Image Processing Reference
In-Depth Information
10.3.2
H
S
OMOGENEOUS
TRAIN
Given tracked landmarks in the form of grid tag intersection points, 2-D motion and
strain can be calculated from a triangulation of points within the heart wall [21].
Assuming a homogeneous strain state within the marker triangle, vectors denoting
line segments between the triangle vertices (
dX
in the undeformed state and
dx
in
the deformed state) are related by the deformation gradient tensor
F
[22]:
dx
=
F
dX
(10.1)
The undeformed state has typically been ED (coinciding with the tag gener-
ation pulse or the first image, thereafter), the deformed any subsequent frame, in
particular end-systole (ES). Given two or more noncollinear line segments
arranged in matrices A
=
[
dX
dX
…
] and B
=
[
dx
dx
…
],
F
can be estimated
1
2
1
2
by linear least squares as
F
=
BA
(AA
)
(10.2)
T
T
−
1
By polar decomposition, the deformation gradient tensor may be separated
into an orthogonal unitary rotation tensor
(which describes rotation about the
triangle centroid) and a positive symmetric stretch tensor
R
U
(which describes
strain):
F
=
R U
(10.3)
The Lagrangian (or Green's) strain tensor
E
can be calculated as [23]
E
=
0.5 (
U
2
−
I
)
=
0.5 (
F
T
F
-
I
)
(10.4)
This strain tensor is related to the change in length of small line segments that
are initially aligned with the material coordinate system axes in the undeformed
state. These homogeneous strain methods extend naturally to three dimensions [24],
in which case four or more points are needed to estimate the homogeneous approx-
imation of
(more points gain more robustness against noise, at the expense of a
larger region in which the strain is assumed homogeneous).
F
10.3.3
N
S
ONHOMOGENEOUS
TRAIN
The assumption of homogeneous strain within a region can lead to errors in strain
calculation if the region is too large. Typically, strain in the heart varies rapidly
in the transmural direction (from outer to inner wall surfaces), and variation also
occurs in the circumferential and longitudinal directions (although less rapidly
in the normal heart) [21,25]. Also, errors in tracking material landmarks are
amplified in the strain calculation (strain can be viewed as the derivative of
displacement and as such suffers from the ill-posed nature of numerical differ-
entiation). In order to reduce these errors, a continuous displacement field may
be fitted to the tracked points [19]. The displacement field can be modeled using
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