Image Processing Reference
In-Depth Information
The weighting function
C
(
x
) can also be determined experimentally. A typical
example is time-sequential imaging, which involves the acquisition of a time
series of images,
ρ
L
(
x
), from the same anatomical site. For many
of this type of imaging experiments, the underlying high-resolution morphology
in the desired image sequence does not change from one image to another. As a
result, it is not necessary to acquire each of these images independently. Specif-
ically, with the GS model, we first acquire one high-resolution (reference) data
set with
N
encodings, followed by a sequence of reduced data set with
M
encod-
ings. In the image reconstruction step, the high-resolution reference image
ρ
1
(
x
),
ρ
2
(
x
),
…
,
ρ
ref
(
x
)
is used as the weighting function for the GS basis functions. That is, we set
C
(
x
)
=
|
ρ
ref
(
x
)|
(2.13)
for the GS model when it is used for image reconstruction from the reduced data
sets. After
C
(
x
) is known, the series coefficients
c
n
are determined by solving a
set of linear equations from the data-consistency constraints. That is,
N
−
∑
2
/
21
Dn k
(
∆
)
=
c D
c
[(
n
−
m
)
∆
k
],
(2.14)
m
mN
=−
/
where
D
c
(
n
∆
k
)
=
F
{
C
(
x
)}(
n
∆
k
).
2.3.3
A
PPLICATION
E
XAMPLES
Constrained image reconstruction has been successfully used in several practical
applications. This section discusses two specific examples: partial Fourier imag-
ing and dynamic imaging.
Example 2.1: Partial Fourier Reconstruction
In partial Fourier imaging,
k
-space is sampled asymmetrically, say,
D
(
n
∆
k
) is
measured for
n
1}. Such a sampling scheme arises
in MRI when a short echo time is used to avoid spin dephasing due to short caused
by local susceptibility changes or uncompensated motion effects. It is sometimes
also used in the phase-encoding direction when an asymmetric set of phase-encoding
measurements is acquired to reduce data acquisition time. Usually,
n
0
is much
smaller than
N
, typically,
n
0
∈
N
data
=
{
−
n
0
,
−
n
0
+
1,
…
,
N
−
T
*
16 or 32 with
n
being on the order of 128. The central
k
-space data are used first to obtain an phase estimate , which is then used as
a constraint to get the final reconstruction. The phase-constrained reconstruction
problem lends itself nicely to the POCS algorithm. Specifically, let
=
ˆ
( )
ϕ
x
ˆ
()}
Ω
1
=
{()|
ρ
x
∠=
ρ
()
x
ϕ
x
(2.15)
and
Ω
=
{()| {()}
ρ
xF x
ρ
=
Dnk n n N
(
∆
),
−
≤
≤
−
1
}.
(2.16)
2
0
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