Biomedical Engineering Reference
In-Depth Information
Figure 13.
Typical definition of the edge functions to control propagation of the moving
front.
boundaries. In addition, the model has a level set formulation, interior contours
are automatically detected, and the initial curve can be anywhere within the image.
The original MS function is defined as follows:
(
u, C
)=
µLength
(
C
)+
λ
F
MS
u
(
x, y
)
|
2
dxdy
Ω
|
u
0
(
x, y
)
−
λ
Ω
\C
|∇
u
(
x, y
)
|
2
dxdy,
+
(20)
where
u
0
approximates original image
u
,
u
o
does not vary too much on each seg-
mented region, and the boundary
C
is becoming as short as possible. In fact,
u
0
is simply a cartoon version of the original image
u
. Basically,
u
0
is a new image
with edges drawn sharply. The objects are drawn smoothly without texture. Such
cartoons are perceived correctly as representing the same scene as a simplification
of the scene containing most of its essential features. In Chan and Vese's ap-
proach [58], a level set approach is proposed to solve the modified Mumford-Shah
functional.
4.4.2. The level set approach to solve the Mumford-Shah function
In order to explain this model clearly, let us first define the evolving curve
C
in Ω as the boundary of an open subset
w
of Ω (i.e.,
w
⊂
Ω, and
C
=
∂w
). In
