Biomedical Engineering Reference
In-Depth Information
Figure 9.
Illustration showing step-up and step-down edges from the observer's point of
view. See attached CD for color version..
Now, if the contour approaches from the region marked A, then to track the
interface between regions A and B it should latch onto a step-up edge along its outer
normal. On the other hand, if it approaches from region B to the same interface,
the contour point needs to snap at a step-down gradient inside the enclosed region
of the contour. In other words, in the latter case the inward normal should see a
step-down gradient. If the orientation of the contour is taken into account, then
the desired edge would either be on the right- or left-hand side of the contour
depending on whether it is considered in a clockwise or anticlockwise sense.
˜
that relates gradient direction with the unit vector orthogonal to the local contour
segment and directed from the “exterior” to the “interior”.
Let
θ
G
(
τ
(
s
i
)) = tan
−
1
∇
y
f
(
τ
(
s
i
))
/
Falcao et al. [21, 22] proposed a solution using the concept of contour orientation
∇
x
f
(
τ
(
s
i
)) be the direction of the
intensity gradient at the point
τ
(
s
i
), where
s
i
indicates the
i
th control point on
contour
∇
x
f
(
τ
(
s
i
)) are the gradients along the
y
- and
x
-
directions, respectively, at the point
.
∇
y
f
(
τ
(
s
i
)) and
τ
τ
(
s
i
). As previously mentioned,
f
(
x, y
)
denotes the image intensity at the location (
x, y
).If
θ
N
(
τ
(
s
i
)) is the normal to
the contour at
τ
(
s
i
), then the energy due to the gradient field is defined as
−
F
gradient
(
τ
(
s
i
))
•
N
π
2
if
|
θ
G
(
τ
(
s
i
))
−
θ
N
(
τ
(
s
i
))
|≤
E
gradient
(
τ
(
s
i
)) =
0
otherwise
(11)
