Biomedical Engineering Reference
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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 + ). 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)
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