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NMR data processing using iterative thresholding and minimum l(1)-norm reconstruction.

Alan S Stern1, David L Donoho, Jeffrey C Hoch

  • 1Rowland Institute at Harvard, Cambridge, MA 02142, USA.

Journal of Magnetic Resonance (San Diego, Calif. : 1997)
|August 29, 2007
PubMed
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Iterative thresholding, using soft thresholding, is shown to be equivalent to minimum L1-norm reconstruction. This finding explains the success of these algorithms and offers more efficient routes for signal processing applications like time series spectrum analysis.

Area of Science:

  • Signal Processing
  • Numerical Analysis
  • Time Series Analysis

Background:

  • Iterative thresholding algorithms have a history in signal processing but lacked theoretical grounding.
  • Previous development was largely heuristic and ad hoc.

Purpose of the Study:

  • To theoretically link iterative thresholding to a fundamental reconstruction principle.
  • To explain the efficacy of existing iterative thresholding methods.
  • To explore efficient reconstruction techniques for underdetermined systems.

Main Methods:

  • Utilized a specific thresholding operation, termed soft thresholding.
  • Demonstrated the equivalence between the fixed point of iterative thresholding and minimum L1-norm reconstruction.
  • Applied the method to spectrum analysis of time series data.

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Main Results:

  • Established that the fixed point of iterative thresholding is equivalent to minimum L1-norm reconstruction.
  • Illustrated the practical application in time series spectrum analysis.
  • Revealed connections to maximum entropy and minimum area methods.

Conclusions:

  • The L1-norm serves as a powerful regularizer for underdetermined systems.
  • This theoretical framework explains the success of iterative thresholding.
  • More efficient reconstruction pathways are identified, with potential applications in Nuclear Magnetic Resonance (NMR).