, 4053–4085. [Google Scholar] [CrossRef] - Tsaig, Y.; Donoho, D.L. Extensions of compressed sensing. Signal Process.
**2006**, 86, 549–571. [Google Scholar] [CrossRef] - Zhang, G.; Jiao, S.; Xu, X. Compressed sensing and reconstruction with semi-hadamard matrices. In Proceedings of the 2010 2nd International Conference on Signal Processing Systems, Dalian, China, 5–7 July 2010; IEEE: Piscataway, NJ, USA, 2010; Volume 1, pp. 194–197. [Google Scholar]
- Yin, W.; Morgan, S.; Yang, J.; Zhang, Y. Practical compressive sensing with Toeplitz and circulant matrices. In Proceedings of the Visual Communications and Image Processing, Huangshan, China, 14 July 2010; Volume 7744, p. 77440K. [Google Scholar]
- Do, T.T.; Tran, T.D.; Gan, L. Fast compressive sampling with structurally random matrices. 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, Las Vegas, NV, USA, 30 March–4 April 2008; IEEE: Piscataway, NJ, USA, 2008; pp. 3369–3372. [Google Scholar]
- Do, T.T.; Gan, L.; Nguyen, N.H.; Tran, T.D. Fast and efficient compressive sensing using structurally random matrices. IEEE Trans. Signal Process.
**2011**, 60, 139–154. [Google Scholar] [CrossRef] - Sarvotham, S.; Baron, D.; Baraniuk, R.G. Compressed sensing reconstruction via belief propagation. Preprint
**2006**, 14. [Google Scholar] - Akçakaya, M.; Park, J.; Tarokh, V. Compressive sensing using low density frames. arXiv
**2009**, arXiv:0903.0650. [Google Scholar] - Gilbert, A.; Indyk, P. Sparse recovery using sparse matrices. Proc. IEEE
**2010**, 98, 937–947. [Google Scholar] [CrossRef] - Baron, D.; Sarvotham, S.; Baraniuk, R.G. Bayesian compressive sensing via belief propagation. IEEE Trans. Signal Process.
**2009**, 58, 269–280. [Google Scholar] [CrossRef] - Akçakaya, M.; Park, J.; Tarokh, V. A coding theory approach to noisy compressive sensing using low density frames. IEEE Trans. Signal Process.
**2011**, 59, 5369–5379. [Google Scholar] [CrossRef] - Baron, D.; Duarte, M.F.; Wakin, M.B.; Sarvotham, S.; Baraniuk, R.G. Distributed compressive sensing. arXiv
**2009**, arXiv:0901.3403. [Google Scholar] - Park, J.Y.; Yap, H.L.; Rozell, C.J.; Wakin, M.B. Concentration of measure for block diagonal matrices with applications to compressive signal processing. IEEE Trans. Signal Process.
**2011**, 59, 5859–5875. [Google Scholar] [CrossRef] - Li, S.; Gao, F.; Ge, G.; Zhang, S. Deterministic construction of compressed sensing matrices via algebraic curves. IEEE Trans. Inf. Theory
**2012**, 58, 5035–5041. [Google Scholar] [CrossRef] - Berinde, R.; Gilbert, A.C.; Indyk, P.; Karloff, H.; Strauss, M.J. Combining geometry and combinatorics: A unified approach to sparse signal recovery. In Proceedings of the 2008 46th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA, 23–26 September 2008; IEEE: Piscataway, NJ, USA, 2008; pp. 798–805. [Google Scholar]
- Calderbank, R.; Howard, S.; Jafarpour, S. Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property. IEEE J. Sel. Top. Signal Process.
