NONUNIFORM SAMPLING + NON-FOURIER SPECTRUM ANALYSIS: AN OVERVIEW
JEFF HOCH
GRASP NMR 2017
NONUNIFORM SAMPLING + NON-FOURIER SPECTRUM ANALYSIS
NUS+NONFOURIER
ω2
- 1mp
p(
H)
ω3 - 13C (ppm)
NUSUS{ Sensitivity
Resolution
Experiment time
US
US 150 days NUS 1.5 days
4D: HCC(CO)NH-TOCSY
LP 64->2048 DFT
NUS 64 out of 2048 MaxEnt
NUS
NONUNIFORM SAMPLING
NMR GAINS A DIMENSION
Jean Jeener
Ampere Summer SchoolBasko Polje, Yugoslavia, 1971
2D NMR via parametric sampling of anindirect dimension
t
t
2
1
f
t
2
1
f
f
2
1
+ Generalizes to higher dimensionality - Limited practicality: high resolution requires long data records
NONUNIFORM SAMPLING
HIGH RESOLUTION REQUIRES LONG EVOLUTION TIMES
Time Frequency
FT
NONUNIFORM SAMPLING
two decaying sinusoids close in frequency
0 20 40 60 80 100 120 140 160 180 200−1
−0.5
0
0.5
1
0 20 40 60 80 100 120 140 160 180 200−1
−0.5
0
0.5
1
0 20 40 60 80 100 120 140 160 180 200−1
−0.5
0
0.5
1
short evolution times high sensitivity low resolution
long evolution times low sensitivity high resolution
GRASP NMR | 2017
Sensitivity: intrinsic signal to noise ratio (iSNR)
Rovnyak et al. Magn.Reson.Chem. 2011
SNR max = 1.26 x T2
signal(tmax
) ⇡R t
max
0
e�tT2
noise(tmax
) ⇡ptmax
9=
; ! SNR(tmax
) ⇡T2
⇣1� e
�tmax
T2
⌘
ptmax
NONUNIFORM SAMPLING
HIGHT-RESOLUTION HIGH-DIMENSION EXPERIMENTS RAPIDLY BECOME IMPRACTICAL
t
1 FID 2 sec.1D
t2
t1
t1: 128 FIDs 8 min. 2D
t1
t2
t3
t1x t2: 128x128=16,384 FIDs 36 hrs.3D
t1x t2 x t3 : 32x32x32=32,768 FIDs 6 days4D
High resolution: a year!
Experiment time depends on the number of samples in the indirect dimensions
NONUNIFORM SAMPLING
NUS: COLLECT @ LONG EVOLUTION TIMES, BUT NOT ALL INTERVENING TIMES
dwell time
dwell time
sample space uniform sampling (US)
nonuniform sampling (NUS)
indel = indirect element
GRASP NMR | 2017
How does NUS spectrum relate to US spectrum?
US time data3D experiment 2D plane colored dot = FID data
sample schedule black dot = 1 white space = 0
NUS time data
US spectrum
NUS spectrum [zero filled]
point-spread function (PSF)
NON-FOURIER SPECTRUM ANALYSIS
A MENAGERIE OF CHOICES
MaxEnt reconstruction Sibisi et al. (1983,1984)Burg MEM Martin; Hoch (1985)
LPSVD van Ormondt et al. (1985)
Iterative thresholding (Jansson-Van Cittert) Scheraga et al. (1989)Complex entropy Hore et al., Hoch et al. (1990)
Filter diagonalization Mandelshtam, Shaka et al. (2000)Multiway decomposition Billeter, Orekhov et al. (2001)
Back projection Kupče & Freeman (2002)
Compressed sensing Donoho et al. (2004)
Wavelet transform, iterative soft thresh Hoch & Stern (1996)
“Multidimensional FT” Kozminski et al. (2006)
LP extrapolation Ni & Scheraga (1983)
Covariance NMR Brüschweiler et al. (2004)
Exponential sampling Laue, Skilling et al. (1987)
Bayesian, MLM Bretthorst; Chylla & Markley (1990; 1993) GFT Szyperski et al. (1993)
APSY, HIFI Wüthrich et al.; Markley Eghbalnia et al. (2005)
NESTA-NMR Byrd et al. (2015)
Now
1983
CAMERA Worley (2016)
Forward MaxEnt (FM) Hyberts, Wagner (2009) SCRUB Coggins, Zhou (2012)
NON-FOURIER SPECTRUM ANALYSIS
Assumptions
Robustness
exponential decay
symmetry
random noise, sparsity
periodicityDFT
MaxEnt, l1-norm, NESTA, CS
MWD
MLM, FDM, LPSVD,Bayesian, CLEAN, SCRUB SMILE
high
low
weak strong
NON-FOURIER SPECTRUM ANALYSIS
REGULARIZATION METHODS
Maximize
Subject to
S=(entropy, l1-norm, l2-norm, etc.)
