Image Processing Reference
In-Depth Information
quantification. More details can be found in recent reviews on these topics
[21-23].
12.2
EXTRACTING INFORMATION
FOR THE FID SIGNAL
The FID signal is usually approximated as a sum of
K-
complex damped sinusoids
according to the following model [22]:
K
∑
xn
()
=
ae e
j
φ
(
−+
d
j
2
π
f
)
n
+
en
()
(12.1)
k
k
k
k
k
=
1
where
a
is the amplitude of the
k
-th sinusoid,
f
its frequency,
d
the damping
k
k
k
factor,
) is assumed to be circular complex white Gaussian
noise (i.e., the real and imaginary parts of the noise are not correlated and
have equal variance). In Equation 12.1,
the phase, and
e
(
n
k
n
=
{1, 2,
…
,
N
} is the discrete-time
index and
the number of observed data points. Each sinusoid is described
by a set of parameters
N
v
=
[
f
,
a
,
d
,
φ
], where the relative frequency
f
is used
k
k
k
k
k
k
to identify the biochemical species and
are the relevant parameters
for metabolite quantification and characterization.
a
and
d
k
k
is proportional to the
number of nuclei contributing to the spectral component at the frequency
a
k
f
k
(number that depends on the metabolite concentration), and
may provide
information about the mobility and macromolecular environment of the
nucleus. When expressed in the frequency domain and ignoring the noise term,
Equation 12.1 becomes
d
k
p
∑
ae
dj
j
φ
k
k
Xf
()
=
(12.2)
+
2
π
(
f
−
f
)
k
k
k
=
1
It consists of a set of spectral peaks of Lorentzian shape. The real part of
X
(
ω
) is known as the
absorption-mode spectrum
(
Figure 12.1
). Taking the real
part of Equation 12.2 and after correct phasing (
φ
=
0), we obtain
k
p
∑
ad
kK
Re{
Xf
(
)}
=
(12.3)
d
+
[(
2
π
f
−
f
]
2
2
k
k
k
=
1
It can be easily observed that
+∞
ad
∫
kk
df
=
a
(12.4)
k
d
+
[(
2
π
f
−
f
]
2
2
−∞
k
k
Thus, the quantification of the metabolite concentrations can be easily
obtained as the integral of spectral lines (Figure 12.1). Manually integrating the
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