Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 193

6

Machine accurate quantification and
illustrated signal-noise separation

6.1 Formulation of the most stringent test for quantifi-

cation in MRS

Time signals with their fundamental harmonics are spectrally analyzed while
solving the quantification problem. The solution of this problem permits re
construction of all the physical complexvalued frequencies and the associated
amplitudes. These spectral parameters are the sole constituents of the funda
mental harmonics of the investigated time signal. Every single harmonic in its
role of a transient possesses its resonant frequency, relaxation time, intensity
and phase. Such elements completely determine the underlying normalmode
damped oscillations in the generic signal. These four realvalued parameters
can be directly deduced from the retrieved complex frequencies and ampli
tudes to give the peak areas of the associated resonance profiles in the cor
responding spectrum. For clinical diagnostics via MRS, the most relevant
quantities are metabolite concentrations that are proportional to the recon
structed peak areas. Therefore, metabolite concentrations of the examined
tissue can be extracted from the spectrally analyzed time signal. Hence the
clinical importance of the quantification problem in MRS.
In this chapter, we shall tackle the most di cult numerical aspects in solv
ing the quantification problem in MRS. The primary goal of chapters 4 and
5 was to establish the methods of rational approximations and their algo
rithms that would be capable of performing a rigorous computation within
finite arithmetics to retrieve exactly all the input spectral parameters of each
resonance from synthesized noiseless time signals. Specifically, the method
ologies expounded in the two preceding chapters would be considered as fully
adequate if they could produce machine accurate output from machine ac
curate input data. A favorable outcome from this evidently most stringent
test would demonstrate that the estimator selected for this reconstruction
problem is extraordinarily accurate, stable and robust even against roundoff
errors from computations.
Conversely, any estimator which does not meet these highest standards im
posed onto spectral analysis with ideal input time signals would hardly have

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