Biomedical Engineering Reference
In-Depth Information
Similarly, the principal curvatures o
f t
he s
urf
ace
S
c
are denoted as
k
1
and
k
2
and
corresponding principal directions are
p
b
and
p
b
(not shown but look similar to that
of in Fig.
1
b).
Consider an ar
bit
rary direction
v
in the tangent plane which is perpendicular to
the normal vector
n
a
as shown in Fig.
1
b along which we would like to compute th
e
normal curvature. The normal curvature on the subsurface
S
c
in the direction of
v
could be computed by Euler's formula:
k
a
¼ k
a
cos
2
a þ k
a
sin
2
a;
(1)
where
is the angle between the maximal principal direction
p
a
and
v
. Similarly,
the normal curvature on the subsurface
S
b
c
in the direction of
v
is
a
k
b
¼ k
b
cos
2
b þ k
b
sin
2
b;
(2)
is the angle between the maximal principal direction
p
b
and
v
.
We then defined the CI by combining the first and second features on
S
c
and
S
c
as given below.
where
b
1
kk
a
n
a
k
b
n
b
k
CI ¼
(3)
Note: For locally flat surfaces, the denominator of (
3
) becomes zero to show that
incongruity is zero or congruity is infinite.
2.2 Application to Medial Tibiofemoral Joint
The MTF cartilage compartments of all the knees were segmented fully automati-
cally using a voxel classification approach [
12
]. The congruity quantification steps
for the MTF joint are presented in detail below.
Initially, the binary segmentations were regularized using mean curvature flow
in level set formulation to reduce the voxellation effects. Mathematically, the level
set formulation described in [
13
] is as follows:
@f
@
t
¼ k
M
jrfj¼ rð
rf
jrfj
Þ
jrfj
(4)
where
and
k
M
is the mean
curvature. In this flow, points with higher curvature move toward points with lower
curvature and thereby ensure the evolution of smooth or physically meaningful
surfaces for computations.
∇ f
is the gradient of the level set representation
f
