NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff...

26
NONUNIFORM SAMPLING + NON-FOURIER SPECTRUM ANALYSIS: AN OVERVIEW JEFF HOCH GRASP NMR 2017

Transcript of NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff...

Page 1: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

NONUNIFORM SAMPLING + NON-FOURIER SPECTRUM ANALYSIS: AN OVERVIEW

JEFF HOCH

GRASP NMR 2017

Page 2: NONUNIFORM SAMPLING + NON-FOURIER … 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

Page 3: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 4: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

NONUNIFORM SAMPLING

HIGH RESOLUTION REQUIRES LONG EVOLUTION TIMES

Time Frequency

FT

Page 5: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 6: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 7: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 8: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 9: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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)

Page 10: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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)

Page 11: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 12: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 13: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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)

Page 14: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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,…

!! = !!!!!!!

!!!

!! = !!!!!!!!!!!/!!!!!!"!!!!!

!!!

Page 15: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 16: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

NON-FOURIER SPECTRUM ANALYSIS

FALSE POSITIVES

l1-norm

entropy

each spectrum agreeswith the time domain datato the same extent

single injected synthetic sinusoid

Page 17: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

NON-FOURIER SPECTRUM ANALYSIS

NONRANDOM NOISE

Hyberts, Arthanari, Robson, Wagner Journal of Magnetic Resonance, Volume 241, 2014, 60–73

hmsIST

Page 18: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

NON-FOURIER SPECTRUM ANALYSIS

FREQUENCY BIAS

‘Mirror-image” LP extrapolation

Page 19: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Copyright/License ► Request permission to reuse

3

fixed-point convex optimization

Page 20: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 21: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 22: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

NON-FOURIER SPECTRUM ANALYSIS

Poverty of RMS difference or l2-norm relative to uniform DFT

Page 23: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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”

Page 24: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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

Page 25: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

NON-FOURIER SPECTRUM ANALYSIS

IN-SITU RECEIVER OPERATING CHARACTERISTIC (IROC)

Matthew Zambrello

Page 26: NONUNIFORM SAMPLING + NON-FOURIER … sampling + non-fourier spectrum analysis: an overview jeff hoch grasp nmr 2017

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