Image Processing Reference
In-Depth Information
APPENDIX A
N
ONTRADITIONAL
S
HAPE
AND
M
OTION
D
ESCRIPTORS
Three-dimensional model-based analysis of left-ventricular shape and motion has
the potential of providing rich morphological and functional information. Current
clinical assessment of cardiac function is based mainly on global parameters such
as LVV and EF. However, several researchers have demonstrated in the past the
importance of local functional indices such as WT and segmental motion analysis
[102,266-268], and local curvature and shape [59-62] as potential cardiac indices.
Unfortunately, most of these studies were based on 2-D imaging techniques.
Although they can indicate major trends about cardiac shape, a 3-D analysis
would be able to better account for the true cardiac geometry. In this section, we
briefly summarize several new indices proposed in the literature for the descrip-
tion of shape and motion. Some of them have been presented as a by-product of
a specific modeling technique, whereas others are easily computable from any
model representation. Therefore, this distinction seems a natural classification.
G
ENERIC
D
ESCRIPTORS
Mean and Gaussian Curvature
The principal curvatures (
k
1
and
k
2
, respectively) measure the maximum and
minimum bending of a regular surface. Rather than using principal curvatures, it
is more common to use two derived quantities known as Gaussian (
K
=
k
1
k
2
) and
mean (H
k
2
)/2) curvatures. By analyzing the signs of the pair (
K
,
H
), it
is possible to locally distinguish between eight surface types [269].
Friboulet et al. [97] have studied the distribution of the Gaussian curvature in
the LV at different phases of the cardiac cycle. From this study it was concluded
that this distribution remains structurally stable over time. Whereas the LV free
wall provides rich and dense curvature information, the curvature at the septal wall
is less suitable for establishing point correspondences. Similar findings were made
by Sacks et al. [115] with respect to the RV free wall: the RV free wall has a
relatively uniform distribution of principal curvatures, and the surface geometry of
the RV free wall does not change significantly from end diastole to end systole.
=
(
k
1
+
Shape Index and Shape Spectrum
Although mean and Gaussian curvatures are related to the concept of curvedness,
there still remains scale information in these shape descriptors. To overcome this
problem, Clarysse et al. [270] have used the
shape index
(
s
) and
curvedness
(
c
), two
parameters that were introduced by Koenderink and van Doorn [271] and are defined
as follows:
kk
kk
+
−
2
s
=
tan
−
1
2
1
(9.11)
π
2
1
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