Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 15

in the input data 1 . Various fitting procedures used in the time and frequency
domains within MRS were reviewed in Refs. [79]-[83]. However, all such fit
tings constitute a naive approach to spectral analysis due the usage of some
ad hoc mathematical formulae that do not stem from any adequate physi
cal description of the underlying dynamics. This basic inadequacy is usually
patched by employing some prior information in order to artificially constrain
the otherwise completely arbitrary variations of adjustable parameters.
Conceptually different from fitting, there exists another strategy for sig
nal processing in MRS. This is quantummechanical spectral analysis which
converts the quantification problem from MRS into the Schrodinger eigen
value problem of the evolution matrix or the Hankel (data) matrix. Here,
diagonalization of such matrices yields the spectral parameters. Rigorously
the same results can alternatively be obtained without diagonalization by
solving the corresponding secular equation through rooting the characteristic
polynomial. Being parametric and noniterative, this kind of estimator can
unequivocally reconstruct the unknown complex frequencies and amplitudes
of each physical resonance. With such processors, these main resonance pa
rameters are retrieved first, and it is only afterwards that the corresponding
component and total shape spectra can be constructed, if desired. In other
words, this represents a completely different methodology from fitting estima
tors that need the precomputed envelope spectrum prior to their attempts
to quantify the encoded data. These two distinctly different approaches, i.e.,
ambiguous fittings and unambiguous quantum estimation are anticipated to
give appreciably different results, particularly regarding closely overlapping
resonances. Experience shows that spectra in MRS are abundant with tightly
overlapped resonances. These latter spectral structures are often of primary
clinical importance.
An example of quantummechanical parametric estimators is the Hankel
Lanczos Singular Value Decomposition (HLSVD) which is in frequent use
within MRS [74]. The HLSVD arranges the encoded raw FID points{c n

}as

a data matrix of the Hankel form{c i+j+1 }, which is subsequently diagonalized

to yield the fundamental complex frequencies for the reconstructed harmon
ics from the time signal. The associated complex amplitudes are extracted
in the HLSVD subsequently through a description of the studied FID by a
linear combination of complex damped exponentials with timeindependent
amplitudes. Such a description yields a system of linear equations for the
sought amplitudes. This latter system uses all the found frequencies (gen
uine and spurious). Such a mix of frequencies is disadvantageous, since the
admixture of even a slight amount of extraneous frequencies could severely
undermine the reliability of the estimates for the amplitudes in the HLVSD.

1 Here, in the context of fitting by, e.g., VARPRO, AMARES, etc., within MRS, “compo-
nents” are the constituent resonances in a total shape spectrum. More generally, “compo-
nents” could be, e.g., the individual species in a chemical or any other compound/sample
or the individual compartments in a multi-compartment tissue/system.