Image Processing Reference
In-Depth Information
Where m,
n, and T take discrete values, and the number of time points sampled is
equal to the resolution along the frequency-encoded direction M. If N phase-encoding
steps are used, Equation 3.3 can be written in matrix form as:
I
I
(, )
(, )
.
.
(, )
(,)
.
11
12
(
kk
11
+
)
(
kM kN
+
)
We
(, )
11
.
.
WMNe
(
, )
x
y
x
y
sG
sG
(, )
(,
1
1
2
k
k
k
y
(
k
12
.
+
k
1
)
(
kM
⋅ +
2
kN
)
WWe
(, )
11
.
.
WMNe
(
, )
x
y
x
y
1
)
k
k
k
y
.
.
.
.
(,
.
.
.
.
IM
I
1
21
sGM
1
)
(
kMk
1
+
1
)
(
kMM
+
kkN
)
We
(, )
11
.
.
WMNe
(
, )
x
y
x
y
k
y
k
k
.
.
.
.
.
.
.
=
.
.
.
.
.
.
.
.
.
(, )
.
.
(,
sG
(
N
, )
1
k
y
(
kkN
1
+
)
(
kM
⋅+
1
kN
)
We
(, )
11
.
.
WMN
(
, )
e
IM
1
x
y
x
y
k
k
.
.
.
.
.
.
.
.
.
.
ssG M
k
(
N
,
)
y
(
kkN
1
+
)
MMNe kMM kN
,) (
+
)
We
(, )
11
.
.
W
(
IMN
)
x
y
x
y
k
k
(3.4)
where the left-hand-side vector [s
-th
coil for all the phase-encoding steps performed (1 to N), and the right-hand-
side vector [I(m,
] contains the time signals received in the
k
k
n)] contains the elements of the image to be reconstructed.
The size of the system of the reconstruction matrix is (M
N
×
M
N ). A slow
way of computing the image I(m,
n) is by inverting the system of equations;
this is never used in practice due to computational inefficiency. In the case
where the sensitivity profile information is unknown, Equation 3.4 can also be
written such that the sensitivity profile information, W
n), is included in
the right-hand-side vector. If K coils are used simultaneously, the system can
be augmented to (K
(m,
k
M
N
×
M
N ), where all the s
vectors are included.
k
3.3.1
F
E
OURIER
NCODING
n) and
is exploited in order to perform computationally efficient image
reconstructions. Once
In the widely used Fourier imaging, the 2DFT relationship between I(m,
sGT
k (,
)
different phase-encoding steps are acquired, at the
Nyquist rate of the spatial resolution in the
N
direction, the linear system of
equations can be solved very efficiently by applying a 2-D Fourier transform
to the acquired matrix yielding the image. When using multiple coils, the
resulting images in all the coils {W
y
I,
W
I,…,W
I} are combined in the least-
1
2
n
squares sense to yield a composite image: (
WW
2
++ +
2
W I . Figure 3.1
2
)
1 2
/
1
2
n
shows images reconstructed from a four-coil array and combined to yield a
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