JEFF HOCH NONUNIFORM SAMPLING + NON-FOURIER SPECTRUM ANALYSIS: AN OVERVIEW

Transcription

1 GRASP NMR 2017 JEFF HOCH NONUNIFORM SAMPLING + NON-FOURIER SPECTRUM ANALYSIS: AN OVERVIEW

2 NONUNIFORM SAMPLING + NON-FOURIER SPECTRUM ANALYSIS NUS+NONFOURIER 4D: HCC(CO)NH-TOCSY Experiment time US 150 days NUS 1.5 days LP 64->2048 DFT NUS 64 out of 2048 MaxEnt Resolution ω2-1 H (pp m ) { Sensitivity US NUS ω 3-13 C (ppm)

3 NONUNIFORM SAMPLING NMR GAINS A DIMENSION Jean Jeener Ampere Summer School Basko Polje, Yugoslavia, 1971 t 2 t 1 2D NMR via parametric sampling of an indirect dimension f 2 t 1 f 1 f 2 + Generalizes to higher dimensionality - Limited practicality: high resolution requires long data records

4 NONUNIFORM SAMPLING HIGH RESOLUTION REQUIRES LONG EVOLUTION TIMES FT Time Frequency

5 NONUNIFORM SAMPLING two decaying sinusoids close in frequency short evolution times high sensitivity low resolution long evolution times low sensitivity high resolution

6 Sensitivity: intrinsic signal to noise ratio (isnr) signal(t max ) R t max e t T 2 0 noise(t max ) p t max 9 = ;! SNR(t max) T 2 1 e t max T 2 p tmax SNR max = 1.26 x T2 GRASP NMR 2017 Rovnyak et al. Magn.Reson.Chem. 2011

7 NONUNIFORM SAMPLING HIGHT-RESOLUTION HIGH-DIMENSION EXPERIMENTS RAPIDLY BECOME IMPRACTICAL t 1 t t 1 t 2 t2 t 3 1D 1 FID 2 sec. 2D t 1 : 128 FIDs 8 min. 3D t 1 x t 2 : 128x128=16,384 FIDs 36 hrs. Experiment time depends on the number of samples in the indirect dimensions 4D t 1 x t 2 x t 3 : 32x32x32=32,768 FIDs 6 days High resolution: a year!

8 NONUNIFORM SAMPLING NUS: LONG EVOLUTION TIMES, BUT NOT ALL INTERVENING TIMES sample space dwell time uniform sampling (US) dwell time indel = indirect element nonuniform sampling (NUS)

9 How does NUS spectrum relate to US spectrum? US time data 3D experiment 2D plane colored dot = FID data US spectrum sample schedule black dot = 1 white space = 0 point-spread function (PSF) NUS time data NUS spectrum [zero filled] GRASP NMR 2017

10 A MENAGERIE OF CHOICES 1983 LP extrapolation Ni & Scheraga (1983) MaxEnt reconstruction Sibisi et al. (1983,1984) Burg MEM Martin; Hoch (1985) LPSVD van Ormondt et al. (1985) Exponential sampling Laue, Skilling et al. (1987) Iterative thresholding (Jansson-Van Cittert) Scheraga et al. (1989) Complex entropy Hore et al., Hoch et al. (1990) Bayesian, MLM Bretthorst; Chylla & Markley (1990; 1993) GFT Szyperski et al. (1993) Wavelet transform, iterative soft thresh Hoch & Stern (1996) Filter diagonalization Mandelshtam, Shaka et al. (2000) Multiway decomposition Billeter, Orekhov et al. (2001) Back projection Kupče & Freeman (2002) Now Covariance NMR Brüschweiler et al. (2004) Compressed sensing Donoho et al. (2004) APSY, HIFI Wüthrich et al.; Markley Eghbalnia et al. (2005) Multidimensional FT Kozminski et al. (2006) Forward MaxEnt (FM) Hyberts, Wagner (2009) SCRUB Coggins, Zhou (2012) NESTA-NMR Byrd et al. (2015) CAMERA Worley (2016)

