Image Processing Reference
In-Depth Information
ρ
Vector of sample values from
Vector of sample values from
Vector of sample values of a regularization image
ρ
()
x
ˆ
()
ˆ
ρ
x
ρ
ρ
r
D
(
k
)
k
-space data sampled at
k
n
n
d
(
x
)
Aliased reconstructed image of the
th coil
d
Vector of
d
(
x
) from the coil array
L
Total number of receiver channels (coils)
R
Acceleration factor or reduction factor in parallel MRI
Sensitivity of
s
()
x
l
th coil
∆
k
Nyquist sampling interval (i.e.,
)
Increased sampling interval used in parallel MRI
∆
k
=
1/
W
k
∆
W
ρ
(
x
) is assumed to be support-limited to |
x
|
<
W
/2
W
W
Reduced FOV due to sub-sampling (
=
W
/
R
)
λ
Regularization parameter.
2.1
INTRODUCTION
This chapter is focused on Fourier transform MRI, in which the imaging equation
can be written, in general, as
w
/
2
∫
Dk
( )
=
ρ
()()
xsxe
−
i
2
π
kx
n
dx
,
(2.1)
n
−
w
/
2
where we explicitly include the sensitivity weighting function
s
(
x
) of the receiver
coil. In conventional Fourier imaging,
) is often ignored because it can be
assumed to be a constant over the field of view (FOV), and
s
(
x
D
(
k
) is usually measured
n
at
being the total number of
encodings acquired. In multichannel Fourier imaging (often known as parallel
imaging), an array of receiver channels (or coils) with sensitivity functions
k
=
n
∆
k
for
n
=
−
N
/2,
−
N
/2
+
1,
…
,
N
/2
−
1, with
N
n
sx
()
is
used to acquire
Dk
n
()
simultaneously for
=
1, 2,
…
,
L
. To increase imaging
ˆ
speed,
Dk
n
()
is measured at
knk
n
=∆
for n
=
−
M
/2
,
−
Ν
/2
+
1,
…
,
M
/2 - 1,
with
ˆ
M
-space signal is measured at a sub-
Nyquist rate in each receiver channel. Before we discuss advanced techniques to
handle the image reconstruction problem associated with these two data acquisition
schemes, a brief review of the popular Fourier reconstruction method is in order.
=
N
/
R
and
∆∆
kRk
=
.
In other words, the
k
2.2
FOURIER RECONSTRUCTION
Given
D
(
n
∆
k
) and
s
(
x
)
=
1, it is well known that
ρ
(
x
) can be reconstructed using
the Fourier series, that is,
∞
∑
1
ρ
()
x
=
∆
k
Dn
(
∆
k e
)
i
2
π
∆
k x
, | |
x
<
.
(2.2)
2
∆
k
n
=−∞
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