Biomedical Engineering Reference
In-Depth Information
of the equation is the distance transform to the surface represented by the initial
condition,
T
0
=
0. The initial surface is computed as the boundary vessel voxels
in the MAP classification. The Fast Marching algorithm is used to compute the
distance transform to this surface. This function is used as initialization to the
level set algorithm.
The level set equation, Eq. (5.16), does not preserve the distance transform
through the iterations. The numerical approximations described below may
cause a numerical deterioration of the solutions if the evolving front
φ
is not
smooth enough. To avoid this phenomenon,
φ
is replaced in each iteration by
a distance transform
φ
with the same level set zero as
φ
. To do this, the Fast
Marching algorithm is used. The values of the voxels where there is a change of
sign define the evolving surface. These voxels are introduced in the Fast March-
ing algorithm as initial condition instead of explicitly computing the zero level
set and then use it as initial condition. The result is also a distance transform to
the evolving surface.
Curvature Constraints.
The mean curvature of a surface is defined in dif-
ferential geometry as the inverse of the radius of the osculant sphere. If the
minimum grid size is
x
, it makes no sense considering osculant spheres of
sizes less than
x
. So, the discrete values of the curvature are limited to the
interval [
−
1
1
x
]. The numerical approximation introduces singularities in the
calculus of the curvature function. To avoid their propagation in the GAR algo-
rithm, a Gaussian smoothing with
σ
=
0
.
8 mm was applied to the curvature.
Parameter Selection.
Parameters
ζ
and
η
control the influence of the region
and boundary based forces in the motion of the surface. The choice of these
parameters depends on the confidence of the user in the different descriptors.
For this application,
ζ
and
η
were chosen equal to 1.0.
Regarding the selection of the edge detector function
g
, there is an interest-
ing property that relates the parameters involved in the level set equation to a
bound on the curvature of the front in evolution. The front evolution is driven
by Eq. (5.15). At the steady-state,
S
t
=
0, so
ζ
(
k
out
−
k
in
)
−
η
(
g
κ
+∇
g
n
)
=
0.
This formula gives upper and lower bounds for the curvature
x
,
·|
k
out
−
k
in
|+
∇
g
g
g
η
(5.19)
|
κ
|≤
In the case of segmentation of narrow vessels, the curvature of the model is
high at certain locations. Therefore, the selection of the parameters has to be
