# A survey on one-bit compressed sensing: theory and applications

- 28 Downloads

## Abstract

In the past few decades, with the growing popularity of compressed sensing (CS) in the signal processing field, the quantization step in CS has received significant attention. Current research generally considers multi-bit quantization. For systems employing quantization with a sufficient number of bits, a sparse signal can be reliably recovered using various CS reconstruction algorithms.

Recently, many researchers have begun studying the one-bit quantization case for CS. As an extreme case of CS, one-bit CS preserves only the sign information of measurements, which reduces storage costs and hardware complexity. By treating one-bit measurements as sign constraints, it has been shown that sparse signals can be recovered using certain reconstruction algorithms with a high probability. Based on the merits of one-bit CS, it has been widely applied to many fields, such as radar, source location, spectrum sensing, and wireless sensing network.

In this paper, the characteristics of one-bit CS and related works are reviewed. First, the framework of one-bit CS is introduced. Next, we summarize existing reconstruction algorithms. Additionally, some extensions and practical applications of one-bit CS are categorized and discussed. Finally, our conclusions and the further research topics are summarized.

## Keywords

compressed sensing one-bit quantization sign information support consistency## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 61302084).

## Supplementary material

## References

- 1.Donoho D L. Compressed sensing. IEEE Transanctions on Information Theory, 2006, 52(4): 1289–1306MathSciNetCrossRefMATHGoogle Scholar
- 2.Lexa M A, Davies M E, Thompson J S. Reconciling compressive sampling systems for spectrally sparse continuous-time signals. IEEE Transactions on Signal Processing, 2012, 60(1): 155–171MathSciNetCrossRefGoogle Scholar
- 3.Xu W B, Li Z L, Tian Y, Wang Y, Lin J R. Perturbation analysis of simultaneous orthogonal matching pursuit. Signal Processing, 2015, 116(C): 91–100CrossRefGoogle Scholar
- 4.Gao K, Batalama S N, Pados D A, Suter B W. Compressive sampling with generalized polygons. IEEE Transactions on Signal Processing, 2011, 59(10): 4759–4766MathSciNetCrossRefGoogle Scholar
- 5.Laska J N, Baraniuk R G. Regime change: bit-depth versus measurement-rate in compressive sensing. IEEE Transactions on Signal Processing, 2012, 60(7): 3496–3505MathSciNetCrossRefGoogle Scholar
- 6.Jacques L, Laska J N, Boufounos P T, Baraniuk R G. Robust 1-bit compressive sensing via binary stable embeddings of sparse vectors. IEEE Transactions on Information Theory, 2011, 59(4): 2082–2102MathSciNetCrossRefMATHGoogle Scholar
- 7.Baraniuk R, Foucart S, Needell D, Plan Y, Wootters M. Exponential decay of reconstruction error from binary measurements of sparse signals. arXiv preprint, arXiv:1407.8246, 2014Google Scholar
- 8.Knudson K, Saab R, Ward R. One-bit compressive sensing with norm estimation. IEEE Transactions on Information Theory, 2014, 62(5): 2748–2758MathSciNetCrossRefMATHGoogle Scholar
- 9.Plan Y, Vershynin R. Robust 1-bit compressed sensing and sparse logistic regression: a convex programming approach. IEEE Transactions on Information Theory, 2013, 59(1): 482–494MathSciNetCrossRefMATHGoogle Scholar
- 10.Ai A, Lapanowski A, Plan Y, Vershynin R. One-bit compressed sensing with non-gaussian measurements. Linear Algebra & Its Applications, 2014, 441(1): 222–239MathSciNetCrossRefMATHGoogle Scholar
- 11.Fang J, Shen Y, Li H. One-bit quantization design and adaptive methods for compressed sensing. Mathematics, 2013Google Scholar
