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

Iterative thresholding algorithms have a long history of application to signal processing. Although they are intuitive and easy to implement, their development was heuristic and mainly ad hoc. Using a special form of the thresholding operation, called soft thresholding, we show that the fixed point of iterative thresholding is equivalent to minimum l(1)-norm reconstruction. We illustrate the method for spectrum analysis of a time series. This result helps to explain the success of these methods and illuminates connections with maximum entropy and minimum area methods, while also showing that there are more efficient routes to the same result. The power of the l(1)-norm and related functionals as regularizers of solutions to underdetermined systems will likely find numerous useful applications in NMR.

[1]  J C Hoch,et al.  Modern spectrum analysis in nuclear magnetic resonance: alternatives to the Fourier transform. , 1989, Methods in enzymology.

[2]  G. L. Bretthorst.  Bayesian analysis. I. Parameter estimation using quadrature NMR models , 1990 .

[3]  A. J. Shaka,et al.  The Multidimensional Filter Diagonalization Method: II. Application to 2D Projections of 2D, 3D, and 4D NMR Experiments , 2000 .

[4]  B. Logan,et al.  Signal recovery and the large sieve , 1992 .

[5]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[6]  S. Gull,et al.  Image reconstruction from incomplete and noisy data , 1978, Nature.

[7]  R. Chylla,et al.  Improved frequency resolution in multidimensional constant-time experiments by multidimensional Bayesian analysis , 1993, Journal of biomolecular NMR.

[8]  Ulrich Günther,et al.  Automated Protein NMR Structure Determination Using Wavelet De-noised NOESY Spectra , 2005, Journal of biomolecular NMR.

[9]  D. van Ormondt,et al.  Improved algorithm for noniterative time-domain model fitting to exponentially damped magnetic resonance signals , 1987 .

[10]  Ulrich Günther,et al.  Using wavelet de-noised spectra in NMR screening. , 2005, Journal of magnetic resonance.

[11]  A S Stern,et al.  Does the maximum entropy method improve sensitivity? , 1990, Proceedings of the National Academy of Sciences of the United States of America.

[12]  David S. Stephenson,et al.  Linear prediction and maximum entropy methods in NMR spectroscopy , 1988 .

[13]  A. Papoulis.  A new algorithm in spectral analysis and band-limited extrapolation. , 1975 .

[14]  P. H. Cittert.  Zum Einfluß der Spaltbreite auf die Intensitätsverteilung in Spektrallinien. II , 1930 .

[15]  D. Donoho,et al.  Uncertainty principles and signal recovery , 1989 .

[16]  Ray Freeman,et al.  Projection-reconstruction technique for speeding up multidimensional NMR spectroscopy. , 2004, Journal of the American Chemical Society.

[17]  William H. Press,et al.  Numerical Recipes: FORTRAN , 1988 .

[18]  P. Jansson,et al.  Resolution Enhancement of Spectra , 1970 .

[19]  I. Johnstone,et al.  Maximum Entropy and the Nearly Black Object , 1992 .

[20]  Keyang Ding,et al.  Novel 2D triple-resonance NMR experiments for sequential resonance assignments of proteins. , 2002, Journal of magnetic resonance.

[21]  R. Newman,et al.  Maximization of entropy and minimization of area as criteria for NMR signal processing , 1988 .

[22]  J. Creyghton,et al.  Application of linear prediction and singular value decomposition (LPSVD) to determine NMR frequencies and intensities from the FID , 1985, Magnetic resonance in medicine.

[23]  Osama A. Ahmed,et al.  New denoising scheme for magnetic resonance spectroscopy signals , 2005, IEEE Transactions on Medical Imaging.

[24]  T. Szyperski,et al.  GFT NMR, a new approach to rapidly obtain precise high-dimensional NMR spectral information. , 2003, Journal of the American Chemical Society.