Biomedical Engineering Reference
In-Depth Information
The mean shift algorithm is a nonparametric technique for estimation of the
density gradient, which was first proposed by Fukunaga
et al.
[39]. The idea
was later generalized by Cheng [40]. The technique was extended to various
applications, amongst them color image segmentation, by Comaniciu
et al.
[12,
13, 20].
10.12 Appendix. Numerical Solution for the
Level Set Implementation of RAGS
Let
φ
i
,
j
denote the value of
φ
at the grid position of
x
i
=
i
x
,
y
i
=
j
y
, where
x
and
y
are the grid steps along
x
and
y
directions respectively. Denote
φ
(
x
i
,
y
j
,
t
n
)by
φ
n
i
,
j
, the time derivative
φ
t
at (
i
,
j
,
t
n
) is approximated by forward
difference as
φ
t
(
i
,
j
,
t
n
)
=
(
φ
n
+
1
i
,
j
)
/
t
, where
t
is a small time interval. As
given in (10.17), the snake evolves according to four forces. However, they can
be categorized into three types based on the nature of their motions.
The first motion is a collapsing one with speed proportional to its curvature. It
is a parabolic contribution to the equation of motion and it can be approximated
with central differences. The curvature
κ
is only dependent on contours; it is
independent of time and spatial position, hence it can also be solved using central
difference approximations. The curvature motion at time
t
is approximated as
i
,
j
−
φ
2
2
+
D
0
y
i
,
j
(
g
(
·
)
κ
|∇
φ
|
)
i
,
j
=
g
(
·
)
i
,
j
K
i
,
j
(
D
0
x
)
1
/
2
,
(10.42)
i
,
j
i
−
1
,
j
)
/
2
x
,
D
0
y
where
D
0
x
i
,
j
−
1
)
/
2
y
, and
K
i
,
j
is the
central difference approximation to the curvature expression given in (10.22):
n
i
+
1
,
j
−
φ
n
n
i
,
j
+
1
−
φ
n
i
,
j
=
(
φ
i
,
j
=
(
φ
i
,
j
,
(
φ
y
)
i
,
j
=
D
0
y
(
φ
x
)
i
,
j
=
D
0
x
(10.43)
i
,
j
,
n
n
i
,
j
+
φ
n
i
−
1
,
j
n
n
i
,
j
+
φ
n
i
,
j
−
1
i
+
1
,
j
−
2
φ
i
,
j
+
1
−
2
φ
(
φ
xx
)
i
,
j
=
φ
(
φ
yy
)
i
,
j
=
φ
,
,
(10.44)
x
2
y
2
n
i
+
1
,
j
+
1
−
φ
n
i
−
1
,
j
+
1
−
φ
n
i
+
1
,
j
−
1
+
φ
n
i
−
1
,
j
−
1
(
φ
xy
)
i
,
j
=
φ
(10.45)
.
4
x
y
The second motion is expanding or shrinking with a spatially constant speed
in its normal direction. It must be approximated through
entropy-satisfying
schemes
[16]. Let
V
0
be the constant speed function regarding
α
g
(
·
). Following
