Biomedical Engineering Reference
In-Depth Information
In their pioneer paper, Osher and Sethian focused on motion under mean
curvature flow where the speed term is expressed as:
∇
φ
|∇
φ
|
V
=
di
v
.
(2.7)
Since its introduction, the concept of deformable models for image segmentation
defined in a level set framework has motivated the development of several fami-
lies of method that include: geometric active contours based on mean curvature
flow, gradient-based implicit active contours and geodesic active contours.
2.2.2
Geometric Active Contours
In their work introducing geometric active contours, Caselles
et al
. [12] proposed
the following functional to segment a given image
I
:
∂
t
=|∇
φ
|
g
(
|∇
I
|
)
di
v
∇
φ
|∇
φ
|
∂φ
(2.8)
+
ν
,
with
1
1
+|∇
G
σ
(
x
,
y
)
∗
I
(
x
,
y
)
|
g
(
|∇
I
(
x
,
y
)
|
)
=
2
,
(2.9)
where
ν
≥
0 and
G
σ
is a Gaussian convolution filter of standard deviation
σ
. The
idea defining geometric deformable models is to modify the initial mean curva-
ture flow of Eq. (2.7) by adding a constant inflation force term
ν
and multiplying
the speed by a term inversely proportional to the smooth gradient of the image.
In this context the model is forced to inflate on smooth areas and to stop at
high-gradient locations as the speed decreases towards zero.
2.2.3
Gradient-Based Level Set Active Contours
In their initial work on applications of the level set framework for segmentation
of medical images, Malladi
et al
. [8] presented a gradient-based speed function
for the general Hamilton-Jacobi type equation of motion in Eq. (2.5).
Their general framework decomposed the speed term into two components:
V
=
V
a
+
V
G
,
(2.10)
where
V
a
is an advection term, independent of the geometry of the front and
V
G
is a remainder term that depends on the front geometry.
