Magnetic resonance in neuro-oncology: Achievements and challenges - Signal Processing in MRS with Biomedical Applications - page 265

of the FPT is quadratic (≈1/N 2 ) or better [17].
By way of an extract from the detailed analysis presented in chapter 7, we
now recapitulate the comparison of the resolution performance of the FPT and
the FFT on two complexvalued time signals c n (0≤n≤N−1,N = 2048)
with bandwidth = 6001.5 Hz encoded at 4T via MRS by Tkac et al. [141].
We show three partial signal lengths at a fixed bandwidth. These data of
full signal length N = 2048 encoded by the group at the Center for Magnetic
Resonance Research, University of Minnesota, Minneapolis, USA [141] have
been kindly made available to us. In Fig. 8.1 , we present the absorption total
shape spectra computed by the FFT (left column) and FPT (right column)
at three signal lengths. At the top of Fig. 8.1 the most dramatic difference
between the FFT and FPT is seen at the shortest signal length (N/16 = 128).
Here, the FFT essentially presents no meaningful spectroscopic information.
In contrast, with the FPT, at N/16 = 128 nearly 90% of the NAA concentra
tion is predicted by the peak at around 2.0 ppm.
On the middle panel at N/4 = 512 the FFT has still not predicted even
70% of the NAA concentration at 2.0 ppm. Moreover, the ratio of creatine
at about 3.0 ppm to choline at about 3.2 ppm appears to be approximately
equal to one, and thus is incorrect. On the other hand, with the FPT at
N/4 = 512, these three major peaks are nearly identical to those at full signal
length. At half signal length N/2 = 1024 on the bottom panel, the FFT
has still not demonstrated the accurate ratio between creatine and choline
at 3.0 ppm and 3.2 ppm, respectively. These two latter metabolites are still
incorrectly appearing as being almost of equal intensity. Furthermore, the
triplet of glutamine and glutamate near 2.4 ppm can be discerned at half
signal length only by the FPT, and not by the FFT.
By contrast, it is seen that at half signal length (N/2 = 1024) the FPT re
solves with fidelity more than twenty metabolites. Furthermore, while the
FFT requires the total signal length (N = 2048) to fully resolve all the
metabolites, the difference between the two FPT spectra at N = 1024 and
N = 2048 is buried entirely in the background noise [20]. In other words,
the FPT total shape spectra at halfsignal length can be treated as fully con
verged. In chapters 3 and 7, detailed presentations are given of the intrinsic
and robust error analysis of the FPT.
It is also important to point out that the FPT is shown in this and many
other examples in chapters 3 and 7 to produce no Gibbs ringing in the process
of converging in a steady fashion as a function of the increased signal length.
This is in sharp contrast to other existing parametric estimators that are
usually unstable as a function of N, typically undergoing wide oscillations
with unacceptable results before eventually converging, if at all.
We have noted that besides the computational e ciency of automatic soft
ware, the main reason for which the FFT gained attractiveness among users
is that it presents no surprises, while steadily converging with increasing sig
nal length. The FPT shares such an advantageous property of the FFT. In
addition, the FPT provides a much faster convergence rate than in the FFT