Fast Open image in new window Minimization by Iterative Thresholding for Multidimensional NMR Spectroscopy

  • Iddo DroriEmail author
Open Access
Research Article
Part of the following topical collections:
  1. Numerical Linear Algebra in Signal Processing Applications


Fast multidimensional NMR is important in chemical shift assignment and for studying structures of large proteins. We present the first method which takes advantage of the sparsity of the wavelet representation of the NMR spectra and reconstructs the spectra from partial random measurements of its free induction decay (FID) by solving the following optimization problem: min Open image in new window subject to Open image in new window , where Open image in new window is a given Open image in new window observation vector, Open image in new window a random sampling operator, Open image in new window denotes the Fourier transform, and Open image in new window an orthogonal 2D wavelet transform. The matrix Open image in new window is a given Open image in new window matrix such that Open image in new window . This problem can be solved by general-purpose solvers; however, these can be prohibitively expensive in large-scale applications. In the settings of interest, the underlying solution is sparse with a few nonzeros. We show here that for large practical systems, a good approximation to the sparsest solution is obtained by iterative thresholding algorithms running much more rapidly than general solvers. We demonstrate the applicability of our approach to fast multidimensional NMR spectroscopy. Our main practical result estimates a four-fold reduction in sampling and experiment time without loss of resolution while maintaining sensitivity for a wide range of existing settings. Our results maintain the quality of the peak list of the reconstructed signal which is the key deliverable used in protein structure determination.


Free Induction Decay Peak List Reconstructed Signal Observation Vector Wavelet Representation 


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Copyright information

© Iddo Drori. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of StatisticsStanford UniversityUSA

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