**2010**, 4, 358–374. [Google Scholar] [CrossRef] - DeVore, R.A. Deterministic constructions of compressed sensing matrices. J. Complex.
**2007**, 23, 918–925. [Google Scholar] [CrossRef] - Nguyen, T.L.N.; Shin, Y. Deterministic sensing matrices in compressive sensing: A survey. Sci. World J.
**2013**, 2013, 192795. [Google Scholar] [CrossRef] - Amini, A.; Montazerhodjat, V.; Marvasti, F. Matrices with small coherence using p-ary block codes. IEEE Trans. Signal Process.
**2011**, 60, 172–181. [Google Scholar] [CrossRef] - Khajehnejad, M.A.; Dimakis, A.G.; Xu, W.; Hassibi, B. Sparse recovery of nonnegative signals with minimal expansion. IEEE Trans. Signal Process.
**2010**, 59, 196–208. [Google Scholar] [CrossRef] - Elad, M. Optimized projections for compressed sensing. IEEE Trans. Signal Process.
**2007**, 55, 5695–5702. [Google Scholar] [CrossRef] - Nhat, V.D.M.; Vo, D.; Challa, S.; Lee, S. Efficient projection for compressed sensing. In Proceedings of the Seventh IEEE/ACIS International Conference on Computer and Information Science (icis 2008), Portland, OR, USA, 14–16 May 2008; IEEE: Piscataway, NJ, USA, 2008; pp. 322–327. [Google Scholar]
- Wu, S.; Dimakis, A.; Sanghavi, S.; Yu, F.; Holtmann-Rice, D.; Storcheus, D.; Rostamizadeh, A.; Kumar, S. Learning a compressed sensing measurement matrix via gradient unrolling. In International Conference on Machine Learning; PMLR: New York, NY, USA, 2019; pp. 6828–6839. [Google Scholar]
- Wu, Y.; Rosca, M.; Lillicrap, T. Deep compressed sensing. In International Conference on Machine Learning; PMLR: New York, NY, USA, 2019; pp. 6850–6860. [Google Scholar]
- Islam, S.R.; Maity, S.P.; Ray, A.K.; Mandal, M. Deep learning on compressed sensing measurements in pneumonia detection. Int. J. Imaging Syst. Technol.
**2022**, 32, 41–54. [Google Scholar] [CrossRef] - Ahmed, I.; Khan, A. Genetic algorithm based framework for optimized sensing matrix design in compressed sensing. Multimed. Tools Appl.
**2022**, 81, 39077–39102. [Google Scholar] [CrossRef] - Pope, G. Compressive Sensing: A Summary of Reconstruction Algorithms. Master’s Thesis, ETH, Swiss Federal Institute of Technology Zurich, Department of Computer Science, Zürich, Switzerland, 2009. [Google Scholar]
- Siddamal, K.V.; Bhat, S.P.; Saroja, V.S. A survey on compressive sensing. In Proceedings of the 2015 2nd International Conference on Electronics and Communication Systems (ICECS), Coimbatore, India, 26–27 February 2015; IEEE: Piscataway, NJ, USA, 2015; pp. 639–643. [Google Scholar]
- Carmi, A.Y.; Mihaylova, L.; Godsill, S.J. Compressed Sensing & Sparse Filtering; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Hameed, M.A. Comparative Analysis of Orthogonal Matching Pursuit and Least Angle Regression; Michigan State University, Electrical Engineering: East Lansing, MI, USA, 2012. [Google Scholar]
- Santosa, F.; Symes, W.W. Linear inversion of band-limited reflection seismograms. SIAM J. Sci. Stat. Comput.
**1986**, 7, 1307–1330. [Google Scholar] [CrossRef] - Donoho, D.L.; Stark, P.B. Uncertainty principles and signal recovery. SIAM J. Appl. Math.
**1989**, 49, 906–931. [Google Scholar] [CrossRef] - Donoho, D.L.; Logan, B.F. Signal recovery and the large sieve. SIAM J. Appl. Math.
**1992**, 52, 577–591. [Google Scholar] [CrossRef] - Donoho, D.L.; Elad, M. Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization. Proc. Natl. Acad. Sci. USA
**2003**, 100, 2197–2202. [Google Scholar] [CrossRef] [PubMed] - Elad, M.; Bruckstein, A.M. A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inf. Theory
**2002**, 48, 2558–2567. [Google Scholar] [CrossRef] - Zhang, Y. Theory of compressive sensing via ℓ1-minimization: A non-rip analysis and extensions. J. Oper. Res. Soc. China
**2013**, 1, 79–105. [Google Scholar] [CrossRef] - Candes, E.J.; Wakin, M.B.; Boyd, S.P. Enhancing sparsity by reweighted ℓ1 minimization. J. Fourier Anal. Appl.
