Biomedical Engineering Reference
In-Depth Information
in which the
s
is a function of
t
and
G
y
¼
0,
S
0
is a constant, and the Larmor frequency asso-
ciated with location
x
n
is
o
n
¼ g
G
x
x
n
ð
16
:
83
Þ
after mixing removes
o
0
at the isocenter.
The one-dimensional Fourier transform of Eq. (16.82) is
1
1
h
i
h
i
Þ
e
i
2p
ft
dt
¼
e
j
t
T
E
j=
T
2
n
e
i
2p
f
n
T
E
e
2pð
f
f
n
Þ
t
S
n
ð
f
,0
Þ¼
s
ð
t
,0
S
dt
ð
16
:
84a
Þ
0
1
1
The form of the complex exponents is recognizable from Eq. (16.77d) as the transform of an
impulse function centered on
. This relation implies that the Fourier transform of the func-
tion in the left brackets is centered on
f
n
. To show this result, it is necessary to determine if
the Fourier transform of the first term in brackets is of the form of Eqs. (16.71a) and (16.71d)
with
f
n
S
0
as the constant,
1
1
h
i
e
i
2p
ft
dt
¼
S
0
Þ
e
i
2p
ft
dt
¼
S
0
e
j
t
T
E
j
T
2
n
W
n
ð
f
,0
Þ¼
w
n
ð
t
,0
ð
16
:
84b
Þ
2
1
þð
2
p
fT
n
Þ
2
1
1
then
S
n
(
f
, 0) can be expressed as
Þ¼
e
i
2p
f
n
T
E
W
n
ð
f
f
n
Þ
S
n
ð
f
,0
ð
16
:
84c
Þ
From the relation for the Larmor frequency, this equation can be rewritten in terms of a
scaled variable for frequency from Eq. (16.83) as
g
G
x
2
Þ¼
e
i
g
G
x
x
n
T
E
W
n
S
n
ð
x
,0
p
ð
x
x
n
Þ
ð
16
:
84d
Þ
The derivation of this key equation reveals a second important principle in magnetic reso-
nance imaging. First, a locally excited resonance encodes a position into a frequency
as
part of the phase of a time waveform. Second, this phase, in terms of Fourier transforms,
ensures that the signal function has a spectrum centered on the frequency associated with
the location of the net magnetization density. Third, through simple scaling, the scaled
spectra are translated into spatial locations along the
f
n
-axis (Figure 16.48b). This process
maps scaled spectral magnitudes into their spatial locations along the
x
-axis.
To advance to two dimensions, the one-dimensional inverse Fourier transforms are com-
bined in a two-dimensional relation:
x
ðð
G
y
Þ¼
``
1
u
Þ
e
i
2p
ft
e
i
2p
uG
y
dfdu
s
ð
t
,
½
I
ð
f
,
u
Þ ¼
I
ð
f
,
ð
16
:
85
Þ
To find the object distribution,
I
, from the measured set of signals from a location (
x
n
,
y
m
),
a forward 2D Fourier transform is
ðð
G
y
Þ
e
i
2p
ft
e
i
2p
uG
y
dtdG
y
I
ð
f
,
u
Þ¼
S
mn
ð
f
,
u
Þ¼
``
½
s
mn
ð
t
,
G
y
Þ ¼
s
mn
ð
t
,
ð
16
:
86
Þ