Biomedical Engineering Reference
In-Depth Information
mean curvature motion is given by the following evolution equation [65]:
div
∇
+
v
,
∂φ
∂t
=
g
(
|∇
φ
f
|
)
·|∇
φ
|·
(13)
|∇
φ
|
where
v
≥
0 is a constraint on the area inside the curve, increasing the propagation
speed;
g
(
|∇
f
|
) is an edge-sensitive speed function and is defined as
1
F
(
x
)=
p
,
(14)
1+
|∇{
G
σ
∗
f
(
x
)
}|
where
p
≥
1. From this argument, it is clear that, if the image gradient
|∇{
G
σ
∗
approaches the local maximum at the object boundaries, the curve gradually
attains zero speed.
f
(
x
)
}|
, i.e.,
F
=0at the boundaries, the evolving curve eventually stops and the final zero
level set Ψ(
x,
Under the ideal condition that
|∇{
G
σ
∗
f
(
x
}|→∞
∞
)=0corresponds to the segmentation result. In practice, it is
impossible for
F
.
Another well-known deformable model [49], the so-called
geodesic snake
model
, is employed in our work for cardiac valve segmentation (see below). It
also uses the image gradient to stop the curve.
=0, and the curve will leak as
t
→∞
Its level set formulation is as
follows:
div
g
(
|∇
∂φ
∂t
=
|∇
∇
φ
φ
|
f
|
)
+
vg
(
|∇
f
|
)
|∇
φ
|
.
(15)
|∇
φ
|
2.2.3. Chan-Vese model
On the other hand, because these classical snake models rely on the edge
function
g
(
|∇
, to stop the curve
evolution, these models can detect only objects with edges defined by a gradient. In
practice, discrete gradients are bounded, and so the stopping function is never zero
on the edges, and the curve may pass through the boundary, even for the geodesic
snake model mentioned above. If the image
f
is very noisy, the isotropic smoothing
Gaussian has to be strong, which will smooth the edges as well. To address these
problems, Chan and Vese [58] recently proposed a different deformable model, i.e.,
a model not based on the gradient of the image
f
for the stopping process. Instead,
the evolvement of the curve is based on the general Mumford-Shah formulation
of image segmentation [66], by minimizing the functional
f
|
), or say, depend on the image gradient
|∇
f
|
(
f, C
)=
µLength
(
C
)+
λ
f
0
|
2
dxdy
+
|
2
dxdy,
(16)
F
MS
Ω
|
f
−
Ω
\C
|∇
f
Ω
→
where
f
0
:
R
is a given image, and
µ
and
λ
are positive parameters. The
solution image
f
is formed by smooth regions
R
i
and sharp boundaries, denoted
here by
C
. A reduced form of this problem is simply the restriction of
F
MS
to
