Skip to content


  • Research Article
  • Open Access

Fast Minimization by Iterative Thresholding for Multidimensional NMR Spectroscopy

EURASIP Journal on Advances in Signal Processing20072007:020248

  • Received: 18 September 2006
  • Accepted: 28 August 2007
  • Published:


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 subject to , where is a given observation vector, a random sampling operator, denotes the Fourier transform, and an orthogonal 2D wavelet transform. The matrix is a given matrix such that . 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


Authors’ Affiliations

Department of Statistics, Stanford University, CA 94305-4065, USA


  1. Kim S, Szyperski T: GFT NMR, a new approach to rapidly obtain precise high-dimensional NMR spectral information. Journal of the American Chemical Society 2003,125(5):1385-1393. 10.1021/ja028197dView ArticleGoogle Scholar
  2. Mobli M, Stern AS, Hoch JC: Spectral reconstruction methods in fast NMR: reduced dimensionality, random sampling and maximum entropy. Journal of Magnetic Resonance 2006,182(1):96-105. 10.1016/j.jmr.2006.06.007View ArticleGoogle Scholar
  3. Rovnyak D, Frueh DP, Sastry M, et al.: Accelerated acquisition of high resolution triple-resonance spectra using non-uniform sampling and maximum entropy reconstruction. Journal of Magnetic Resonance 2004,170(1):15-21. 10.1016/j.jmr.2004.05.016View ArticleGoogle Scholar
  4. Schmieder P, Stern AS, Wagner G, Hoch JC: Application of nonlinear sampling schemes to COSY-type spectra. Journal of Biomolecular NMR 1993,3(5):569-576.View ArticleGoogle Scholar
  5. Schmieder P, Stern AS, Wagner G, Hoch JC: Improved resolution in triple-resonance spectra by nonlinear sampling in the constant-time domain. Journal of Biomolecular NMR 1994,4(4):483-490. 10.1007/BF00156615View ArticleGoogle Scholar
  6. Drori I, Cohen-Or D, Yeshurun H: Fragment-based image completion. ACM Transactions on Graphics (TOG) 2003,22(3):303-312. 10.1145/882262.882267View ArticleGoogle Scholar
  7. Drori I, Lischinski D: Fast multiresolution image operations in the wavelet domain. IEEE Transactions on Visualization and Computer Graphics 2003,9(3):395-411. 10.1109/TVCG.2003.1207446View ArticleGoogle Scholar
  8. Gilbert AC, Guha S, Indyk P, Muthukrishnan S, Strauss M: Near-optimal sparse Fourier representations via sampling. Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 2002, Montreal, Quebec, Canada 152-161.Google Scholar
  9. Donoho DL: Compressed sensing. IEEE Transactions on Information Theory 2006,52(4):1289-1306.MathSciNetView ArticleMATHGoogle Scholar
  10. Candès EJ, Romberg J, Tao T: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory 2006,52(2):489-509.View ArticleMathSciNetMATHGoogle Scholar
  11. Berlekamp E, McEliece R, van Tilborg H: On the inherent intractability of certain coding problems. IEEE Transactions on Information Theory 1978,24(3):384-386. 10.1109/TIT.1978.1055873MathSciNetView ArticleMATHGoogle Scholar
  12. Chen SS, Donoho DL, Saunders MA: Atomic decomposition by basis pursuit. SIAM Journal of Scientific Computing 1998,20(1):33-61. 10.1137/S1064827596304010MathSciNetView ArticleMATHGoogle Scholar
  13. Donoho DL:For most large underdetermined systems of equations, the minimal -norm near-solution approximates the sparsest near-solution. Communications on Pure and Applied Mathematics 2006,59(7):907-934. 10.1002/cpa.20131MathSciNetView ArticleMATHGoogle Scholar
  14. Efron B, Hastie T, Johnstone I, Tibshirani R: Least angle regression. The Annals of Statistics 2004,32(2):407-499. 10.1214/009053604000000067MathSciNetView ArticleMATHGoogle Scholar
  15. Tibshirani R: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society 1996,58(1):267-288.MathSciNetMATHGoogle Scholar
  16. Donoho DL, Elad M, Temlyakov VN: Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Transactions on Information Theory 2006,52(1):6-18.MathSciNetView ArticleMATHGoogle Scholar
  17. Donoho DL, Huo X: Uncertainty principles and ideal atomic decomposition. IEEE Transactions on Information Theory 2001,47(7):2845-2862. 10.1109/18.959265MathSciNetView ArticleMATHGoogle Scholar
  18. Donoho DL, Tsaig Y:Fast solution of -norm minimization problems when the solution may be sparse. In Tech. Rep. 2006-18. Department of Statistics, Stanford University, Stanford, Calif, USA; 2006.Google Scholar
  19. Saunders MA, Kim B: PDCO: primal-dual interior method for convex objectives.
  20. Ye Y: Interior Point Algorithms: Theory and Analysis. John Wiley & Sons, New York, NY, USA; 1997.View ArticleMATHGoogle Scholar
  21. Lustig M, Donoho D, Pauly JM: Rapid MR imaging with compressed sensing and randomly under-sampled 3DFT trajectories. Proceedings of the International Society for Magnetic Resonance in Medicine (ISMRM '06), May 2006, Seattle, Wash, USAGoogle Scholar
