Biomedical Engineering Reference
In-Depth Information
10.4.1 Gradient Flow Force: A Summary
As mentioned earlier, the gradient flows impose local constraints while the re-
gion force contributes global constraints. Within a homogeneous region of an
image, measured by region segmentation, the snake evolves mainly according
to gradient flows. The first gradient flow is the weighted length gradient flow,
which is given by (10.7). It is composed of two terms. The first is the weighted
curvature term,
g
(
|∇
I
|
)
κ N
, which smooths the active contour and also shrinks
it. The second term, (
∇
g
(
|∇
I
|
)
·
N
)
N
, is on the normal factor of the gradient
of the weighting function. Unlike the curvature, the vector field
∇
g
(
|∇
I
|
)is
static. The direction and strength of this field depend on position only, and is
independent of time and contour.
The second gradient flow,
g
(
|∇
I
|
)
c
N
, is introduced by constant motion
which locally minimizes area (see [14] for proof). It helps the snake shrink
or expand toward object boundaries and accelerates its convergence speed.
For all these forces, the weighting function
g
can be defined as any decreasing
function of the image
I
edge map
f
such that
g
→
0as
f
→∞
. When dealing
with gray level images, the solution (as used in this work) is straightforward:
1
1
+
f
.
(10.9)
f
=|∇
(Gauss
∗
I
)
|
and
g
=
This monotonically decreasing nature is illustrated in Fig. 10.2. As for color
images, the edge function
f
becomes a little more intricate (an example function
will be presented in Section 10.6). However, the derivation of the decreasing
function
g
can remain the same.
10.4.2 Diffused Region Force
The aim of generating a region force is to empower the snake with a global view
of image features. A typical region segmentation method splits an image into
several regions, giving the segmentation map
S
. From this, the region map
R
is generated by computing the gradient of
S
. The gradient computation is the
same as the edge computation stage for generating gradient forces. Then, we
compute the gradient
∇
R
of this region map, resulting in region constraints in the
vicinity of the region boundaries. Having slithered across a homogeneous region
reliant on the gradient flow forces, if the snake tries to step from one region into
another, it must concur with the region force in
∇
R
since it breaks the region
