### Abstract

Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size δ. The robust recovery result of Candès et ale and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, ℓ^{1} recovery yields an approximation of the original sparse signal with an accuracy of O (δ). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in R^{N}, the reconstruction accuracy can be improved by a factor of (m/k)^{(r-1/2)α} for any 0 < α < 1 if m ≳_{γ} k(log N)^{1/(1-α)} (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via γth-order Sobolev dual frames.

Original language | English (US) |
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Title of host publication | 2010 44th Annual Conference on Information Sciences and Systems, CISS 2010 |

DOIs | |

State | Published - 2010 |

Event | 44th Annual Conference on Information Sciences and Systems, CISS 2010 - Princeton, NJ, United States Duration: Mar 17 2010 → Mar 19 2010 |

### Other

Other | 44th Annual Conference on Information Sciences and Systems, CISS 2010 |
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Country | United States |

City | Princeton, NJ |

Period | 3/17/10 → 3/19/10 |

### Fingerprint

### ASJC Scopus subject areas

- Information Systems
- Information Systems and Management

### Cite this

*2010 44th Annual Conference on Information Sciences and Systems, CISS 2010*[5464825] https://doi.org/10.1109/CISS.2010.5464825

**Sigma delta quantization for compressed sensing.** / Gunturk, C. Sinan; Lammers, Mark; Powell, Alex; Saab, Rayan; Yilmaz, Özgür.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2010 44th Annual Conference on Information Sciences and Systems, CISS 2010.*, 5464825, 44th Annual Conference on Information Sciences and Systems, CISS 2010, Princeton, NJ, United States, 3/17/10. https://doi.org/10.1109/CISS.2010.5464825

}

TY - GEN

T1 - Sigma delta quantization for compressed sensing

AU - Gunturk, C. Sinan

AU - Lammers, Mark

AU - Powell, Alex

AU - Saab, Rayan

AU - Yilmaz, Özgür

PY - 2010

Y1 - 2010

N2 - Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size δ. The robust recovery result of Candès et ale and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, ℓ1 recovery yields an approximation of the original sparse signal with an accuracy of O (δ). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in RN, the reconstruction accuracy can be improved by a factor of (m/k)(r-1/2)α for any 0 < α < 1 if m ≳γ k(log N)1/(1-α) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via γth-order Sobolev dual frames.

AB - Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size δ. The robust recovery result of Candès et ale and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, ℓ1 recovery yields an approximation of the original sparse signal with an accuracy of O (δ). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in RN, the reconstruction accuracy can be improved by a factor of (m/k)(r-1/2)α for any 0 < α < 1 if m ≳γ k(log N)1/(1-α) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via γth-order Sobolev dual frames.

UR - http://www.scopus.com/inward/record.url?scp=77953706903&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953706903&partnerID=8YFLogxK

U2 - 10.1109/CISS.2010.5464825

DO - 10.1109/CISS.2010.5464825

M3 - Conference contribution

SN - 9781424474172

BT - 2010 44th Annual Conference on Information Sciences and Systems, CISS 2010

ER -