Biomedical Engineering Reference
In-Depth Information
diffusion term in the segmentation model. In fact,
g
(
|∇
,
u
t
=
ε
2
+
|∇
u
ε
2
+
|∇
∇
I
0
|
)
|
2
∇·
u
G
σ
∗
(2)
u
|
2
where the Evans-Spruck regularization [51]
|
ε
=
ε
2
+
|∇
|∇
u
|≈|∇
u
u
|
2
(3)
g
0
·∇
is used, gives the same advection term
u
as (1). The parameter
ε
shifts the
model from the mean curvature motion of level sets (
ε
=0) to the mean curvature
flow of graphs (
ε
=1). This means that either level sets of the segmentation func-
tion move in the normal direction proportionally to the (mean) curvature (
ε
=0),
or the graph of the segmentation function itself moves (as a 2D surface in 3D space
in segmentation of 2D images, or a 3D hypersurface in 4D space in segmentation
3D images) in the normal direction proportionally to the mean curvature. In both
cases large variations in the graph of the segmentation function outside edges are
smoothed due to large mean curvature. On edges the advection dominates, so all
the level sets that are close to the edge are attracted from both sides to this edge
and a shock (steep gradient) is subsequently formed. For example, if the initial
“point-of-view” surface, as plotted in the top right portion of Figure 3, illustrating
the 2D situation, is evolved by Eq. (2), the so-called subjective surface is formed
finally (see Figure 3, bottom right), and it is easy to use one of its level lines, e.g.,
(max(
u
) + min(
u
))
/
2, to get the boundary of the segmented object.
In the next example we illustrate the role of the regularization parameter
ε
.
The choice of
ε
=1is not appropriate for segmentation of an image object with
a gap, as seen in Figure 4 (top). However, decreasing
ε
, i.e., if we go closer to
the level set flow Eq. (1), we get very good segmentation results for that image
containing a circle with a large gap, as presented in Figure 4 (middle and bottom).
If the image is noisy, the motion of the level sets to the shock is more irregular,
but finally the segmentation function is smoothed and flattened as well. For a
comprehensive overview of the role of all the model parameters, the reader is
referred, e.g., to [52].
The subjective surface segmentation (2) is accompanied byDirichlet boundary
conditions:
−∇
u
(
t, x
)=
u
D
in [0
,T
]
×
∂
Ω
,
(4)
IR
d
,
where
∂
Ω is a Lipschitz continuous boundary of a computational domain Ω
⊂
d
=3, and with initial condition
u
(0
,x
)=
u
0
(
x
)
in Ω
.
(5)
We assume that the initial state of the segmentation function is bounded, i.e.,
u
0
∈
L
∞
(Ω). The segmentation is an evolutionary process given by the solution
