Journal of Fourier Analysis and Applications

, Volume 19, Issue 6, pp 1255–1273 | Cite as

Near-Optimal Encoding for Sigma-Delta Quantization of Finite Frame Expansions

Article
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Abstract

In this paper we investigate encoding the bit-stream resulting from coarse Sigma-Delta quantization of finite frame expansions (i.e., overdetermined representations) of vectors. We show that for a wide range of finite-frames, including random frames and piecewise smooth frames, there exists a simple encoding algorithm—acting only on the Sigma-Delta bit stream—and an associated decoding algorithm that together yield an approximation error which decays exponentially in the number of bits used. The encoding strategy consists of applying a discrete random operator to the Sigma-Delta bit stream and assigning a binary codeword to the result. The reconstruction procedure is essentially linear and equivalent to solving a least squares minimization problem.

Keywords

Vector quantization Frame theory Rate-distortion theory Random matrices Overdetermined systems Pseudoinverses 

AMS Subject Classification

42C15 94A12 94A34 65F20 15B52 68Q17 
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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of Electrical and Computer EngineeringMichigan State UniversityEast LansingUSA
  3. 3.Department of MathematicsUniversity of California, San DiegoSan DiegoUSA

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