Biomedical Engineering Reference
In-Depth Information
interested in the change of the front. It is not important to get the solution at
points far away from the front, so the solution is important at the points near the
front. The points (highlighted in Fig. 9.23(b)) are called the narrow band points.
The change of the level set function at these points only is considered. Other
points (outside the narrow band) are called the far away points and they are given
large positive or large negative values to be out of interest (not processed), and
it speeds up the iterations. The use of the delta function defined by Eq. 9.64 is
very important to give the narrow band points.
9.5.6 Reinitialization
The existence of the front means that the level set function has positive and
negative parts, then it has negative and positive values including zeroes. The
level set function with this property is called a signed distance function. This
property should be kept through the iterations in order not to lose the front.
There are different solutions for this problem [54]. We will discuss only the
solution introduced by Osher
et al.
[55]. It was proved that recomputing the level
set function by solving Eq. 9.67 frequently enough will maintain the function as
signed distance function:
∂φ
∂
t
=
sign (
φ
)(1
−|∇
φ
|
)
,
(9.67)
where it contains the sign function sign. When the level set function is negative,
the information flows one way and when it is positive, the information flows the
other way. The net effect is to “straighten out” the level set function on either
sides of the zero level set,
0
=
sign (
φ
)(1
−|∇
φ
|
)
.
(9.68)
By solving this equation, the derivative of
φ
with respect to time will vanish
resulting in Eq. 9.68.
|∇
φ
|=
1 denotes the measure for signed distance function.
9.6 Application: MRA Data Segmentation
Using Level Sets
The human cerebrovascular system is a complex three-dimensional anatomical
structure. Serious types of vascular diseases such as carotid stenosis, aneurysm,
