Image Processing Reference
In-Depth Information
3.3
FORMULATION OF THE PROBLEM
The starting point of all parallel imaging techniques in MRI is the equation
describing the received signal. When using an array of N coils, the general
equation of the signal received in a coil “k” can be written as:
∫∫
∫
i
i
i
s G G t
( ,
i
i
,)
=
I xyzW xyzP xyz
(,,)
(,,)
L
(,,)
e
jG tGy Gz
γ
(
+
τ
+
τ
)
dxdydz
(3.1)
1
2
k
k
where
I
(x,
y,
z) represents the image that we seek to identify; W
(x,
y,
z)
k
represents the 3-D sensitivity profile of coil “k”; P
L
(x, y, z) represents the RF
selective excitation profile during the
L
-th excitation;
GGG
i
,
i
,
i
represent the
values of the gradients applied, respectively, in the x,
y, and z directions during
the
i
-th acquisition, assuming x as the frequency-encoded direction; and
τ
and
1
τ
represent the duration of the phase-encoding gradients in the y and z directions,
respectively. This equation lumps the T 1, T 2, and spin-density dependencies
into the element designating the image I(x,
2
z). In addition, it takes into account
the possibility of acquiring multiple echoes (index “i”) for each excitation profile
(index “ L”), and allows for the total flexibility in the design of the excitation
profile P
y,
L
(x,
y,
z). In Fourier imaging, P
L
(x,
y,
z) is kept constant for all values
of “m,”
τ
=
τ
=
τ
, unless otherwise specified, and
G
i
is kept constant for all
1
2
values of “i.”
Equation 3.1 is a linear equation relating the time signal of an acquired echo
to the image, through the parameters of the imaging pulse sequence. The time
signal is sampled, and each time sample provides one linear equation of the image
I(x,
P independent
equations to resolve it. For the sake of simplicity, we will describe 2-D imaging
in the following text; the same analysis can be extended to the 3-D case by adding
the
y,
z). If I(x,
y,
z) is an M
×
N
×
P matrix, then we need M
×
N
×
dimension.
In 2-D imaging, RF excitation is used to select a slice in the volume. This is
equivalent to setting the profile P
z
z) to a value of 1 for the desired slice,
and 0, otherwise. For the sake of the discussion, let us assume that the slice
selection is performed along the z direction, for a position z
L
(x,
y,
. This means that
0
P
L
(x,
y,
z)
=
1 when
z
=
z
, and P
L
(x,
y,
z)
=
0 for all other values of z. Equation 3.1
0
would become:
∫∫
i
s
(, )
G
i
t
=
I x y W
(,
)(,
x y e
)
(
jGxtGy
x
γ
+
τ
)
dxd
y
(3.2)
k
y
k
for the imaged slice at z
=
z
. With a change of variables, k
=
jG
and
0
x
x
kjG
i
=
i
τ
,
this equation can be discretized to yield:
M
N
∑∑
1
(
)
=
i
skT
k
i
,
ImnWmne
(
,
)
(
,
)
(
kmT k
x
+
n
)
(3.3)
k
m
=
n
=
1
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