Image Processing Reference
In-Depth Information
1
2
kk
+
2
2
c
=
1
2
(9.12)
2
whereas
c
is inversely proportional to the object size,
s
defines a continuous
distribution of surface types ranging from cuplike umbilic (
s
=
−
1) to peaklike
umbilic (
s
1) points. It can be shown that whereas the shape index is invariant
by homothecy, the curvedness is not. In this way, shape information and size can
be easily decoupled.
The shape spectrum [272],
=
(
h
,
t
), is a global shape index defined as the
fractional area of the LV with shape index value
h
at time
t
γ
1
∫∫
γ
(,)
ht
=
δ
(()
s
x
−
h ds
)
(9.13)
A
s
where
A
∫∫
s
dS
is the total area of the surface
S
,
dS
is a small region around the
point
x
, and
=
) is the 1-D Dirac delta function. Cardiac deformation can be
analyzed by tracking the shape index and curvedness of similar shape patches
(SSPs) over time. SSPs are connected surface patches whose points have similar
shape indices, i.e., the shape index falls within a given range
s
δ
(
⋅
s
. Clarysse
et al. have shown the potential applicability of these indices by analyzing phan-
toms of normal and diseased LVs. An LV model of dilated cardiomyopathy, and
a model of an ischemic LV (both akinetic and hypokinetic in the left anterior
coronary territory) were generated using 4-D spherical harmonics. The curvedness
spectrum was significantly altered by both pathologies, even when they were
localized (as in the ischemic models). Reduction of the global function in the
dilated myocardium had no significant repercussion on the shape index spectra.
This could be an indicator that this pathology mostly affects the magnitude of
motion only. An alternative to global analysis is to track the curvature parameters
in predetermined regions. Clarysse et al. tracked three reference points over time:
the apex, a point in the anterior wall, and a point in the cup of the pillar anchor.
Using the local temporal variation of the curvedness and shape index, it was possible
to distinguish between the normal and the diseased model. A potential problem
with this technique is the reliable tracking of SSPs. If local deformations are too
large, the trace of points might be lost.
±
∆
L
OCAL
S
TRETCHING
Mishra et al. [96] have presented a computational scheme to derive local epicar-
dial stretching under conformal motion. In conformal motion, it is assumed that
motion can be described by a spatially variant but locally isotropic strething
factor. I
n
particular, for any two corresponding patches before and after motion,
P
and
P
,
the local stretching factor,
τ
, can be computed from the change in
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