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Solving 2D Fredholm Integral from Incomplete Measurements Using Compressive Sensing.

Alexander Cloninger1, Wojciech Czaja1, Ruiliang Bai2

  • 1Department of Mathematics, Norbert Wiener Center, University of Maryland, College Park, MD 20742.

SIAM Journal on Imaging Sciences
|July 16, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces a novel algorithm to accelerate nuclear magnetic resonance spectroscopy (NMR) by reconstructing data from limited measurements. The method uses compressive sensing principles to efficiently acquire and process NMR data, significantly reducing acquisition time.

Keywords:
Fredholm integralcompressive sensingmatrix completionnuclear magnetic resonancetight frame

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Area of Science:

  • Computational physics
  • Spectroscopy
  • Applied mathematics

Background:

  • Nuclear Magnetic Resonance (NMR) spectroscopy is a powerful analytical technique.
  • Traditional NMR data acquisition can be time-consuming, limiting its application.
  • Accelerating NMR data acquisition is crucial for real-time analysis and high-throughput screening.

Purpose of the Study:

  • To develop an algorithm for solving 2D Fredholm integral equations of the first kind.
  • To accelerate data acquisition in NMR spectroscopy.
  • To leverage compressive sensing principles for efficient data recovery.

Main Methods:

  • An algorithm is presented to solve 2D Fredholm integral equations with tensor product structure.
  • Compressive sensing arguments are incorporated to reconstruct data from limited measurements.
  • The Venkataramanan-Song-Hürlimann algorithm is used for zeroth-order regularization minimization.

Main Results:

  • The algorithm recovers a compressed data matrix from measurements forming a tight frame.
  • It is demonstrated that recovery is possible from as few as 10% of total measurements.
  • The approach shows realistic potential for speeding up NMR data acquisition.

Conclusions:

  • The proposed algorithm effectively reconstructs data from limited measurements using compressive sensing.
  • This method offers a significant speed-up for nuclear magnetic resonance spectroscopy.
  • The technique is validated on simulated data, showing its practical applicability.