  22. Lustig M, Santos JM, Donoho DL, Pauly JM: Rapid MR angiography with randomly under-sampled 3DFT trajectories and non-linear reconstruction. Proceedings of the 9th Annual Scientific Sessions of the Society for Cardiovascular Magnetic Resonance (SCMR '06), January 2006, Miami, Fla, USAGoogle Scholar
  23. Lustig M, Santos JM, Lee J-H, Donoho DL, Pauly JM: Compressed sensing for rapid MR imaging. Proceedings of the Signal Processing with Adaptative Sparse Structured Representations (SPARS '05), November 2005, Rennes, FranceGoogle Scholar
  24. Donoho DL, Logan BF: Signal recovery and the large sieve. SIAM Journal on Applied Mathematics 1992,52(2):577-591. 10.1137/0152031MathSciNetView ArticleMATHGoogle Scholar
  25. Candès EJ, Tao T: Decoding by linear programming. IEEE Transactions on Information Theory 2005,51(12):4203-4215. 10.1109/TIT.2005.858979View ArticleMathSciNetMATHGoogle Scholar
  26. Rudelson M, Vershynin R: Geometric approach to error-correcting codes and reconstruction of signals. International Mathematics Research Notices 2005, 2005: 4019-4041. 10.1155/IMRN.2005.4019MathSciNetView ArticleMATHGoogle Scholar
  27. Drori I, Stodden VC, Hurowitz EH:Fast minimization for genome-wide analysis of mRNA lengths. Proceedings of the IEEE International Workshop on Genomic Signal Processing and Statistics (GENSIPS '06), May 2006, College Station, Tex, USA 19-20.Google Scholar
  28. Freeman R, Kupče E: New methods for fast multidimensional NMR. Journal of Biomolecular NMR 2003,27(2):101-114. 10.1023/A:1024960302926View ArticleGoogle Scholar
  29. Malmodin D, Billeter M: High-throughput analysis of protein NMR spectra. Progress in Nuclear Magnetic Resonance Spectroscopy 2005,46(2-3):109-129. 10.1016/j.pnmrs.2005.01.002View ArticleGoogle Scholar
  30. Hoch JC, Stern AS: NMR Data Processing. John Wiley & Sons, New York, NY, USA; 1996.Google Scholar
  31. Hiller S, Fiorito F, Wüthrich K, Wider G: Automated projection spectroscopy (APSY). Proceedings of the National Academy of Sciences of the United States of America 2005,102(31):10876-10881. 10.1073/pnas.0504818102View ArticleGoogle Scholar
  32. Kupče E, Freeman R: Fast reconstruction of four-dimensional NMR spectra from plane projections. Journal of Biomolecular NMR 2004,28(4):391-395.View ArticleGoogle Scholar
  33. Orekhov VY, Ibraghimov I, Billeter M: Optimizing resolution in multidimensional NMR by three-way decomposition. Journal of Biomolecular NMR 2003,27(2):165-173. 10.1023/A:1024944720653View ArticleGoogle Scholar
  34. Donoho DL, Tsaig Y, Drori I, Starck J-L: Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit. In Tech. Rep. 2006-02. Department of Statistics, Stanford University, Stanford, Calif, USA; 2006.Google Scholar
  35. Sardy S, Bruce AG, Tseng P: Block coordinate relaxation methods for nonparametric wavelet denoising. Journal of Computational and Graphical Statistics 2000,9(2):361-379. 10.2307/1390659MathSciNetGoogle Scholar
  36. Sardy S, Bruce AG, Tseng P: Robust wavelet denoising. IEEE Transactions on Signal Processing 2001,49(6):1146-1152. 10.1109/78.923297View ArticleGoogle Scholar
  37. Daubechies I, Defrise M, De Mol C: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics 2004,57(11):1413-1457. 10.1002/cpa.20042MathSciNetView ArticleMATHGoogle Scholar
  38. Elad M: Why simple shrinkage is still relevant for redundant representations? IEEE Transactions on Information Theory 2006,52(12):5559-5569.MathSciNetView ArticleMATHGoogle Scholar
  39. Elad M, Matalon B, Zibulevsky M: Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization. Journal on Applied and Computational Harmonic Analysis 2006.Google Scholar
  40. Figueiredo MAT, Nowak RD: An EM algorithm for wavelet-based image restoration. IEEE Transactions on Image Processing 2003,12(8):906-916. 10.1109/TIP.2003.814255MathSciNetView ArticleMATHGoogle Scholar
  41. Figueiredo MAT, Nowak RD: A bound optimization approach to wavelet-based image deconvolution. Proceedings of the IEEE International Conference on Image Processing (ICIP '05), September 2005, Genova, Italy 2: 782-785.Google Scholar
  42. Paige CC, Saunders MA: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Transactions on Mathematical Software (TOMS) 1982,8(1):43-71. 10.1145/355984.355989MathSciNetView ArticleMATHGoogle Scholar
  43. Pati YC, Rezaiifar R, Krishnaprasad PS: Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition. Proceedings of the 27th Asilomar Conference on Signals, Systems & Computers, November 1993, Pacific Grove, Calif, USA 1: 40-44.View ArticleGoogle Scholar
  44. Tropp JA: Greed is good: algorithmic results for sparse approximation. IEEE Transactions on Information Theory 2004,50(10):2231-2242. 10.1109/TIT.2004.834793MathSciNetView ArticleMATHGoogle Scholar
  45. Lee PT, Li J, Chapman MR, et al.: SESAME: an experiment management system for biomolecular NMR.
  46. Daubechies I: Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, Pa, USA; 1994.Google Scholar