Image Processing Reference
In-Depth Information
1.9.3
P
HASE
E
NCODING
Considering the one-dimensional case after a RF pulse, if we turn on a gradient
G
y
for a short interval t
y
and then we turn it off, the local signal under the influence
of this gradient is:
ρ
()
ye
−+
i B
γ
yG
)
t
0
≤≤
t
t
"
"
0
y
y
ds t, y
()
=
(1.38)
−
iyGt
γ
ρ
()
ye
yy
e
−
iBt
γ
t
≥
t
0
y
where
ρ
(y) is the spin distribution along y. From Equation 1.38, during the interval
t
y
the local signal is frequency encoded; as a result of this frequency
encoding, signals from different y positions accumulate different phase angles
after a time interval t
y
. Therefore, the signal collected after t
y
will bear an initial
phase angle
0
≤
t
≤
ϕ
(y)
=−γ
yG
yy
(1.39)
(y) is linearly related to the signal location
y
, the signal is said to
be
phase encoded
, the gradient G
y
is called
phase-encoding gradient
, and t
y
is
the
phase-encoding interval.
Phase encoding along an arbitrary direction can be also done for a multi-
dimensional object by turning on G
x
, G
y
, and G
z
simultaneously during the
phase-encoding period G
phas
Because
ϕ
=
(G
x
, G
y
, G
z
) for 0
≤
t
≤
t
y
; the initial angle is
ϕ
r
G
phas
t
y
. Similar to frequency encoding, the received signal is the sum
of all the local phase-encoded signals and is given by:
(
r
)
=
−γ
∞
&
'
∞
)
*
∫
∫
−
irG
γ
t
phas G
s t
()
=
ds r t
(,)
=
c
ρ
()
r e
dr
e
−
i
ω
0
t
(1.40)
(
+
−∞
−∞
where the carrier signal exp(
−
i
ω
0
t) is removed after signal demodulation.
1.9.4
P
HASE
H
ISTORY
OF
M
AGNETIZATION
V
ECTORS
DURING
P
HASE
E
NCODING
Let us consider the evolution of the phase angle of magnetization vectors in
the transverse plane as a function of a different phase-encoding gradient ampli-
tude G
′
y
=
mG
y
by varying m; we call this a
phase-encoding step
. Referring to
the scheme of
Figure 1.14
, each phase-encoding step corresponds to a different
value of m, that is, a different amplitude of the phase-encoding gradient. We
can write the sequence of phase shifts added to a magnetization vector at the
location
y
0
as
ϕ
m
(y
0
)
=
−γ
y
0
mG
y
t
y
. Therefore, the expression for a set of different
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