**2008**, 14, 877–905. [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] - Huggins, P.S.; Zucker, S.W. Greedy basis pursuit. IEEE Trans. Signal Process.
**2007**, 55, 3760–3772. [Google Scholar] [CrossRef] - Biegler, L.T. Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2010. [Google Scholar]
- Tibshirani, R. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B
**1996**, 58, 267–288. [Google Scholar] [CrossRef] - Fu, W.J. Penalized regressions: The bridge versus the lasso. J. Comput. Graph. Stat.
**1998**, 7, 397–416. [Google Scholar] - Maleki, A.; Anitori, L.; Yang, Z.; Baraniuk, R.G. Asymptotic analysis of complex LASSO via complex approximate message passing (CAMP). IEEE Trans. Inf. Theory
**2013**, 59, 4290–4308. [Google Scholar] [CrossRef] - Candes, E.J.; Tao, T. Decoding by linear programming. IEEE Trans. Inf. Theory
**2005**, 51, 4203–4215. [Google Scholar] [CrossRef] - Candes, E.; Romberg, J. l1-Magic: Recovery of Sparse Signals Via Convex Programming. Available online: www.acm.caltech.edu/l1magic/downloads/l1magic.pdf (accessed on 14 April 2005).
- Candes, E.; Tao, T. The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Stat.
**2007**, 35, 2313–2351. [Google Scholar] - Meenakshi, S.B. A survey of compressive sensing based greedy pursuit reconstruction algorithms. Int. J. Image Graph. Signal Process.
**2015**, 7, 1–10. [Google Scholar] [CrossRef] - Akhila, T.; Divya, R. A survey on greedy reconstruction algorithms in compressive sensing. Int. J. Res. Comput. Commun. Technol.
**2016**, 5, 126–129. [Google Scholar] - Tropp, J.A. Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inf. Theory
**2004**, 50, 2231–2242. [Google Scholar] [CrossRef] - Mallat, S.G.; Zhang, Z. Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process.
**1993**, 41, 3397–3415. [Google Scholar] [CrossRef] - Pati, Y.C.; Rezaiifar, R.; Krishnaprasad, P.S. Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In Proceedings of the 27th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 1–3 November 1993; IEEE: Piscataway, NJ, USA, 1993; pp. 40–44. [Google Scholar]
- DeVore, R.A.; Temlyakov, V.N. Some remarks on greedy algorithms. Adv. Comput. Math.
**1996**, 5, 173–187. [Google Scholar] [CrossRef] - Wen, J.; Zhou, Z.; Wang, J.; Tang, X.; Mo, Q. A sharp condition for exact support recovery with orthogonal matching pursuit. IEEE Trans. Signal Process.
**2016**, 65, 1370–1382. [Google Scholar] [CrossRef] - Wang, J. Support recovery with orthogonal matching pursuit in the presence of noise: A new analysis. arXiv
**2015**, arXiv:1501.04817. [Google Scholar] - 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] - Needell, D.; Vershynin, R. Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit. IEEE J. Sel. Top. Signal Process.
**2010**, 4, 310–316. [Google Scholar] [CrossRef] - Needell, D.; Tropp, J.A. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal.
**2009**, 26, 301–321. [Google Scholar] [CrossRef] - Dai, W.; Milenkovic, O. Subspace pursuit for compressive sensing signal reconstruction. IEEE Trans. Inf. Theory
**2009**, 55, 2230–2249. [Google Scholar] [CrossRef] - Blumensath, T.; Davies, M.E. Iterative thresholding for sparse approximations. J. Fourier Anal. Appl.
