Data Processing Methods in MRS - Advance Image Processing in Magnetic Resonance Imaging - page 410

β G are the weighted factors used to create the mixture of L and G
curves and w ( f ) is a white Gaussian noise. The values of

where β L and

β G can be either
fixed a priori or included in the parameters to be estimated. Also, the term
is added to account for possible baseline distortions that may be derived
from underlying signals or unwanted resonances. A polynomial can be used to
model the baseline, p being the polynomial order and c p its coefficients.
The estimation of model parameters v k is obtained by solving a classical non-
linear least-square (NLLS) problem, by minimizing the following figure of merit

β L and

p

∑ = cf

p

p 1 p

−
∑

N 1

) ˆ ( | Xn f ( ∆ − Xn f ∆ ) |

2

n = 0

(12.10)

2

−
∑

N 1 K P

∑ ∑

= Xn f ( ∆ ) − ( β L ( v , n ∆ f ) + β Gn ( v , ∆ f )) + c f

p

Lk k Gk k p

n = 0 k = 1 p = 1

ˆ (

where is the difference between the actual and model values of
the spectrum. Both real and imaginary parts of X ( f ) are considered in the fitting.
Using vector notation, J can be rewritten in a more compact form:

Xn f ( ∆ ) − Xn f ∆ )

ˆ

J =− || XX || (12.11)

where || || represents the Euclidean norm, X = [ X (0), X ( ∆ f ), … , X (( N − 1) ∆ f )] T ,

ˆ ˆ (), ˆ ( , ˆ ((

and T indicates the matrix transpose.
Because Equation 12.11 is nonlinear in the parameter, the Levenberg-Marquardt
[32] algorithm is often used to solve the problem [15]. Several fitting algorithms
operating in the frequency domain have been developed in recent years [15].
Some of them [33,34] allow for the inclusion of a priori knowledge between
spectral components. In fact, known relationships sometimes do exist between
spectral lines, such as known amplitude and damping ratios, and frequency and
phase shifts. These relationships are translated into constraints among spectral
parameters, which may be included in Equation 12.11, thus reducing the number
of parameters to be fitted. This operation dramatically increases estimation accu-
racy and decreases computation time. Prior knowledge is particularly important
to resolve overlapping peaks or to impose common line widths in noise spectra
to improve accuracy of the other estimates.

XX Xf = [ 0 ∆ … , XN − 1 ) ∆ f )]

T

12.5 TIME-DOMAIN METHODS

Quantitative evaluation of metabolite parameters may be obtained by processing
the time-domain FID signals. Two main categories of methods have been pro-
posed: black-box methods and interactive approaches [22]. The former are based