Biomedical Engineering Reference
In-Depth Information
The authors studied the design of the speed term to stop the front propagation
at high-gradient locations depending on the value of
V
G
.
In the first case, for
V
G
=
0 they proposed the following speed term:
(
M
1
−
M
2
)
(
|∇
G
σ
I
|−
M
2
)
V
a
V
=
−
V
a
+
,
(2.11)
where (
M
1
,
M
2
) are the maximum and minimum values of the smooth gradient
image
∇
G
σ
I
.
In the case where
V
G
=
0, the speed term needs to be multiplied by a gradient-
based term to stop the front evolution, as follows:
1
V
=
∇
G
σ
I
×
(
V
a
+
V
G
)
.
(2.12)
1
+
Numerical schemes for approximation of spatial derivatives with theses speed
terms must respect the appropriate entropy condition for propagating fronts as
discussed in detail in [13] and [14]. This entropy condition ensures that the prop-
agating front corresponds to the boundary of an expanding region. An analogy
invoked by Sethian to illustrate the entropy principle is to consider the moving
front as a source for a burning flame and expand the flame thus ensuring that
once a point in the domain is ignited, it stays burnt. The entropy principle puts
some constraints in the choice of particular numerical schemes for temporal
and spatial derivatives of the level set function. In their work, Malladi
et al.
[8]
used a forward difference in time, upwind scheme for the constant inflation term
and central differences for the remainder term (see Fig. 2.3).
A second issue with this framework arises from the fact the image-based
speed terms are only defined on the zero-level of the moving front, as it was
designed to stop the evolution of this level at high-gradient locations. On the
other hand, the energy functional is defined over the entire domain and the
speed term must have a consistent definition over all values of the level set
function. This is done by extending the speed term from its values defined only
on the level zero. There are several methods available to perform the extension.
One of the most popular methods assigns the values of the closest point on the
level zero to a given point of the domain.