**2008**, 14, 629–654. [Google Scholar] [CrossRef] - Blumensath, T.; Davies, M.E. Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal.
**2009**, 27, 265–274. [Google Scholar] [CrossRef] - Blumensath, T.; Davies, M.E. Normalized iterative hard thresholding: Guaranteed stability and performance. IEEE J. Sel. Top. Signal Process.
**2010**, 4, 298–309. [Google Scholar] [CrossRef] - Chartrand, R. Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process. Lett.
**2007**, 14, 707–710. [Google Scholar] [CrossRef] - Wen, J.; Li, D.; Zhu, F. Stable recovery of sparse signals via lp-minimization. Appl. Comput. Harmon. Anal.
**2015**, 38, 161–176. [Google Scholar] [CrossRef] - Kanevsky, D.; Carmi, A.; Horesh, L.; Gurfil, P.; Ramabhadran, B.; Sainath, T.N. Kalman filtering for compressed sensing. In Proceedings of the 2010 13th International Conference on Information Fusion, Edinburgh, UK, 26–29 July 2010; IEEE: Piscataway, NJ, USA, 2010; pp. 1–8. [Google Scholar]
- Chartrand, R.; Staneva, V. Restricted isometry properties and nonconvex compressive sensing. Inverse Probl.
**2008**, 24, 035020. [Google Scholar] [CrossRef] - Chartrand, R.; Yin, W. Iteratively reweighted algorithms for compressive sensing. In Proceedings of the 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, Las Vegas, NA, USA, 30 March –4 April 2008; IEEE: Piscataway, NJ, USA, 2008; pp. 3869–3872. [Google Scholar]
- Wipf, D.P.; Rao, B.D. Sparse Bayesian learning for basis selection. IEEE Trans. Signal Process.
**2004**, 52, 2153–2164. [Google Scholar] [CrossRef] - Ji, S.; Xue, Y.; Carin, L. Bayesian compressive sensing. IEEE Trans. Signal Process.
**2008**, 56, 2346–2356. [Google Scholar] [CrossRef] - Ji, S.; Carin, L. Bayesian compressive sensing and projection optimization. In Proceedings of the 24th International Conference on Machine Learning, Corvallis, OR, USA, 20–24 June 2007; pp. 377–384. [Google Scholar]
- Bernardo, J.M.; Smith, A.F.M. Bayesian Theory; Wiley: New York, NY, USA, 1994. [Google Scholar]
- Yin, W.; Osher, S.; Goldfarb, D.; Darbon, J. Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing. SIAM J. Imaging Sci.
**2008**, 1, 143–168. [Google Scholar] [CrossRef] - Osher, S.; Burger, M.; Goldfarb, D.; Xu, J.; Yin, W. An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul.
**2005**, 4, 460–489. [Google Scholar] [CrossRef] - Cai, J.F.; Osher, S.; Shen, Z. Linearized Bregman iterations for compressed sensing. Math. Comput.
**2009**, 78, 1515–1536. [Google Scholar] [CrossRef] - Goldstein, T.; Osher, S. The split Bregman method for ℓ1-regularized problems. SIAM J. Imaging Sci.
**2009**, 2, 323–343. [Google Scholar] [CrossRef] - Rudin, L.I.; Osher, S.; Fatemi, E. Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom.
**1992**, 60, 259–268. [Google Scholar] [CrossRef] - Chambolle, A. An algorithm for total variation minimization and applications. J. Math. Imaging Vis.
**2004**, 20, 89–97. [Google Scholar] - Yang, J.; Zhang, Y.; Yin, W. An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput.
**2009**, 31, 2842–2865. [Google Scholar] [CrossRef] - Geman, D.; Yang, C. Nonlinear image recovery with half-quadratic regularization. IEEE Trans. Image Process.
**1995**, 4, 932–946. [Google Scholar] [CrossRef] - Li, C. An Efficient Algorithm for Total Variation Regularization with Applications to the Single Pixel Camera and Compressive Sensing; Rice University: Houston, TX, USA, 2010. [Google Scholar]
- Wang, J.; Kwon, S.; Shim, B. Generalized orthogonal matching pursuit. IEEE Trans. Signal Process.
