Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 18

tions since, as mentioned, changing some free parameters could easily lead to
large and unphysical variations in the other adjustable parameters. Such a ba
sic defect of fitting is then generalized by the users of fittings to state that all
the adequate algorithms for quantification should introduce prior information.
This generalization is misleading. In principle, certain physically motivated
prior information could be imposed upon the Pade estimation, if needed.
In fact, the adequate prior information to the Schrodinger eigenproblem is
already inherently introduced by the proper initial or boundary condition.
Moreover, according to the main working postulate of quantum mechanics,
the reconstructed total Schrodinger eigenstate contains the whole informa
tion about the studied system. Hence, there is no need to impose any other
prior information, since the correct solution of the Schrodinger eigenproblem
leads to the true eigenspectrum with the sought complexvalued fundamental
frequencies and amplitudes. Should some of these spectral parameters obey
certain relationships that are known to exist prior to quantification, such re
lationships would also be inherently ingrained in the investigated FID and,
hence, extractable from the retrieved Schrodinger eigenspectrum with no im
posed constraints. In parametric methods, prior information is used through
a structure of the given time signal. The FPT is a parametric estimator
which is also a quantummechanical eigenproblem solver [5]. This method
employs directly or indirectly the Schrodinger equation, which can be solved
exactly in a finitedimensional space through the Krylov or the Lanczos basis
set functions [5].
Within the FPT, computation is done by systematically enlarging the di
mension of the Schrodinger state space (also carrying the name of Krylov) with
the purpose of monitoring and detecting constancy of the retrieved spectral
parameters. This represents a procedure of proven validity for demonstrating
that the information uncovered from the given FID is complete, without the
need to introduce any constraint by some prior information. Therefore, prior
information used in fitting techniques becomes superfluous and unsubstan
tiated by quantummechanical spectral analysis, which quantifies MRS data
without any fitting, as has previously been demonstrated within the FPT
[5, 24] and this will also be illustrated in the present book. Even describ
ing FIDs from MRS by linear combinations of complex damped exponentials
with both stationary and nonstationary amplitudes, as done in the FPT, can
not be considered as an imposed, constrained relationship. This comes from
the fact that precisely such a relationship follows directly from the two main
entities of quantum physics - the exact evolution operator and the related
autocorrelation functions.
Recent vigorous advances of the Pade methodology in MRS and MRI have
made it evident that the FPT is a very reliable and robust signal processor,
which gives the full and intrinsically crossvalidated spectral information of
great significance for medical diagnostics [16]-[37], as will also be thoroughly
reviewed in this topic. Such a finding from the MR literature is reminiscent
of the wellestablished status of the general Pade theory over the years across