- 12.Boufounos P T, Baraniuk R G. 1-bit compressive sensing. In: Proceedings of the 42nd Annual Conference on Information Sciences and Systems. 2008, 16–21Google Scholar
- 13.Hale E T, Yin W, Zhang Y. A fixed-point continuation method for l1-regularized minimization with applications to compressed sensing. CAAM Technical Report TR07-07. 2007Google Scholar
- 14.Laska J N, Wen Z, Yin W, Baraniuk R G. Trust, but verify: fast and accurate signal recovery from 1-bit compressive measurements. IEEE Transactions on Signal Processing, 2011, 59(11): 5289–5301MathSciNetCrossRefGoogle Scholar
- 15.Boufounos P T. Greedy sparse signal reconstruction from sign measurements. In: Proceedings of the Conference Record of the 43rd Asilomar Conference on Signals, Systems and Computers. 2009, 1305–1309Google Scholar
- 16.Yan M, Yang Y, Osher S. Robust 1-bit compressive sensing using adaptive outlier pursuit. IEEE Transanctions on Signal Processing, 2012, 60(7): 3868–3875MathSciNetCrossRefGoogle Scholar
- 17.Movahed A, Panahi A, Durisi G. A robust rfpi-based 1-bit compressive sensing reconstruction algorithm. In: Proceedings of the IEEE Information Theory Workshop. 2012, 567–571Google Scholar
- 18.Yang Z, Xie L, Zhang C. Variational bayesian algorithm for quantized compressed sensing. IEEE Transactions on Signal Processing, 2013, 61(11): 2815–2824MathSciNetCrossRefGoogle Scholar
- 19.Li FW, Fang J, Li H B, Huang L. Robust one-bit bayesian compressed sensing with sign-flip errors. IEEE Signal Processing Letters, 2015, 22(7): 857–861CrossRefGoogle Scholar
- 20.Zhou T Y, Tao D C. 1-bit hamming compressed sensing. In: Proceedings of IEEE International Symposium on Information Theory Proceedings. 2012, 1862–1866Google Scholar
- 21.Tian Y, Xu W B, Wang Y, Yang H W. A distributed compressed sensing scheme based on one-bit quantization. In: Proceedings of the 79th Vehicular Technology Conference. 2014, 1–6Google Scholar
- 22.Xiong J P, Tang Q H, Zhao J. 1-bit compressive data gathering for wireless sensor networks. Journal of Sensors, 2014, 2014(7): 177–183Google Scholar
- 23.Shen Y N, Fang J, Li H B. One-bit compressive sensing and source location is wireless sensor networks. In: Proceedings of IEEE China Summit and International Conference on Signal and Information Processing. 2013, 379–383Google Scholar
- 24.Feng C, Valaee S, Tan Z H. Multiple target localization using compressive sensing. In: Proceedings of IEEE Global Telecommunications Conference. 2009, 1–6Google Scholar
- 25.Chen C H, Wu J Y. Amplitude-aided 1-bit compressive sensing over noisy wireless sensor networks. IEEE Wireless Communications Letters, 2015, 4(5): 473–476CrossRefGoogle Scholar
- 26.Meng J, Li H S, Han Z. Sparse event detection in wireless sensor neworks using compressive sensing. In: Proceedings of the 43rd Annual Conference on Information Sciences and Systems. 2009, 181–185Google Scholar
- 27.Sakdejayont T, Lee D, Peng Y, Yamashita Y, Morikawa H. Evaluation of memory-efficient 1-bit compressed sensing in wireless sensor networks. In: Proceedings of the Hummanitarian Technologhy Conference. 2013, 326–329Google Scholar
- 28.Lee D, Sasaki T, Yamada T, Akabane K, Yamaguchi Y, Uehara K. Spectrum sensing for networked system using 1-bit compressed sensing with partial random circulant measurement matrices. In: Proceedings of the Vehicular Technology Conference. 2012, 1–5Google Scholar
- 29.Fu N, Yang L, Zhang J C. Sub-nyquist 1 bit sampling system for sparse multiband signals. In: Proceedings of the 22nd European Signal Processing Conference. 2014, 736–740Google Scholar
