Foundations of Computational Mathematics

, Volume 13, Issue 1, pp 1–36 | Cite as

Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements

  • C. S. Güntürk
  • M. Lammers
  • A. M. Powell
  • R. Saab
  • Ö. Yılmaz
Article

Abstract

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y=Φx of a k-sparse signal x∈ℝN, where Φ satisfies the restricted isometry property, then the approximate recovery x# via 1-minimization is within O(δ) of x. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth-order ΣΔ (Sigma–Delta) quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduced approximation error that is of the order δ(k/m)(r−1/2)α for any 0<α<1, if mr,αk(logN)1/(1−α). The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x whose magnitudes are suitably bounded away from zero on their support.

Keywords

Quantization Finite frames Random frames Alternative duals Compressed sensing 

Mathematics Subject Classification

41A46 94A12 

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Copyright information

© SFoCM 2012

Authors and Affiliations

  • C. S. Güntürk
    • 1
  • M. Lammers
    • 2
  • A. M. Powell
    • 3
  • R. Saab
    • 4
  • Ö. Yılmaz
    • 5
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.University of North CarolinaWilmingtonUSA
  3. 3.Vanderbilt UniversityNashvilleUSA
  4. 4.Duke UniversityDurhamUSA
  5. 5.University of British ColumbiaVancouverCanada

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