Journal of Fourier Analysis and Applications

, Volume 15, Issue 4, pp 488–501 | Cite as

Painless Reconstruction from Magnitudes of Frame Coefficients

  • Radu Balan
  • Bernhard G. Bodmann
  • Peter G. Casazza
  • Dan Edidin
Article

Abstract

The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present linear reconstruction algorithms for tight frames associated with projective 2-designs in finite-dimensional real or complex Hilbert spaces. Examples of such frames are two-uniform frames and mutually unbiased bases, which include discrete chirps. The number of operations required for reconstruction with these frames grows at most as the cubic power of the dimension of the Hilbert space. Moreover, we present a very efficient algorithm which gives reconstruction on the order of d operations for a d-dimensional Hilbert space.

Keywords

Frames Reconstruction without phase Projective 2-designs 

Mathematics Subject Classification (2000)

42C15 05B20 

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  • Radu Balan
    • 1
  • Bernhard G. Bodmann
    • 2
  • Peter G. Casazza
    • 3
  • Dan Edidin
    • 3
  1. 1.Mathematics Department and CSCAMMUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA
  3. 3.Department of MathematicsUniversity of MissouriColumbiaUSA

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