11 high DFT periodicity MaxEnt, l1-norm, NESTA, CS random noise, sparsity MWD symmetry Robustness low MLM, FDM, LPSVD, Bayesian, CLEAN, SCRUB SMILE exponential decay weak strong Assumptions

12 REGULARIZATION METHODS Data d Trial spectrum f Mock data (= idft(f)) Maximize S=(entropy, l 1 -norm, l 2 -norm, etc.) Subject to noise 0 Bayesian limit MINT, FM limit In the MINT limit, regularization approaches are approximately linear (preserve norms, lineshapes) However this corresponds to statistical over-fitting, prone to false positives

13 MULTI-WAY DECOMPOSITION (OREKHOV, BILLETER) S(f 1, f 2, f 3 ) = s(f 1 ) s(f 2 ) s(f 3 ) Also known as PARAFAC; PCA is a 2D equivalent also (Denk & Wagner; Rinaldi et al.) Less restrictive than the exponential decay assumption Random noise does not have a decomposition Unique decomposition restricted to 3+ dimensions

14 NON-FOURIER SPECTRUM ANALYSIS EXPONENTIAL DECAY (LORENTZIAN LINE SHAPE) MODEL LP extrapolation, LPSVD, HSVD, Burg MaxEnt, FDM, MLM, CLEAN, LP, LPSVD, HSVD, Burg maximum entropy Bayesian, MLM, FDM, CLEAN,!!!! =!!!!!!!!!!! =!!!!!!!!!!!/!!!!!!"!!!!!!! Being parametric, they yield frequency lists directly Prone to bias, false positives when S/N is low or signals non-ideal

15 PITFALLS convergence fixed-point methods exhibit premature convergence nonrandom noise false positives exponential decay models view everything as a peak: false positives bias violated assumptions can lead to biased (but visually pleasing) results frequency bias: LP extrapolation can lead to frequency errors (RDCs, missed correlations) phase distortions: some regularization functionals are not phase-insensitive non-uniqueness strictly convex regularizations functionals guarantee uniqueness; l 1 -norm is not strictly convex

16 FALSE POSITIVES single injected synthetic sinusoid l 1 -norm entropy each spectrum agrees with the time domain data to the same extent

17 NONRANDOM NOISE hmsist Hyberts, Arthanari, Robson, Wagner Journal of Magnetic Resonance, Volume 241, 2014, 60 73

18 FREQUENCY BIAS Mirror-image LP extrapolation

19 FIXED-POINT METHODS: PREMATURE CONVERGENCE 3 fixed-point convex optimization Page 1 of 2

20 UNIVERSAL BEHAVIOR: TRUST REGIONS Undersampling Theories CS Theory Phase Transition, (, )PhaseDiagram N uniform sample grid 1 Fixed experiment time n subsamples Sparsity ratio 0.8 Sampling coverage n/n Sparsity ratio ε Failure 0.2 Success Sampling coverage Hatef Monajemi Stanford University, CA May 25, / 23 δ

21 COMPARISONS AND QUALITY METRICS with nonlinear methods: RMS differences, SNR not reliable for comparisons comparison of spectra that differ in their agreement with the empirical data (idft of the spectrum compared to the data) are difficult, if not dubious

22 Poverty of RMS difference or l 2 -norm relative to uniform DFT

23 IN-SITU ANALYSIS: QUALITY METRICS FOR ARBITRARY SIGNAL PROCESSING METHODS Ground truth Synthetic signals added to time-domain data Volume of synthetic peak measured in spectrum Amplitude of synthetic signals

24 SNR SENSITIVITY Nonlinear spectral estimates can improve S/N without improving sensitivity Noise and signals are both scaled down Peaks separated by a noise threshold on the left are still separated on the right

25 IN-SITU RECEIVER OPERATING CHARACTERISTIC (IROC) Matthew Zambrello

26 Published 2017 New Developments in NMR Fast NMR Data Acquisition Beyond the Fourier Transform All methods in NMRbox Edited by Mehdi Mobli and Jeffrey C Hoch