Biomedical Engineering Reference
In-Depth Information
Figure 5.
Implicit level set surface
ψ
is the dotted line superimposed over the image grid.
The location of the surface is interpolated by the image pixel values. The grid pixels closest
to the implicit surface are shown in gray.
where
x
i
is
i
th component of
. Since
x
N
=(
ψ
x
1
,ψ
x
2
,ψ
x
3
,...,ψ
x
N
)
·
(
x
1
t
,x
2
t
,x
3
t
,...,x
N
t
)=
F
(
x
(
t
)
|∇
ψ
|
,
i
=1
(5)
we have the evolution equation for
ψ
, namely,
ψ
t
+
F
|∇
ψ
|
=0
,
(6)
with a given value of
ψ
(
x
,
t
=0). Combining Eqs. (4) and (5), the final curve
evolution equation is given as
∂ψ
∂t
=
F
(
κ
)
|∇
ψ
|
,
(7)
where
ψ
is the level set function, and
F
(
κ
) is the speedwithwhich the front (or zero
level curve) propagates. This fundamental equation describes the time evolution of
the level set function
ψ
in such a way that the zero level curve of this evolving func-
tion is always identified with the propagating interface (see Suri et al. [16]). The
