Image Processing Reference
In-Depth Information
Note that we can shift the filtered k-space harmonic peak back to the origin
of k-space using a shift operator (i.e., a convolution in k-space with
k
e
): a
delta function at
k
e
). This has the effect of multiplying the harmonic image by
exp( jk
e
⋅
δ
(
f
−
x), giving
M
(( )
(exp(
Xx
0
H
()
x
=
j
e
kXx
⋅
(
−
)))
(10.13)
2
This is an image whose phase is proportional to displacement
u
wrapped into
the range [
).
The spatial resolution of the harmonic phase is determined by the size of the filter
used to isolate the spectral peak. If only 32
−π
,
π
32 pixels are included in the k-space
filter, then the resolution of the harmonic image is approximately 32
×
×
32 [30].
10.4.2
K
INEMATICS
For small motions (less than
/k
e
), the component of displacement
u
in the tag
encoding direction at a spatial point
x
can be calculated as the difference in
harmonic phase at
x
between the deformed and undeformed time points. For
larger displacements, aliasing occurs, and a phase unwrapping procedure must
be employed.
Because the harmonic phase is linearly related to material coordinate
X
, the
spatial derivative of the harmonic phase is simply related to the inverse of the
deformation gradient tensor
F
:
π
=
∂
∂
X
x
i
[]
F
−
1
(10.14)
ij
j
The Eulerian (or Almansi's) strain tensor
e
is given by
1
2
e
=−
−−
(
I
FF
T
1
)
(10.15)
where
F
−
T
is (
F
−1
)
T
. This strain tensor is related to the change in length of line
segments that are currently aligned with the spatial coordinate system axes at the
deformed time t. Its relationship to the Lagrangian strain tensor E is given by
1
2
e
=−+
−
(
I
REI
(
2
)
1
T
)
(10.16)
A 3-D analysis can be performed using the procedure outlined in Subsection
10.2.5, where
n
is a unit vector in the direction of the tagging gradient
g
and the
displacements
u
are directly obtained from the (unwrapped) HARP phase offset
(Equation 13).
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