Biomedical Engineering Reference
In-Depth Information
If the set of signals is taken to be simply the product of the one-dimensional signals for
this simplified example
h
i
e
jð
t
T
E
Þ=
T
2
mn
j
e
j
o
n
ð
t
T
E
Þ
e
i
2p
y
m
G
y
s
mn
ð
t
,
G
y
Þ¼
S
ð
16
:
87
Þ
0
then inserting this set into the 2D transform of Eq. (16.86) yields
g
G
x
2
y
Þ¼
e
i
g
G
x
x
n
T
E
W
mn
S
mn
ð
x
,
p
ð
x
x
n
Þ
½
2
p=ðg
t
p
Þdð
y
y
m
Þ
ð
16
:
88
Þ
In this case, as depicted in Figure 16.48c, the intersection of these one-dimensional func-
tions places a scaled spectrum at the location (
x
n
,
y
m
). The role of
u
m
is to localize the posi-
tion of the spectrum along
,is
displayed as amplitude mapped on a gray scale with full white as maximum amplitude
in the image plane at location
y
through scaling. In an MRI image, this spectrum,
S
mn
. What are displayed, then, are not the detected time signals
but the amplitudes of their spectra, which through encoding methods are mapped into their
proper spatial locations from anatomy.
In reality, during actual measurements, what are sensed are the signals from all the
active net magnetization vectors from all the locations being scanned. The detected voltage
is the sum of all the signals picked up in a specific
z
plane
z
X
V
ð
t
,
G
y
Þ¼
,
n
s
m
,
n
ð
t
,
G
y
Þ
ð
16
:
89
Þ
m
where the signals are not the ideal forms derived in Eqs. (16.87) and (16.88) but signals
reflecting the characteristics and organization of tissue structures.
A convenient way of discussing the signals in terms of Fourier transforms is to describe the
acquired signals as belonging to a
k
-space domain and the transformed signals as being in the
space (
) domain. As shown in the following, this perspective is simply accomplished as a
change in variables in the previous transform equations. The simplest place to start is in the
exponential arguments of the transform equations, Eqs. (16.83) and (16.80a), that contain the
key variables. These variables are set equal to the desired new ones:
2
x, y
p
f
n
t
¼
k
x
x
n
ð
16
:
90a
Þ
2
p
uG
y
¼
k
y
y
m
ð
16
:
90b
Þ
From previous definitions, it is straightforward to show that
k
x
¼ g
G
x
t
ð
16
:
91a
Þ
k
y
¼g
G
y
t
p
ð
16
:
91b
Þ
or
t
¼
k
x
=g
G
x
ð
16
:
92a
Þ
G
y
¼
k
y
=g
t
p
ð
16
:
92b
Þ
Equations (16.92a) and (16.92b) can be substituted in Eq. (16.85) to yield
ðð
g
t
p
2
I
x
y
g
G
x
2
g
G
x
2
g
t
p
2
e
ik
x
x
e
ik
y
y
dxdy
s
ð
k
x
=g
G
x
,
k
y
=g
t
p
Þ¼
,
ð
16
:
93
Þ
p
p
p
p
