Biomedical Engineering Reference
In-Depth Information
Sethian's
upwinding
finite difference scheme, the solution is given by
⎧
⎨
(
V
0
|
φ
|
)
i
,
j
=
V
0
i
,
j
[max(
D
−
x
+
min(
D
+
x
i
,
j
,
0)
2
,
0)
2
i
,
j
+
max(
D
−
y
+
min(
D
+
y
i
,
j
,
0)
2
i
,
j
)
2
]
1
/
2
if
V
0
i
,
j
≥
0
,
(10.46)
⎩
(
V
0
|
φ
|
)
i
,
j
=
V
0
i
,
j
[max(
D
+
x
+
min(
D
−
x
i
,
j
,
0)
2
i
,
j
,
0)
2
+
max(
D
+
y
+
min(
D
−
y
i
,
j
,
0)
2
i
,
j
)
2
]
1
/
2
otherwise
i
,
j
)
/
x
,
D
+
y
where
D
+
x
n
i
+
1
,
j
−
φ
n
n
i
,
j
+
1
−
φ
n
i
,
j
)
/
y
and
D
−
x
n
i
,
j
−
i
,
j
=
(
φ
i
,
j
=
(
φ
i
,
j
=
(
φ
i
−
1
,
j
)
/
x
,
D
−
y
i
,
j
−
φ
i
,
j
−
1
)
/
y
are the forward and backward differences,
φ
i
,
j
=
(
φ
respectively.
The external forces left in (10.17) contribute the third underlying static ve-
locity field for snake evolution. Their direction and strength are based on spatial
position, but not on the snake. This motion can be numerically approximated
as follows. Let
U
(
x
,
y
,
t
) denote the underlying static velocity field according to
β
R
−∇
g
(
·
). We check the sign of each component of
U
and construct one-sided
upwind differences in the appropriate (upwind) direction [16]:
(
U
·∇
φ
)
i
,
j
=
max(
u
i
,
j
,
0)
D
−
x
i
,
j
+
min(
u
i
,
j
,
0)
D
+
x
i
,
j
,
0)
D
−
y
i
,
j
,
0)
D
+
y
i
,
j
n
i
,
j
n
i
,
j
+
max(
v
+
min(
v
,
(10.47)
where
U
=
(
u
,v
). Thus, (10.17) is numerically solved using the schemes de-
scribed above.
Questions
1.
What are the advantages of geometric snakes over their parametric coun-
terparts?
2.
Which are some of the key papers on the geometric snake?
3.
How do I diffuse the region segmentation map?
4.
Describe howweighting functions p
(
·
)
and q
(
·
)
behave in vector diffusion?
5.
What are the parameters in RAGS?
6.
How do I choose the parameter values?
7.
What are some of the disadvantages of RAGS?
