Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 19

interdisciplinary scientific and engineering research, including physics, chem
istry, biology, signal processing, system theory, circuit theory, speech pattern
recognition, etc. A recent review of the versatile use of the PA in such vastly
different research branches has been provided in Ref. [5]. Further, relevance
of the FPT for medical diagnostics, particularly in clinical oncology has also
been been thoroughly analyzed [7]. We reemphasize that the PA and FPT
belong to the same general Pade methodology. It has been shown in quantum
chemistry that the FPT can carry out parametric analysis of theoretically gen
erated and experimentally measured FIDs through extremely accurate com
putation of complex frequencies and amplitudes of the reconstructed physical
resonances [5, 28, 34]. Such a conclusion is also relevant to medicine, es
pecially when quantifying FIDs that are encoded from patients via MRS or
MRSI. In these reconstructions, the quantification problem is recognized as
the harmonic inversion problem and/or the complexvalued power moment
problem [5]. The task of this inverse problem is to reconstruct the proper
number K of physical resonances, as well as their true fundamental complex
frequencies and amplitudes{ω k ,d k }(1≤k≤K) that build the given FID,
which is the only input data available.
Regarding the quantification problem on synthesized FIDs, we will select
the input complex frequencies and amplitudes according to other similar data
in the MRS literature [88]. In order to secure a robust error analysis and cross
validation of the obtained results, both versions of the FPT via the FPT (+)
and FPT (−) will be employed throughout. One of the main objectives of the
exemplified quantifications in MRS is to assess the overall performance of the
FPT for parametric estimation of fully controlled theoretical FID with and
without noise. The FPT is scrutinized rigorously against these entry data by
testing for the possibility of reconstructing all the spectral parameters exactly.
These testings also aim at verifying whether the FPT is stable and reliable
while retrieving the total number of resonances and their complex frequencies
and amplitudes by exhausting only a portion of the full signal length N.
A successful outcome of such a testing would have far reaching ramifications
beyond onedimensional (1D) MRS and thus also become important for two
dimensional (2D) MRS, 2D MRSI as well as threedimensional (3D) MRSI
and MRI. In particular, for 2D MRS, one of the total acquisition times is a
fraction of the corresponding value from 1D MRS. Therefore, whenever the
2D FFT is employed to compute crosscorrelation plots using in vivo 2D FIDs
encoded via 2D MRS, lowresolution results are obtained along the frequency
axis associated with the short total acquisition time [89]. In such cases, no
appreciable improvement in the results of the FFT is achieved by zero filling
of the FID on the short time axis. Hence the need for signal processors that
possess extrapolation features. Such features are automatically built into the
FPT via modeling the response function by polynomial quotients. These
latter rational functions describe adequately and realistically the manner in
which the examined physical system responds to external perturbations. As
opposed to the FFT, the FPT predicts that the system's answer to these