Image Processing Reference
In-Depth Information
Reconstructed
0
Raw Data
a
1
Harmonic Data
even lines
1
a
1
Reconstructed
k−space Data
k
y
k
y
k
x
k
x
0
a
L
odd lines
k
y
1
a
L
k
x
k
y
k
y
k
x
k
x
FIGURE 3.4
Data flow and reconstruction in SMASH imaging.
combined to yield spatial harmonics in the
y
direction. Note that the harmonic
fit is approximately, but not exactly, sinusoidal—leading to the problems in
reconstruction illustrated in the Examples section.
Figure 3.4 shows the signal flow for the SMASH algorithm. Here each coil
is used to collect regularly subsampled k-space sets with a skip factor of two.
Reconstructing these sets yields aliased images, weighted by the coil sensitivity
profiles. The acquired signals are combined linearly to form the harmonics as
described in Equation 3.8. Reconstruction of each harmonic set separately also
yields aliased images, where the coil weighting has been removed. Finally, the
harmonic sets are combined into a k-space representation, filling odd lines from
one harmonic and even lines from the other. Note that the chosen acceleration
factor in SMASH imaging determines the number of harmonic fits needed to
generate the composite k-space data.
3.3.3.3.1.1 Better Harmonic Fitting in SMASH
The quality of SMASH imaging is a function of the accuracy of the fits computed
in Equation 3.6; variations on the design of this fit have been proposed in order
to maximize its accuracy.
Better fits covering the full FOV of the image can be
found when Equation 3.6 is expanded to include the x direction.
K
∑
W
Comp
=
a
k
m
⋅
W
(, )
x y
≈
A B x
⋅
()
⋅
e
jm k y
(
⋅
∆
⋅
)
(3.9)
y
m
k
k
=
1
where each location along the frequency-encoded axis x would have a set of
parameters associated with it. If N is the resolution along the frequency-
encoded direction, then N different SMASH reconstructions can be performed
a(x)
k
m
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