Biomedical Engineering Reference
In-Depth Information
where
M
1
and
M
2
are the maximum and minimum values of the magnitude of the
image gradient
|∇
G
σ
∗
I
(
x, y
)
|
. When the Gaussian smoothed gradient reaches
M
1
, the value of the speed function will be 0 and the zero level set will provide
the contour of the object.
By integrating (36) and (37) together, we have
,Ψ(
s,
0)=Ψ
0
(
s
)
.
(38)
Applying the FDM, the numerical solution of (38) is given by the following iterative
equation:
γ
∂
∂t
=
|∇
Ψ
|
γ
M
1
−
−
M
2
(
|∇
G
σ
∗
I
(
x, y
)
|−
M
2
)
γ
,
γ
M
1
−
n
+1
=Ψ
n
Ψ
+∆
t
·
|∇
Ψ
|
−
M
2
(
|∇
G
σ
∗
I
(
x, y
)
|−
M
2
)
(39)
Ψ(
s,
0)=Ψ
0
(
s
)
,
where ∆
t
is the iteration step. Equation (39) can be implemented iteratively.
Figure 5 shows the experimental results applying the algorithm introduced in
this section. From the experiments we conclude that level set-based deformable
models are quite insensitive to initial positions and can handle topological changes
successfully. (Similar as in the previous section, all the segmentation results are
obtained after applying the color gradients in [17]; otherwise, the result will not
converge to the nuclei if applying the normal gradient used in [5].)
4.2. Geodesic Deformable Models
In almost parallel efforts, the geodesic model was proposed by Caselles [6].
Equation (1) of the traditional deformable model can be rewritten in a more general
form:
1
E
(
C
)=
(
E
int
(
C
(
s
)) +
E
ext
(
C
(
s
)))
ds,
(40)
0
where
C
is a specific allowed space of curves. By applying Fermat's principle, it
is equivalent to solve the following intrinsic problem:
1
C
(
s
)
|
arg min
C
g
(
|∇
I
(
C
(
s
))
|·|
ds
)
,
(41)
0
where
I
is the image, and the function
g
is an edge detector defined as
1
g
(
|∇
I
(
C
(
s
))
|
=
p
,
(42)
1+
|∇
G
σ
∗
I
(
x, y
)
|
where
p
is the rank with
p>
0.
