Journal of Mathematical Chemistry

, Volume 45, Issue 3, pp 563–597 | Cite as

The general concept of signal–noise separation (SNS): mathematical aspects and implementation in magnetic resonance spectroscopy

Original Paper

Abstract

Magnetic resonance spectroscopy (MRS) and spectroscopic imaging (MRSI) are increasingly recognized as potentially key modalities in cancer diagnostics. It is, therefore, urgent to overcome the shortcomings of current applications of MRS and MRSI. We explain and substantiate why more advanced signal processing methods are needed, and demonstrate that the fast Padé transform (FPT), as the quotient of two polynomials, is the signal processing method of choice to achieve this goal. In this paper, the focus is upon distinguishing genuine from spurious (noisy and noise-like) resonances; this has been one of the thorniest challenges to MRS. The number of spurious resonances is always several times larger than the true ones. Within the FPT convergence is achieved through stabilization or constancy of the reconstructed frequencies and amplitudes. This stabilization is a veritable signature of the exact number of resonances. With any further increase of the partial signal length N, towards the full signal length N, i.e., passing the stage at which full convergence has been reached, it is found that all the fundamental frequencies and amplitudes “stay put”, i.e., they still remain constant. Moreover, machine accuracy is achieved here, proving that when the FPT is nearing convergence, it approaches straight towards the exact result with an exponential convergence rate (the spectral convergence). This proves that the FPT is an exponentially accurate representation of functions customarily encountered in spectral analysis in MRS and beyond. The mechanism by which this is achieved, i.e., the mechanism which secures the maintenance of stability of all the spectral parameters and, by implication, constancy of the estimate for the true number of resonances is provided by the so-called pole-zero cancellation, or equivalently, the Froissart doublets. This signifies that all the additional poles and zeros of the Padé spectrum will cancel each other, a remarkable feature unique to the FPT. The FPT is safe-guarded against contamination of the final results by extraneous resonances, since each pole due to spurious resonances stemming from the denominator polynomial will automatically coincide with the corresponding zero of the numerator polynomial, thus leading to the pole-zero cancellation in the polynomial quotient of the FPT. Such pole-zero cancellations can be advantageously exploited to differentiate between spurious and genuine content of the signal. Since these unphysical poles and zeros always appear as pairs in the FPT, they are viewed as doublets. Therefore, the pole-zero cancellation can be used to disentangle noise as an unphysical burden from the physical content in the considered signal, and this is the most important usage of the Froissart doublets in MRS. The general concept of signal–noise separation (SNS) is thereby introduced as a reliable procedure for separating physical from non-physical information in MRS, MRSI and beyond.

Abbreviations

Ala

Alanine

AMARES

Advanced Method for Accurate Robust and Efficient Spectral fitting

Asp

Aspartate

au

Arbitrary units

BPH

Benign prostatic hypertrophy

Cho

Choline

Cr

Creatine

Crn

Creatinine

CT

Computerized tomography

FID

Free induction decay

FFT

Fast Fourier transform

FPT

Fast Padé transform

GABA

Gamma amino butyric acid

Glu

Glutamate

Gln

Glutamine

Glc

Glucose

GPC

Glycerophosphocholine

1H MRS

Proton magnetic resonance spectroscopy

HLSVD

Hankel–Lanczos Singular Value Decomposition

Ins

Inositol

Iso

Isoleucine (Iso)

Lac

Lactate

LCModel

Linear Combination of Model in vitro Spectra

Lip

Lipid

Lys

Lysine

Met

Methionine

MR

Magnetic resonance

MRI

Magnetic resonance imaging

MRS

Magnetic resonance spectroscopy

MRSI

Magnetic resonance spectroscopic imaging

ms

Milliseconds

NAA

N-acetyl aspartate

NAAG

N-acetyl aspartyl glutamate

NMR

Nuclear magnetic resonance

PA

Padé approximant

PCho

Phosphocholine

PCr

Phoshocreatine

PET

Positron emission tomography

ppm

Parts per million

PSA

Prostate specific antigen

RT

Radiation therapy

SNR

Signal-to-noise ratio

SNS

Signal–noise separation

Tau

Taurine

TE

Echo time

Thr

Threonine

Val

Valine

VARPRO

Variable Projection Method

ww

Wet weight

1D

One dimensional

2D

Two dimensional

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Oncology–PathologyKarolinska InstituteStockholmSweden
  2. 2.Institute for Prevention ResearchUniversity of Southern California Keck School of MedicineLos AngelesUSA

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