≈ noise
Data d Trial spectrum f Mock data (= iDFT(f))
Bayesian limit
≈ 0 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
NON-FOURIER SPECTRUM ANALYSIS
MULTI-WAY DECOMPOSITION (OREKHOV, BILLETER)
also (Denk & Wagner; Rinaldi et al.)
Less restrictive than the exponential decay assumptionRandom noise does not have a decompositionUnique decomposition restricted to 3+ dimensions
Also known as PARAFAC; PCA is a 2D equivalent
S(f1, f2, f3) = s(f1)⍟s(f2)⍟s(f3)
NON-FOURIER SPECTRUM ANALYSIS
EXPONENTIAL DECAY (LORENTZIAN LINE SHAPE) MODEL
▸ LP extrapolation, LPSVD, HSVD, Burg MaxEnt, FDM, MLM, CLEAN,…
Being parametric, they yield frequency lists directly Prone to bias, false positives when S/N is low or signals non-ideal
⦿ LP, LPSVD, HSVD, Burg maximum entropy
⦿ Bayesian, MLM, FDM, CLEAN,…
!! = !!!!!!!
!!!
!! = !!!!!!!!!!!/!!!!!!"!!!!!
!!!
NON-FOURIER SPECTRUM ANALYSIS
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; l1-norm is not strictly convex
NON-FOURIER SPECTRUM ANALYSIS
FALSE POSITIVES
l1-norm
entropy
each spectrum agreeswith the time domain datato the same extent
single injected synthetic sinusoid
NON-FOURIER SPECTRUM ANALYSIS
NONRANDOM NOISE
Hyberts, Arthanari, Robson, Wagner Journal of Magnetic Resonance, Volume 241, 2014, 60–73
hmsIST
NON-FOURIER SPECTRUM ANALYSIS
FREQUENCY BIAS
‘Mirror-image” LP extrapolation
NON-FOURIER SPECTRUM ANALYSIS
FIXED-POINT METHODS: PREMATURE CONVERGENCE
6/9/14, 10:10 PMPubMed Central, 3: J Magn Reson. Oct 2007; 188(2): 295–300. Published online Aug 3, 2007. doi: 10.1016/j.jmr.2007.07.008
Page 1 of 2http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3199954/figure/F3/
<< Prev 3 Next >>PMC full text: J Magn Reson. Author manuscript; available in PMC Oct 24, 2011.Published in final edited form as:
J Magn Reson. Oct 2007; 188(2): 295–300.Published online Aug 3, 2007. doi: 10.1016/j.jmr.2007.07.008
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3
fixed-point convex optimization
NON-FOURIER SPECTRUM ANALYSIS
UNIVERSAL BEHAVIOR: TRUST REGIONSUndersampling Theories CS Theory
Phase Transition, (✏, �) Phase Diagram
δ
ε
0.0 0.2 0.4 0.6 0.8 1.0
0
0.2
0.4
0.6
0.8
1
Failure
Success
Hatef Monajemi Stanford University, CA May 25, 2016 20 / 23
Sampling coverage
Spar
sity
ratio
Fixed experiment timeN uniform sample grid
n subsamples
Sparsity ratio
Sampling coverage n/N
NON-FOURIER SPECTRUM ANALYSIS
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
NON-FOURIER SPECTRUM ANALYSIS
Poverty of RMS difference or l2-norm relative to uniform DFT
NON-FOURIER SPECTRUM ANALYSIS
IN-SITU ANALYSIS: QUALITY METRICS FOR ARBITRARY SIGNAL PROCESSING METHODS
Synthetic signals added to time-domain data Volume of synthetic peak
measured in spectrum
Amplitude of synthetic signals
“Ground truth”
NON-FOURIER SPECTRUM ANALYSIS
SNR ≠ SENSITIVITY
Nonlinear spectral estimates can improveS/N without improving sensitivity
Noise and signals are both scaled downPeaks separated by a noise threshold on the left are still separated on the right
NON-FOURIER SPECTRUM ANALYSIS
IN-SITU RECEIVER OPERATING CHARACTERISTIC (IROC)
Matthew Zambrello
NON-FOURIER SPECTRUM ANALYSIS
New Developments in NMR
Fast NMR Data AcquisitionBeyond the Fourier Transform
Edited by Mehdi Mobli and Jeffrey C Hoch
Published 2017
All methods in NMRbox