Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 17

Unification of these two computationally different algorithms into a single
concept and methodology yields the clause 'Pade transform' in the acronym
FPT. Further, the adjective 'fast' in the FPT stands to indicate that the enve
lope spectra from the FPT can optionally be computed by the fast N(ln 2 N) 2
Euclid algorithm [86, 87]. In its role of a transform, the FPT does not need to
reconstruct the spectral parameters first. Such an application of the FPT can
be implemented through, e.g., acceleration of Fourier sequences by the well
known Wynn ε−nonlinear recursion. The Wynn algorithm has been invented
to alleviate a cumbersome, direct evaluation of quotients of Hankel determi
nants that enter the definition of the Shanks transform (ST) [5]. The Shanks
transform is an extension of the Aitken 2 −extrapolation from one to an ar
bitrary number of harmonics (further details are given in chapter 2). When
a sequence or a series to be nonlinearly transformed is comprised of partial
sums, as is the case in the acceleration of Fourier sequences by the Wynn
algorithm in the frequency domain, the Shanks transform is identical to the
Pade approximant. In such a case, the Wynn ε−algorithm is simply a recur
sive generation of the Pade approximant in a frequencybyfrequency sweeping
throughout a chosen window in the spectrum or in the whole Nyquist range.
In the present context of MRS, the Wynn recursion is extremely e cient if
applied to many frequencies because the Fourier sequences to be accelerated
are very short, containing barely a dozen precomputed FFT spectra of in
creasing length, N = 2 s (s = 0, 1,...) [12]. These sequences are short, since
the full length of a typical FID encoded at clinical scanners by means of MRS
for a chosen bandwidth does not ordinarily go beyond N = 2048, and this
corresponds to 2 s for s = 11.
By construction, the FPT is set up to work for both Lorentzian (non
degenerate) and nonLorentzian (degenerate) spectra. The latter degenerate
spectra include peaks due to multiple roots of the denominator polynomial
Q K in the defining quotient from the FPT. Crucially, being a rational response
function, the FPT is simultaneously an interpolator and extrapolator. A par
ticularly important feature of the FPT is extrapolation, since it leads directly
to resolution enhancement. Moreover, unlike the resolution ω min = 2π/T in
the FFT, resolution ω ave in the FPT is not restricted critically by T. Instead
ω ave depends upon the average density of resonances in the given frequency
window. In most situations, we have ω ave < ω min . Such a circumstance per
mits resolution enhancement below the RayleighFourier limit 2π/T [5, 13].
Among users of fitting prescriptions for in vivo MRS, there is a perception
that, in general, successful quantification should include some kind of prior
information. This is certainly untrue for nonfitting algorithms such as the
FPT or HLSVD. The real reason for which some prior information is used at
all in fitting recipes should be clarified. Ample evidence proves that fittings in
MRS customarily employ certain chosen prior information (e.g., a fixed ratio
among the heights or widths of some resonant peaks or the like), exclusively to
constrain the possibly wide variations of the otherwise free adjustable param
eters. Without such constraints, fittings usually yield inadequate reconstruc