Biomedical Engineering Reference
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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
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