- 30.Alberti G, Franceschetti G, Schirinzi G, Pascazio V. Time-domain convolution of one-bit coded radar signals. IEE Proceedings F- Radar and Signal Processing, 1991, 138(5): 438–444CrossRefGoogle Scholar
- 31.Franceschetti G, Merolla S, Tesauro M. Phase quantized sar signal processing: Theory and experiments. IEEE Transactions on Aerospace and Electronic Systems, 1999, 35(1): 201–214CrossRefGoogle Scholar
- 32.Dong X, Zhang Y H. A map approach for 1-bit compressive sensing in synthetic aperture radar imaging. IEEE Geoscience and Remote Sensing Letters, 2015, 12(6): 1237–1241CrossRefGoogle Scholar
- 33.Allstot E G, Chen A Y, Dixon A M R, Gangopadhyay D, Mitsuda H, Allstot D J. Compressed sensing of ECG bio-signals using one-bit measurement matrices. In: Proceedings of the 9th IEEE International New Circuits and Systems Conference. 2011, 213–216Google Scholar
- 34.Haboba J, Mangia M, Rovatti R, Setti G. An architecture for 1-bit localized compressive sensing with applications to eeg. In: Proceedings of IEEE Biomedical Circuits and Systems Conference. 2011, 137–140Google Scholar
- 35.Candes E J. The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique, 2008, 346(9–10): 589–592MathSciNetCrossRefMATHGoogle Scholar
- 36.Wright S J, Nowak R D, Figueiredo M A T. Sparse reconstruction by separable approximation. IEEE Transactions on Signal Processing, 2009, 57(7): 2479–2493MathSciNetCrossRefGoogle Scholar
- 37.Kaasschieter E F. Preconditioned conjugate gradients for solving singular systems. Journal of Computational and Applied Mathematics, 1988, 24(1–2): 265–275MathSciNetCrossRefMATHGoogle Scholar
- 38.Blumensath T, Davies M E. Iterative hard thresholding for compressed sensing. Applied and computational harmonic analysis, 2009, 27(3): 265–274MathSciNetCrossRefMATHGoogle Scholar
- 39.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. 1993, 40–44CrossRefGoogle Scholar
- 40.Needell D, Vershynin R. Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Foundations of Computational Mathematics, 2009, 9(3): 317–334MathSciNetCrossRefMATHGoogle Scholar
- 41.Needell D, Tropp J A. Cosamp: iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis, 2009, 26(3): 301–321MathSciNetCrossRefMATHGoogle Scholar
- 42.Do T T, Lu G, Nguyen N, Tran T D. Sparsity adaptive matching pursuit algorithm for practical compressed sensing. In: Proceedings of the 42nd Asilomar Conference on Signals, Systems and Computers. 2008, 581–587Google Scholar
- 43.Ji S H, Xue Y, Carin L. Bayesian compressive sensing. IEEE Transactions on Signal Processing, 2008, 56(6): 2346–2356MathSciNetCrossRefGoogle Scholar
- 44.Chen S S, Donoho D L, Saunders MA. Atomic decomposition by basis pursuit. SIAM Review, 2001, 43(1): 129–159MathSciNetCrossRefMATHGoogle Scholar
- 45.Tibshirani R. Regression shrinkage and selection via the lasso: a retrospective. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2011, 73(3): 273–282MathSciNetCrossRefGoogle Scholar
- 46.Candes E J, Romberg J K, Tao T. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 2006, 59(8): 1207–1223MathSciNetCrossRefMATHGoogle Scholar
- 47.Shen Y, Fang J, Li H, Chen Z. A one-bit reweighted iterative algorithm for sparse signal recovery. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing. 2013, 5915–5919Google Scholar
- 48.Zhang L J, Yi J F, Jin R. Efficient algorithms for robust one-bit compressive sensing. In: Proceedings of the 31st International Conference on Machine Learning. 2014, 820–828Google Scholar
- 49.Wang H, Wan Q. One bit support recovery. In: Proceedings of the 6th International Conference on Wireless Communications Networking and Mobile Computing. 2010, 1–4Google Scholar