**2012**, 60, 6202–6216. [Google Scholar] [CrossRef] - Rangan, S. Generalized approximate message passing for estimation with random linear mixing. In Proceedings of the 2011 IEEE International Symposium on Information Theory Proceedings, St. Petersburg, Russia, 31 July–5 August 2011; IEEE: Piscataway, NJ, USA, 2011; pp. 2168–2172. [Google Scholar]
- Khajehnejad, M.A.; Xu, W.; Avestimehr, A.S.; Hassibi, B. Weighted ℓ1 minimization for sparse recovery with prior information. In Proceedings of the 2009 IEEE International Symposium on Information Theory, Seoul, Republic of Korea, 28 June–3 July 2009; IEEE: Piscataway, NJ, USA, 2009; pp. 483–487. [Google Scholar]
- De Paiva, N.M.; Marques, E.C.; de Barros Naviner, L.A. Sparsity analysis using a mixed approach with greedy and LS algorithms on channel estimation. In Proceedings of the 2017 3rd International Conference on Frontiers of Signal Processing (ICFSP), Paris, France, 6–8 September 2017; IEEE: Piscataway, NJ, USA, 2017; pp. 91–95. [Google Scholar]
- Kwon, S.; Wang, J.; Shim, B. Multipath matching pursuit. IEEE Trans. Inf. Theory
**2014**, 60, 2986–3001. [Google Scholar] [CrossRef] - Wen, J.; Zhou, Z.; Li, D.; Tang, X. A novel sufficient condition for generalized orthogonal matching pursuit. IEEE Commun. Lett.
**2016**, 21, 805–808. [Google Scholar] [CrossRef] - Sun, H.; Ni, L. Compressed sensing data reconstruction using adaptive generalized orthogonal matching pursuit algorithm. In Proceedings of the 2013 3rd International Conference on Computer Science and Network Technology, Dalian, China, 12–13 October 2013; IEEE: Piscataway, NJ, USA, 2013; pp. 1102–1106. [Google Scholar]
- Huang, H.; Makur, A. Backtracking-based matching pursuit method for sparse signal reconstruction. IEEE Signal Process. Lett.
**2011**, 18, 391–394. [Google Scholar] [CrossRef] - Gilbert, A.C.; Strauss, M.J.; Tropp, J.A.; Vershynin, R. Algorithmic linear dimension reduction in the l_1 norm for sparse vectors. arXiv
**2006**, arXiv:cs/0608079. [Google Scholar] - Blanchard, J.D.; Tanner, J.; Wei, K. CGIHT: Conjugate gradient iterative hard thresholding for compressed sensing and matrix completion. Inf. Inference A J. IMA
**2015**, 4, 289–327. [Google Scholar] [CrossRef] - Zhu, X.; Dai, L.; Dai, W.; Wang, Z.; Moonen, M. Tracking a dynamic sparse channel via differential orthogonal matching pursuit. In Proceedings of the MILCOM 2015–2015 IEEE Military Communications Conference, Tampa, FL, USA, 26–28 October 2015; IEEE: Piscataway, NJ, USA, 2015; pp. 792–797. [Google Scholar]
- Karahanoglu, N.B.; Erdogan, H. Compressed sensing signal recovery via forward–backward pursuit. Digit. Signal Process.
**2013**, 23, 1539–1548. [Google Scholar] [CrossRef] - Gilbert, A.C.; Muthukrishnan, S.; Strauss, M. Improved time bounds for near-optimal sparse Fourier representations. In Wavelets XI; International Society for Optics and Photonics: Washington, DC, USA, 2005; Volume 5914, p. 59141A. [Google Scholar]
- Foucart, S. Hard thresholding pursuit: An algorithm for compressive sensing. SIAM J. Numer. Anal.
