Digital Signal Processing Reference
In-Depth Information
names, e.g., spectral analysis, harmonic analysis or complexvalued moment
problem (mathematics), harmonic inversion (physics, chemistry), quantifica
tion (medicine: MRS, MRSI), etc.
We underscore the key role played by signal processing in MRS. This is due
to the fact that the most clinically relevant quantitative information about the
examined tissue cannot be extracted from encoded FIDs without analytical
methods. These mathematical methods are needed to properly interpret the
measured time signals. Such a theoretical interpretation of the encoded data,
through solving the underlying inverse problem, yields the sought information
about metabolite relaxation times, concentrations and the like. All of this
is accomplished via an adequate reconstruction of the frequencydependent
parameters for the spectral profiles of the unknown fundamental components
that are ingrained in the encoded FID. Generally, an inverse problem consists
of finding the causes from the observation or measurement of the effects of
the investigated phenomenon. In fact, nearly all important measurements in
science belong to the category of inverse problems. These inverse problems
are extremely abundant in medicine. They are the essence of many powerful
diagnostic tools without which modern hospitals and clinics would be basically
nonfunctional. The most prominent examples of these versatile diagnostic
modalities, that originate from basic research in physics and chemistry, are
NMR, MRI, MRS, MRSI, CT, PET, SPECT, US, etc. The common feature
of these powerful diagnostic tools is the impossibility for direct interpretations
of the measurements. Acquisition of the needed information requires theory
whose mathematical methods are capable of solving reconstruction or inverse
problems that are inherent in all the mentioned diagnostic modalities.
Surprisingly, for too long a time now, signal processing in MRS has been
dominated by the FFT. Such an occurrence is at variance with the quan
tification problem, which is the very goal of MRS. As noted, the FFT is a
nonquantifying estimator of envelope spectra. This qualitative information is
usually complemented in MRS by fitting either encoded FIDs or Fourier spec
tra. These fittings employ some free, LSadjusted parameters in attempts
to quantify the given MRS data. However, none of the fitting algorithms
from MRS and elsewhere can provide the unequivocal solution for the men
tioned inverse problem. Namely, fittings cannot yield the unique estimates
for the sought spectral parameters that are the positions, widths, heights and
phases of the peak profiles, nor about the total number of resonances. As seen
herein, this is the case because a given total shape spectrum can be fitted by
several different sets of component shape spectra. Each set contains a num
ber of metabolites. However, any fixed resonance can be of vastly different
spectral characteristics when passing from one to another set of component
shape spectra. Thus, these fittings are nonunique in MRS and in other
research fields. This typically leads to two di
culties in practice, such as
overfitting and underfitting. Overfitting retrieves some nonexistent, i.e.,
spurious metabolites. Underfitting fails to reconstruct some of the existing,
i.e., genuine metabolites. In both cases, a fitting bias exists, entailing guessing
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