Biomedical Engineering Reference
In-Depth Information
This is the basic equation of the level set [4], and the zero level set denotes the
object contour curve:
Γ(
t
)=
{
(
x, y
)
|
Φ(
x, y, t
)=0
}
.
(6)
The function Φ is usually calculated based on the signed distance measure to the
initial front. Applying to the image domain, it can be simply the Euclidean distance
between the curve and one image point. That is,
Φ(
x, y
)=
±
d
(
x, y
)
,
(7)
where
d
(
x, y
) is the Euclidean distance from the point to the boundary, and the
sign is chosen such that points inside the boundary have a negative sign and those
outside have a positive sign.
Now the evolution of the boundary is defined via a partial differential equation
on the zero level set of Φ:
∂
∂t
=
−
F
|∇
Φ
|
.
(8)
Here,
F
is a known function, and is determined by the local curvature
κ
at the zero
level set, i.e.,
F
=
F
(
κ
)
,
(9)
where the curvature
κ
is given by
φ
xx
φ
y
−
2
φ
x
φ
y
φ
xy
+
φ
yy
φ
x
(
φ
x
+
φ
y
)
3
/
2
κ
=
∇·
∇
φ
=
,
(10)
|∇
|
φ
3.2. Speed Function
Consider a boundary separating one region from another (either a curve in 2D
or a surface in 3D), and imagine that this interface moves in a direction normal to
itself with a known speed function
F
. The purpose is to track the motion of this
interface as it evolves [4], as illustrated in Figure 2. Sethian [4, 20] represents the
speed function
F
as
F
=
F
(
L, G, I
)
,
(11)
where
L
is the
local information
, which is determined by the local geometric
properties, such as curvature and normal direction;
G
is the
global property
of
the front that depends on the shape and position of the front;
I
represents the
independent properties
that are independent of the front, such as an underlying
fluid velocity that passively transports the front.
It is difficult to design a generic model for the speed function
F
, because it
is closely related to the applications. In this section, we give a brief view of two
important terms in the speed function. First we examine the curvature term in
Eq. (9), and then we look at the speed term introduced by Caselles et al. [16] and
Malladi et al. [9] for image segmentation purposes.
