Biomedical Engineering Reference
In-Depth Information
where the speed terms (
V
in
,
V
out
) are functions of the surface normal direction
(e.g., curvature of the surface), image-derived information and the distance be-
tween the two surfaces. When this distance is within the desired range, the two
surfaces propagate according to the first two terms of the speed term. When
the distance is out of the desired range, the speed term based on the distance
controls the deformation as to correct the surface positions.
When defining the initial level set function as the signed distance function
to its level zero, and ensuring that the distance function is preserved during the
deformation process of the front through reinitialization, the distance of any
point on the inner surface to the outer surface is directly read as the value of
the outer level set function and vice versa.
Defining the speed terms as:
V
in
=
F
in
h
(
φ
out
)
V
out
=
F
out
h
(
φ
in
)
(2.23)
with
(
F
in
,
F
out
)
speed terms derived from image and curvature properties and
h
() a smooth approximating the windowing step function defined for a range
of distance [
d
1
d
2
] that is equal to one inside this interval and 0 outside.
Zeng
et al.
[33] applied this framework for the segmentation of brain cortical
gray matter (GM) surfaces. In this application, the speed terms were defined as:
V
inside
=
S
−
(
I
−
I
in
)
+
S
+
(
φ
out
+
ε
)
V
outside
=
S
−
(
I
−
I
out
)
+
S
+
(
φ
in
−
ε
)
,
(2.24)
where
I
is the intensity of the MRI,
I
in
is a threshold value corresponding to the
white matter and
I
out
a threshold value corresponding to the gray matter,
ε
is the
desired thickness of the gray matter layer,
S
−
,
S
+
are two sigmoid functions,
respectively, decreasing and increasing with bounded value between [
−
1
,
1].If
the curve evolution is implemented with Eq. (2.5), the magnitude of the gradients
(
|∇
φ
in
|
,
|∇
φ
out
|
) will increase and the estimation of the distance between the
zero-levels of the two functions will be overestimated, leading (
φ
in
,φ
out
) to get
closer as they evolve and eventually collide until the level set functions are
reinitialized. Results are illustrated on three regions of interest from three MRI
slices and show very interesting results but no quantitative evaluation of the
accuracy of the method was performed.
