Biomedical Engineering Reference
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Figure 4. The sign of ψ ( x, t ) is decided by whether the point is in the interior (minus) or
exterior (plus) of the zero level set.
normal direction, etc. The main idea is to embed this propagating interface as the
zero level set of a higher-dimensional function ψ . Let ψ ( x
,t =0), where
is a
x
N be defined by
point in
ψ ( x
,t =0)= ±
d,
(1)
where d is the signed distance from
x
to γ ( t =0), and the plus (minus) sign is
chosen if the point
is outside (inside) the initial front γ ( t =0), as shown in
Figure 4. The best discrete approximation of the surface is therefore the set of grid
positions closest to the zero-crossings in the image, as shown in Figure 5. Thus,
an initial function is ψ ( ρ, t =0):
x
N
with the property that
γ ( t =0)=( x |
ψ ( x
,t =0)=0) .
(2)
The goal is to produce an equation for the evolving function ψ ( x
,t ), which
contains the embedded motion of γ ( t ) as the level set ψ =0. Let
x ( t ), t
[0 ,
),
be the path of a point on the propagating front, i.e.,
x ( t =0)is a point on the
initial front γ ( t =0), and
x t normal to the front at
x ( t ) (see Mulder et al. [35]). Since the evolving function ψ is always zero on the
propagating front, then
x t = F ( x ( t )) with vector
ψ ( x ( t ) ,t )=0 .
(3)
According to the chain rule,
N
ψ t +
ψ x i x i t =0 ,
(4)
i =1
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