- 50.Plan Y, Vershynin R. One-bit compressed sensing by linear programming. Communications on Pure and Applied Mathematics, 2013, 66(8): 1275–1297MathSciNetCrossRefMATHGoogle Scholar
- 51.North P, Needell D. One-bit compressive sensing with partial support. In: Proceedings of the 6th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing. 2015, 349–352Google Scholar
- 52.Fu X, Han F M, Zou H X. Robust 1-bit compressive sensing against sign flips. In: Proceedings of IEEE Global Communications Conference. 2014, 3121–3125Google Scholar
- 53.Zayyani H, Korki M, Marvasti F. Dictionary learning for blind one bit compressed sensing. IEEE Signal Processing Letters, 2016, 23(2): 187–191CrossRefGoogle Scholar
- 54.Huang X L, Shi L, Yan M, Suykens J A K. Pinball loss minimization for one-bit compressive sensing. Mathematics, 2015Google Scholar
- 55.Yan M. Restoration of images corrupted by impulse noise and mixed gaussian impulse noise using blind inpainting. SIAM Journal on Imaging Sciences, 2013, 6(3): 1227–1245MathSciNetCrossRefMATHGoogle Scholar
- 56.Tipping M. Sparse bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 2001, 1(3): 211–244MathSciNetMATHGoogle Scholar
- 57.Chen S, Banerjee A. One-bit compressed sensing with the k-support norm. In: Proceedings of the 18th International Conference on Artificial Intelligence and Statistics. 2015, 138–146Google Scholar
- 58.Zhou T Y, Tao D C. k-bit hamming compressed sensing. In: Proceedings of IEEE International Symposium on Information Theory. 2013, 679–683Google Scholar
- 59.Baron D, Wakin M B, Duarte M F, Sarvotham S, Baraniuk R G. Distributed compressed sensing. Preprint, 2012, 22(10): 2729–2732Google Scholar
- 60.Yin H P, Li J X, Chai Y, Yang S X. A survey on distributed compressed sensing theory and applications. Frontiers of Computer Science, 2014, 8(6): 893–904MathSciNetCrossRefGoogle Scholar
- 61.Tian Y, Xu W B, Wang Y, Yang H W. Joint reconstruction algorithms for one-bit distributed compressed sensing. In: Proceedings of the 22nd International Conference on Telecommunications. 2015, 338–342Google Scholar
- 62.Nakarmi U, Rahnavard N. Joint wideband spectrum sensing in frequency overlapping cognitive radio networks using distributed compressive sensing. In: Proceedings of the Military Communications Conference. 2011, 1035–1040Google Scholar
- 63.Li Z L, Xu WB, Wang Y, Lin J R. A tree-based regularized orthogonal matching pursuit algorithm. In: Proceedings of the 22nd International Conference on Telecommunications. 2015, 343–347Google Scholar
- 64.He L H, Carin L. Exploiting structure in wavelet-based bayesian compressive sensing. IEEE Transactions on Signal Processing, 2009, 57(9): 3488–3497MathSciNetCrossRefGoogle Scholar
- 65.Allstot E G, Chen A Y, Dixon A M R, Gangopadhyay D. Compressive sampling of ecg bio-signals: quantization noise and sparsity considerations. In: Proceedings of the IEEE Biomedical Circuits and Systems Conference. 2010, 41–44Google Scholar
- 66.Haboba J, Rovatti R, Setti G. Rads converter: an approach to analog to information conversion. In: Proceedings of the 19th IEEE International Conference on Electronics, Circuits and Systems. 2012, 49–52Google Scholar
- 67.Movahed A, Reed M C. Iterative detection for compressive sensing: Turbo cs. In: Proceedings of IEEE International Conference on Communications. 2014, 4518–4523Google Scholar
- 68.Yamada T, Lee D, Toshinaga H, Akabane K, Yamaguchi Y, Uehara K. 1-bit compressed sensing with edge detection for compressed radio wave data transfer. In: Proceedings of the 18th Asia-Pacific Conference on Communications. 2012, 407–411Google Scholar