**2011**, 49, 2543–2563. [Google Scholar] [CrossRef] - Gilbert, A.C.; Strauss, M.J.; Tropp, J.A.; Vershynin, R. One sketch for all: Fast algorithms for compressed sensing. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, San Diego, CA, USA, 11–13 June 2007; pp. 237–246. [Google Scholar]
- Tanner, J.; Wei, K. Normalized iterative hard thresholding for matrix completion. SIAM J. Sci. Comput.
**2013**, 35, S104–S125. [Google Scholar] [CrossRef] - Mileounis, G.; Babadi, B.; Kalouptsidis, N.; Tarokh, V. An adaptive greedy algorithm with application to nonlinear communications. IEEE Trans. Signal Process.
**2010**, 58, 2998–3007. [Google Scholar] [CrossRef] - Lee, J.; Choi, J.W.; Shim, B. Sparse signal recovery via tree search matching pursuit. J. Commun. Netw.
**2016**, 18, 699–712. [Google Scholar] [CrossRef] - Rangan, S.; Schniter, P.; Fletcher, A.K. Vector approximate message passing. IEEE Trans. Inf. Theory
**2019**, 65, 6664–6684. [Google Scholar] [CrossRef] - Daubechies, I.; Defrise, M.; De Mol, C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. J. Issued Courant Inst. Math. Sci.
**2004**, 57, 1413–1457. [Google Scholar] [CrossRef] - Beck, A.; Teboulle, M. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci.
**2009**, 2, 183–202. [Google Scholar] [CrossRef] - Donoho, D.L.; Maleki, A.; Montanari, A. Message-passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA
**2009**, 106, 18914–18919. [Google Scholar] [CrossRef] [PubMed] - Montanari, A.; Eldar, Y.C.; Kutyniok, G. Graphical models concepts in compressed sensing. Compress. Sens.
**2012**, 394–438. [Google Scholar] - Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R. Least angle regression. Ann. Stat.
**2004**, 32, 407–499. [Google Scholar] [CrossRef] - Figueiredo, M.A.T.; Nowak, R.D.; Wright, S.J. Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process.
**2007**, 1, 586–597. [Google Scholar] [CrossRef] - Gorodnitsky, I.F.; Rao, B.D. Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm. IEEE Trans. Signal Process.
**1997**, 45, 600–616. [Google Scholar] [CrossRef] - Do, T.T.; Gan, L.; Nguyen, N.; Tran, T.D. Sparsity adaptive matching pursuit algorithm for practical compressed sensing. In Proceedings of the 2008 42nd Asilomar Conference on Signals, SYSTEMS and computers, Pacific Grove, CA, USA, 26–29 October 2008; IEEE: Piscataway, NJ, USA, 2008; pp. 581–587. [Google Scholar]
- Blumensath, T.; Davies, M.E. Gradient pursuits. IEEE Trans. Signal Process.
**2008**, 56, 2370–2382. [Google Scholar] [CrossRef] - Gu, H.; Yaman, B.; Moeller, S.; Ellermann, J.; Ugurbil, K.; Akçakaya, M. Revisiting ℓ1-wavelet compressed-sensing MRI in the era of deep learning. Proc. Natl. Acad. Sci. USA
**2022**, 119, e2201062119. [Google Scholar] - Adler, A.; Boublil, D.; Elad, M.; Zibulevsky, M. A deep learning approach to block-based compressed sensing of images. arXiv
**2016**, arXiv:1606.01519. [Google Scholar] - Xie, Y.; Li, Q. A review of deep learning methods for compressed sensing image reconstruction and its medical applications. Electronics
**2022**, 11, 586. [Google Scholar] [CrossRef] - Zonzini, F.; Carbone, A.; Romano, F.; Zauli, M.; De Marchi, L. Machine learning meets compressed sensing in vibration-based monitoring. Sensors
**2022**, 22, 2229. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**CSSPI results of different measurement matrices at different sampling ratios. (

