Image Processing Reference
In-Depth Information
inconsistency between the dynamic and reference data sets will result in data trun-
cation artifact and, as a result, dynamic image features are produced only at low
resolution. With RIGR, image reconstruction is done using the GS model described
in Section 2.2, in which the basis functions are determined by the reference data
and the coefficients are determined by the dynamic data. This reconstruction algo-
rithm can overcome the limited resolution problem with the keyhole method. It has
been shown that with multiple references, RIGR can reconstruct dynamic features
in a resolution close to that of the reference image [11].
2.4
REGULARIZED IMAGE RECONSTRUCTION
IN PARALLEL MRI
2.4.1
B
ASIC
R
ECONSTRUCTION
M
ETHODS
The Fourier image of the
th channel (ignoring the data truncation effects) is
given by
−
∑
ρ
0
R
1
ˆ
)(
ˆ
),
dx
()
=
(
xmWsxmW
−
−
(2.22)
m
=
ˆ
for
…
=
1,, , ,
L
and
WWxW
/
2
−<<
/
2
. Assuming that
R
≤
L
, we can solve
for
(
x
) pixel by pixel from the earlier equations. More specifically, rewriting
Equation 2.22 in matrix form
ρ
S
ρ =
d
(2.23)
where
ˆ
)
ˆ
)
sx
()
sx W
(
−
sx
(
−
(
R
−
1
)
W
"
"
"
1
1
1
ˆ
)
ˆ
)
sx
()
sx W
(
−
ssx R
(
−−
(
1
)
W
S
=
2
2
2
ˆ
)
ˆ
sx
()
sxW
(
−
sx R
(
−
(
−
1
)
W
)
!
L
L
1
ρ
x
xW
()
dx
dx
()
()
1
"
"
"
"
"
"
"
"
"
ˆ
)
ρ
(
−
2
ρ
=
,
and
d
=
.
ˆ
)
dx
L
()
ρ
(
xR W
−−
(
1
)
!
!
Equation 2.23 is known as the sensitivity encoding (SENSE) reconstruction formula
[12], which can be derived from Papoulis' generalized sampling theorem [13].
Clearly, perfect reconstruction of
ρ
(
x
) requires: (a) precise knowledge of
sx
( )
to
ˆ
form, (b) to be nonsingular for
WWx
/
2
−<<
W
/
2
, and (c)
dx
()
to be noiseless
and not corrupted by the data truncation artifact.
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