Biomedical Engineering Reference
In-Depth Information
its squared norm given by
2
2
∂
∂
u
i
∂
2
d
=
∂
u
j
du
i
du
j
.
(10.27)
i
=
1
j
=
1
Using standard Riemannian geometry notation, let
s
ij
=
∂
∂
u
i
·
∂
∂
u
j
, such that
du
1
du
2
T
s
11
du
1
du
2
2
2
s
12
2
d
s
ij
du
i
du
j
=
(10.28)
=
.
s
21
s
22
i
=
1
j
=
1
2
(
v
) indicates the rate of change of
the image in the direction of
v
. The extrema of the quadratic form are obtained in
the directions of the eigenvectors of the metric tensor
s
ij
, and the corresponding
eigenvalues are
For a unit vector
v
=
(cos
θ,
sin
θ
), then
d
(
s
11
−
s
22
)
2
s
11
+
s
22
±
+
4
s
12
(10.29)
λ
±
=
2
with eigenvectors (cos
θ
±
,
sin
θ
±
) where the angles
θ
±
are given by
1
2
s
12
s
11
−
s
22
2
arctan
θ
+
=
(10.30)
.
θ
−
=
θ
+
+
2
The maximal and minimal rates of change are the
λ
+
and
λ
−
eigenvalues
respectively, with corresponding directions of change being
θ
+
and
θ
−
. The
strength of an edge in a vector-valued case is not given simply by the rate
of maximal change
λ
+
, but by the difference between the extrema. Hence, a
good approximation function for the vector edge magnitude should be based on
f
=
f
(
λ
+
,λ
−
). Now RAGS can be extended to the region-aided geometric color
snake by selecting an appropriate edge function
f
col
. The edge stopping function
g
col
is defined such that it tends to 0 as
f
col
→∞
. The following functions can
be used (cf. (10.9)):
1
1
+
f
col
.
f
col
=
λ
+
−
λ
−
and
g
col
=
(10.31)
Then replacing
g
col
(
·
) for the edge stopping term
g
(
·
) in (10.17), we have the
color RAGS snake:
C
t
=
[
g
col
(
|∇
I
|
)(
κ
+
α
)
−∇
g
col
(
|∇
I
|
)
·
N
+
β
R
·
N
]
N.
(10.32)