Data Processing Methods in MRS - Advance Image Processing in Magnetic Resonance Imaging - page 412

and where K < L < N , L being the guessed order of prediction. From a theoretical
point of view, in the absence of noise, the matrix A should have rank K . This fact
is only “approximately” true when the signal is embedded in noise. Using this
fact, Kumaresan and Tufts [36] proposed to replace the matrix A in Equation
12.14 by its K -rank approximant obtained by singular value decomposition
(SVD). The approximant is built by retaining only the K dominant singular values
of A and setting the others to zero. Therefore, instead of solving Equation 12.14
directly, a modified version (in which matrix A is substituted by its approximant)
is used. In this case, the most significant contribution of the noise can be removed,
and it can be demonstrated that frequencies and damping factors are obtained in
a more robust way [35,36]. However, the amplitudes are still unknown. In the
LPSVD approach, the problem is solved by substituting the estimated frequencies
and damping factors in Equation 12.1 and writing this equation down for all N
data points. In this way, we obtain a set of linear equations in the unknown
parameters [35]. The latter can therefore be estimated by solving
another linear system analogous to Equation 12.14. An example of metabolite

α = ae

k

j k φ

quantification using the LPSVD method is shown in Figure 12.3 . It is worth

noting that the method is able to detect and separate small metabolite contributions
that were not detectable in the absorption-mode spectra.
Several variants of the method cited in the preceding text have been applied
in the analysis of MRS data in an effort to reduce computation time and improve
accuracy in parameter estimation. Details can be found in various articles [22].
These methods try to overcome the limitations imposed by the fixed truncation
of SVD [37,38] or by polynomial rooting and root selection [39,40]. In Reference
41, frequency localization is improved by applying the KT approach in different
subband signals obtained by wavelet packet decomposition of the original FID.
With interactive methods, the estimation of model parameters v k is obtained
by solving a classical NLLS problem. A maximum likelihood (ML) estimate of
v k is achieved under the hypothesis of white Gaussian noise by minimizing the
following figure of merit (in analogy with the figure of merit introduced for
frequency-domain fitting in Equation 12.10 and Equation 12.11):

2

N N K

∑ ∑ ∑

|() ˆ ()|

ˆ ||

J = x n − x n

2

= x n () − a e

j φ

e

( −+ d j 2 π f ) n

=− || xx (12.16)

k k k

k

n = 1 n = 1 k = 1

() ˆ ()

where is the so-called prediction error (i.e., the difference between
actual sample and model-predicted sample). Minimization of Equation 12.16 can
be obtained by using the classical Levenberg-Marquardt algorithm [32] or more
sophisticated NLLS methods [42,43]. In addition, the problem can be simplified
by using the variable projection method [44]. A visual inspection of Equation 12.1
reveals that the model can be split into linear and nonlinear parts, thus obtaining

xn − xn

K

∑ ακ
1

ˆ ()

xn = ( , , ) f d n (12.17)

k k k

k =