Biomedical Engineering Reference
In-Depth Information
what follows, inside(
C
) denotes the region
w
, and outside(
C
) denotes the region
Ω
\
w
. The method is the minimization of an energy based-segmentation. Assume
that the image
u
0
is formed by two regions of approximatively piecewise-constant
intensities, of distinct values
u
i
0
and
u
o
0
. Assume further that the object to be
detected is represented by the region with the value
u
i
0
. Denote its boundary by
u
i
0
u
o
0
C
0
. Then we have
u
0
≈
inside the object [or inside (
C
0
)], and
u
0
≈
outside
the object [or outside (
C
0
)]. The following fitting term is defined as:
c
1
|
2
dxdy
F
1
(
C
)+
F
2
(
C
)=
|
u
0
(
x, y
)
−
inside(
C
)
outside(
C
)
|
c
2
|
2
dxdy,
+
u
0
(
x, y
)
−
(21)
where
C
is any other variable curve, and the constants
c
1
,c
2
, depending on
C
, are
the averages of
u
0
inside
C
and, respectively, outside
C
. In this simple case, it is
obvious that
C
0
, the boundary of the object, is the minimizer of the fitting term:
in
C
{
F
1
(
C
)+
F
2
(
C
)
}≈
0
≈
F
1
(
C
0
)+
F
2
(
C
0
)
.
(22)
If the curve
C
is outside the object, then
F
1
(
C
)
>
0 and
F
2
(
C
)
≈
0. If the
curve
C
is inside the object, then
F
1
(
C
)
≈
0 and
F
2
(
C
)
>
0. If the curve
C
is
both inside and outside the object, then
F
1
(
C
)
>
0 and
F
2
(
C
)
>
0. The fitting
term is minimized when
C
=
C
0
, i.e., the curve
C
is on the boundary of the object
(Figure 14).
In order to solve more complicated segmentation, we require regularizing
terms, like the length of the curve
C
, or the area of the region inside
C
. A new
energy functional
F
(
c
1
,c
2
,C
) is defined as
F
(
c
1
,c
2
,C
)=
µ
·
Length(
C
)+
ν
·
Area(inside(
C
))+
λ
1
c
1
|
2
dxdy
+
inside(
C
)
|
u
0
(
x, y
)
−
(23)
λ
2
c
2
|
2
dxdy,
outside(
C
)
|
u
0
(
x, y
)
−
where
µ
≥
0,
ν
≥
0,
λ
1
,
λ
2
≥
0. (In Chan-Vese's approach,
λ
1
=
λ
2
=1and
ν
=0). Correspondingly, the level set based on
C
is defined as
C
=
∂w
=
{
(
x, y
)
∈
Ω:Φ(
x, y
)=0
}
inside(
C
)=
w
=
{
(
x, y
)
∈
Ω:Φ(
x, y
)
>
0
}
.
(24)
outside(
C
)=Ω
\
w
=
{
(
x, y
)
∈
Ω:Φ(
x, y
)
<
0
}
