Biomedical Engineering Reference
In-Depth Information
Figure 2.
Evolution only by advection leads to attracting a curve (initial ellipse) to spurios
edges, but adding a regularization term related to the curvature of evolving curve, the edge
is found smoothly also in the case of a 2D noisy image (right).
as
N
v
=
g
0
k
+
∇
g
0
·
of the segmentation curve, where
k
is its curvature and
N
is its normal vector.
Similarly, the geometrical equation for the moving segmentation surface has the
form
N,
v
=
g
0
H
+
∇
g
0
·
where
H
is its mean curvature and
N
is its normal vector. The level set formulation
of either such curve or surface evolution is given by ([43, 44, 45, 46, 47])
.
∇
u
|∇
.
g
0
∇
u
|∇
,
u
t
=
g
0
|∇
g
0
.
u
|∇
+
∇
∇
u
=
|∇
u
|∇
(1)
u
|
u
|
where the moving curve or surface is given by the same evolving level line and,
respectively, level surface of the level set function
u
.
There is still a practical problem with the previous approach. It gives satis-
factory results if the initial segmentation curve or surface belongs to the vicinity
of an edge; otherwise, it is difficult to drive an arbitrary initial state there. An im-
portant observation, leading to the subjective surface method ([48, 49, 50]), is that
Eq. (1) moves not only one particular level set, but all the level sets, by the above
mentioned advection-diffusion mechanism. So we can consider the evolution of
the whole (hyper)surface
u
, which we call the
segmentation function
, composed
by those level sets. Moreover, we are a bit free in choosing the precise form of the