**a**) The three curve graphs show reconstruction PSNR, SSIM, and running time. (

**b**) The reconstructed images under sampling ratios of 20%, 40%, 60%, and 80%.

**Figure 3.**CSSPI results of different measurement matrices under different SNRs. (

**a**) The three curve graphs show reconstruction PSNR, SSIM, and time. (

**b**) The reconstructed images with SNR = 30 dB, 20 dB, 10 dB, and 0 dB.

**Figure 5.**The “Cameraman” images were reconstructed by partial reconstruction algorithms at different sampling rates.

**Figure 7.**Comparison of the “cameraman” images reconstructed by different reconstruction algorithms from single-pixel measurements contaminated with Gaussian noise.

**Figure 9.**Comparison of images reconstructed by different measurement matrices at different sampling ratios. (

**a**) Imaging results of “four bars”. (

**b**) Imaging results of “ladybug”.

**Figure 10.**Comparison of images reconstructed by different reconstruction algorithms at different sampling ratios. (

**a**) Imaging results of “four bars”. (

**b**) Imaging results of “ladybug”.

Works | Content |
---|---|

Our works | -Reviews the concept of CSSPI. -Summarizes the main measurement matrices in CSSPI. -Summarizes the main reconstruction algorithms in CSSPI. -The performance of measurement matrices and reconstruction algorithms in CSSPI is discussed in detail through simulations and experiments. -The advantages and disadvantages of mainstream measurement matrices and reconstruction algorithms in CSSPI are summarized. |

Existing works | Refs. [40,41]: Review the development of CS. |

Refs. [42,43]: Review the main measurement matrices in CS. | |

Refs. [44,45,46]: Review the reconstruction algorithms in CS. | |

Refs. [2,47]: Review the development of SPI. | |

Refs. [48,49]: Review the algorithms of SPI. For CS, only the TVAL3 algorithm is involved. |

Type of Measurement Matrix | Definition | Advantages | Disadvantages | References | |
---|---|---|---|---|---|

Random Matrix | Gaussian | -Each coefficient obeys a random distribution separately. | -The RIP property is satisfied with high probability. -Fewer measurements and noise robustness. | -Large storage space. -Difficult to implement in hardware. -No explicit constructions. | [62] |

Bernoulli | [63] | ||||

Semi-deterministic random matrix | Toeplitz and Circulant | -Each coefficient is generated in a particular way. | -Easy hardware implementation and robustness. -Sparse random matrix retains the advantage of an unstructured random matrix. | -High uncertainty. -More measurements. -For a particular type of signal. | [67,68] |

Sparse random | [70,71,72,73] | ||||

Partial orthogonal matrix | Partial Fourier | -Some rows are randomly selected from the orthogonal matrix. | -It is fast to generate and easy to save. -Easy hardware implementation and robustness. | -Fourier matrix needs more recovery time and measurement times. -The dimension limit of the Hadamard matrix. | [47] |

Partial Hadamard | [65,66] |

Algorithm | Advantages | Disadvantages | References | |
---|---|---|---|---|

Convex | Dantzig | -Fewer measurements. -Noise robustness. | -High computational complexity. -Slower, not suitable for large-scale problems. | [109] |

BPDN | [101,104,105,106] | |||

BP | [101,102] | |||

DLP | [107,108] | |||

Greedy | OMP | -Easier to implement and faster. -IHT and CoSaMP can add/discard entries per iteration. - Noise robustness. | -A priori knowledge of signal sparsity is required. -The sparsity of the solution cannot be guaranteed. -More measurements. | [113,114,115,116,117] |

CoSaMP | [120] | |||

IHT | [122,123,124] | |||

SP | [121] | |||

TV | TVAL3 | -Preserves the sharp edges and prevents blurring. -Noise robustness. -TVAL3 and TV-Qc are faster. | -Not linear. -Not differentiable. -Dantzig selector is slower. | [142] |

TV-DS | [109] | |||

TV-QC | [108] | |||

Non-Convex | BCS | -More sparse solution. -Faster, suitable for large-scale problems. | -Rely more on the prior knowledge of the signal. -High computational complexity. | [130,131,132,133] |

IRLS | -Fewer measurements than the convex algorithm. -Can be implemented under weaker RIP. | -High computational complexity. -Slower, not suitable for large-scale problems. | [128,129] | |

Bregman | SB | -Faster, noise robustness. -Small memory footprint. | -More measurements than the convex algorithm. | [137] |

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Zhao, W.; Gao, L.; Zhai, A.; Wang, D.
Comparison of Common Algorithms for Single-Pixel Imaging via Compressed Sensing. *Sensors* **2023**, *23*, 4678.
https://doi.org/10.3390/s23104678

**AMA Style**

Zhao W, Gao L, Zhai A, Wang D.
Comparison of Common Algorithms for Single-Pixel Imaging via Compressed Sensing. *Sensors*. 2023; 23(10):4678.
https://doi.org/10.3390/s23104678

**Chicago/Turabian Style**

Zhao, Wenjing, Lei Gao, Aiping Zhai, and Dong Wang.
2023. "Comparison of Common Algorithms for Single-Pixel Imaging via Compressed Sensing" *Sensors* 23, no. 10: 4678.
https://doi.org/10.3390/s23104678