Biomedical Engineering Reference
In-Depth Information
(a)
(b)
(c)
Figure 10.
Illustration of gradient orientation on snake deformation: (a) contour deforma-
tion without gradient orientation information; (b) segmentation of region when foreground
intensity is less than background and (c) when foreground intensity is greater than back-
ground intensity.
In both (b) and (c) the contour has properly latched onto the desired
boundary. See attached CD for color version. Reprinted with permission from [24].
Copyright c
2004, SPIE.
The normal
θ
N
(
τ
(
s
i
)) at a point
τ
(
s
i
) on the contour
at
s
i
is defined as
τ
θ
N
(
τ
(
s
i
)) =
θ
C
(
τ
(
s
i
)) +
λ
π
2
,
(12)
where
θ
C
(
τ
(
s
i
)) is the tangent at the point
τ
(
s
i
). The value of
λ
can be +1 or -1
depending on the desired direction of the normal. If the object intensity is greater
than the background, then for contour in the counterclockwise direction the value
of lambda should be 1, indicating an inward normal. This ensures that the contour
will only be able to see the gradient, which is in the direction of the inward normal.
But if it encounters a step-up gradient, then the difference between the two gradient
angles and the growing normal will be more than
π
2
, so that the step-up gradient
will be invisible to the growing contour. The reverse is the case if the contrast is
changed. The effect of using the gradient orientation information within the snake
framework is shown in Figure 10b,c. Snake deformation without the orientation
information is illustrated in Figure 10a, which shows the final contour alternating
between the two disconnected strong edges [24].
3.2. Convergence to Concavities
One of the major challenges faced in the initial phases of snake formulation
was its inability to converge into concavities. Xu and Prince [25] designed a new
external static force field vector
F
ext
(
x, y
)=
v
(
x, y
), called the gradient vector
flow field. This field originates from the edges and makes the snake converge to the
gradient concavities. The gradient vector flow field is the vector field
v
(
x, y
)=
[
u
(
x, y
)
,v
(
x, y
)], which minimizes the energy functional
µ
(
u
x
+
u
y
+
v
x
+
v
y
)+
|∇
2
dxdy,
|
2
|
v
−∇
E
=
g
g
|
(13)
where
g
is the intensity gradient.
