Biomedical Engineering Reference
In-Depth Information
Remark: In the case of higher-dimensional HJ equations with symmetric Hamil-
tonians, numerical schemes can be built by replicating in each space variable
what we did for the one-dimensional case. Hence, the following equation with
convex ( H is convex
H smooth and 2 H ( p )
∂p i ∂p j 0( p
=( p 1 ,p 2 ,...,p n )) or
H ( λp +(1
λq ))
λH ( p )+(1
λ ) H ( q ) ,
0
λ
1 ,
p, q
R
) Hamil-
tonian
U t + H ( U x ,U y ,U z )=0
can be approximated by
U n +1
i,j,k
U i,j,k tg ( D −x U i,j,k ,
D + x U i,j,k ,D −y U i,j,k ,D + y U i,j,k ,D −z U i,j,k ,D + z U i,j,k ) ,
=
where
U i− 1 ,j,k
U i,j,k
U i− 1 ,j,k
U i,j,k
D −x U i,j,k =
, and D + x U i,j,k =
.
x
x
The same thing applies for the other space variables, with ∆ y and ∆ z the spacings
in the y and z directions.
Example — first-order accuracy scheme : The 2D version of level set Eq. (7)
can be approximated as
1
2 ,
φ n +1
i,j
= φ i,j t (max( F i,j , 0) + + min( F i,j , 0) )
where
i,j , 0) 2 + max( D −y
i,j , 0) 2 + min( D + y
1
2 ,
+ = [max( D −x
i,j , 0) 2 + min( D + x
i,j , 0) 2 ]
i,j , 0) 2 + max( D + y
i,j , 0) 2 + min( D −y
1
2 .
= [max( D + x
i,j , 0) 2 + min( D −x
i,j , 0) 2 ]
where we have used shorthand notation in which D + α
i,j = D + α φ i,j with α =
x, y . Note that this scheme chooses the appropriate “upwinding” [37] direction
depending on the sign of the speed fuction F .
Finally, all of the second-order curvature-dependent terms can be approxi-
mated, for instance, using central differences:
φ i +1 ,j 2 φ i,j + φ i 1 ,j
(∆ x ) 2
2 φ
∂x 2
=
,
φ i +1 ,j +1 + φ i− 1 ,j− 1
φ i +1 ,j− 1
φ i− 1 ,j +1
2 φ
∂x∂y
=
.
4(∆ x ) 2
The same applies for the other derivatives and for higher dimensions. For more
details on the numerical techniques for level sets, see [40, 41, 42]. Estimating
the region parameters based on the level set function will be presented in the next
section so that they can be placed in the main evolution equation.
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