Biomedical Engineering Reference
In-Depth Information
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Figure 4.12:
A contour map of the distance from the origin on the manifold
3
4
sin(3
π
x
) sin(3
π
y
), computed using the ordered upwind method. Reprinted
with permission from [101].
z
=
The solution is to do both the fast marching method with the velocity exten-
sion at the same time:
F
∇
φ
=
1
,
(4.40)
∇
F
·∇
φ
=
0
.
(4.41)
The discretization of these two equations is the same as before, but the solution
method requires some explanation. Again, suppose the values of
φ
and
F
are
already determined at
x
i
−
1
,
j
and
x
i
,
j
+
1
. Then Eqs. 4.40 and 4.41 become
F
i
,
j
((
D
x
φ
i
,
j
)
2
+
(
D
y
φ
i
,
j
)
2
)
=
1
,
(4.42)
(
D
x
F
i
,
j
)(
D
x
φ
i
,
j
)
+
(
D
y
F
i
,
j
)(
D
y
φ
i
,
j
)
=
0
.
(4.43)
These equations correspond to Eqs. 4.22 and 4.34 respectively, where the un-
knowns are
F
i
,
j
and
φ
i
,
j
, and the remainder of the terms are known. This pair