- 69.Mo J, Schniter P, Prelcic N G, Heath R W. Channel estimation in millimeter wave mimo systems with one-bit quantization. In: Proceedings of the 48th Asilomar Conference on Signals, Systems and Computers. 2014, 957–961Google Scholar
- 70.Luo C. A low power self-capacitive touch sensing analog front end with sparse multi-touch detection. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing. 2014, 3007–3011Google Scholar
- 71.Chen H, Shao H. Sparse recovery-based doa estimator with signaldependent dictionary. In: Proceedings of the 8th International Conference on Signal Processing and Communication Systems. 2014, 1–4Google Scholar
- 72.Gupta A, Nowak R, Recht B. Sample complexity for 1-bit compressed sensing and sparse classification. In: Proceedings of IEEE International Symposium on Information Theory Proceedings. 2010, 1553–1557Google Scholar
- 73.Candes E, Romberg J, Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006, 52(2): 489–509MathSciNetCrossRefMATHGoogle Scholar
- 74.Rauhut H. Circulant and toeplitz matrices in compressed sensing. Mathematics, 2009Google Scholar
- 75.Candes E J, Wakin M B. An introduction to compressive sampling. IEEE Signal Processing Magzine, 2008, 25(2): 21–30CrossRefGoogle Scholar
- 76.Candes E, Tao T. Decoding by linear programming. IEEE Transactions on Information Theory, 2005, 51(12): 4203–4215MathSciNetCrossRefMATHGoogle Scholar
- 77.Scarlett J, Evans J S, Dey S. Compressed sensing with prior information: information-theoretic limits and practical decoders. IEEE Transactions on Signal Processing, 2013, 61(2): 427–439MathSciNetCrossRefGoogle Scholar
- 78.Eldar Y C, Kuppinger P, Bolcskei H. Compressed sensing of blocksparse signals: uncertainty relations and efficient recovery. IEEE Transactions on Signal Processing, 2010, 58(6): 3042–3054MathSciNetCrossRefGoogle Scholar
- 79.Yu X H, Baek S J. Sufficient conditions on stable recovery of sparse signals with partial support information. IEEE Signal Processing letters, 2013, 20(5): 539–542CrossRefGoogle Scholar
- 80.Friedlander M P, Mansour H, Saab R, Yilmaz O. Recovering compressively sampled signals using partial support information. IEEE Transactions on Information Theory, 2012, 58(2): 1122–1134MathSciNetCrossRefMATHGoogle Scholar
- 81.Miosso C J, Borries R V, Pierluissi J H. Compressive sensing with prior information: requirements and probabilities of reconstruction in l1- minimization. IEEE Transactions on Signal Processing, 2013, 61(9): 2150–2164MathSciNetCrossRefGoogle Scholar
- 82.Davenport M A, Wakin M B. Analysis of orthogonal matching pursuit using the restricted isometry property. IEEE Transactions on Information Theory, 2010, 56(9): 4395–4401MathSciNetCrossRefMATHGoogle Scholar
- 83.Liu E, Temlyakov V N. The orthogonal super greedy algorithm and applications in compressed sensing. IEEE Transactions on Information Theory, 2012, 58(4): 2040–2047MathSciNetCrossRefMATHGoogle Scholar
- 84.Ding J, Chen L M, Gu Y T. Perturbation analysis of orthogonal matching pursuit. IEEE Transactions on Signal Processing, 2013, 61(2): 398–410MathSciNetCrossRefGoogle Scholar
- 85.Mo Q, Shen Y. A remark on the restricted isometry property in orthogonal matching pursuit. IEEE Transactions on Infirmation Theroy, 2012, 58(6): 3654–3656MathSciNetCrossRefMATHGoogle Scholar
- 86.Maleh R. Improved RIP analysis of orthogonal matching pursuit. Computer Science, 2011Google Scholar
- 87.Dai W, Milenkovic O. Subspace pursuit for compressive sensing: Closing the gap between performance and complexity. IEEE Transactions on Infirmation Theroy, 2008, 55(5): 2230–2249CrossRefMATHGoogle Scholar