Biomedical Engineering Reference
In-Depth Information
3. The third type of motion in (10.17) is contributed by the underlying static
velocity field, the direction and strength of which are based on spatial
position. It is independent of the shape and position of the snake. The
motion of contours under this velocity field can be numerically approxi-
mated through
upwind schemes
by checking the sign of each component
of the velocity field and constructing one-sided upwind differences in the
appropriate direction. For a positive speed component, backward differ-
ence approximation is used, otherwise forward difference approximation
should be applied.
By using these approximation schemes, (10.17) can be numerically imple-
mented. The detailed numerical solutions for RAGS are presented in Appendix
A. For general numerical solution to level sets, including concepts such as
en-
tropy condition
and
upwind scheme
, the interested reader is referred to works
by Sethian [16, 17] and by Osher
et al.
[18].
10.6 Region-Aided Geometric Snake on
Vector-Valued Images
The theory of boundary detection by the geometric or geodesic snake can be
applied to any general “edge detector” function. The stopping function
g
should
tend to zero when reaching edges.
When dealing with gray level images, the decreasing function
g
can be easily
derived from the edge detector
f
, as shown in (10.9). We use a similar stopping
function for edges obtained directly from vector-valued images such as a color
image.
A consistent extension of scalar gradients based on a solid theoretical foun-
dation has been presented by di Zenzo [19]. This extension has been applied in
the active contour literature to both geometric and parametric snakes.
In a vector-valued image the vector edge is considered as the largest differ-
ence between eigenvalues in the tensor metric. Let
(
u
1
,
u
2
):
m
be an
m-band image for
i
=
1
,
2
,...,
m
. For color images,
m
=
3. A point in the image
is considered as a vector in
2
→
m
. The distance between two points,
P
=
(
u
1
,
u
2
)
and
Q
=
(
u
1
,
u
2
), is given by
=
(
P
)
−
(
Q
). When this distance tends to
be infinitesimal, the difference becomes the differential
d
=
i
=
1
∂
∂
u
i
du
i
with
