Biomedical Engineering Reference
In-Depth Information
3.3. Narrow-Band Algorithm and Implementation
Malladi et al. [9] implemented the level set approach using the
Narrow-Band
Method
, which solves the initial value problem of level set methods. They then
sped up the boundary-value problem by also employing a
Fast Marching Method
[41], which requires that the speed
F
should be strictly positive or negative. In
this chapter, we focus on Narrow-Band methods with our particular application.
Instead of using all the grid points, the key idea of the Narrow-Band approach
is to constrain the computation only to the pixels that are close to the zero level
set. The pseudocode of this method is:
1.
Initialize
the signed distance function Φ(
x,
0) of the initial contour Γ;
2.
Find
the narrow band points: determine those points
X
i
whose distance
|
Φ(
x, t
)
|
is less than the specified narrow bandwidth, and mark them as
the narrow band points;
3.
Update
: resolve level set Eq. (20), and track the zero level set curve; update
the level set function value
|
Φ(
x, t
+∆
t
)
|
in the narrow band;
4.
Reinitialize
: reinitialize the narrow band when the zero level set reaches
the boundary of the narrow band. Repeat steps 3 and 4.
5.
Convergence test
: check whether the iteration converges or not. If so,
stop; otherwise, enter the calculation of the next step, and go to step 3.
In step 1 the signed distance is always obtained by the closest distance from
the narrow band points to the zero level set. The sign is chosen such that the
inner part of the zero level set is negative, and the outer part is positive. In this
way, the signed distance function can be seen as the projection of
φ
from the
one-dimensional higher space to the image space.
In step 3, when solving propagation Eq. (20), the normal
n
and curvature
F
(
κ
)
terms actually are only defined on the zero level set. We need to extend the speed
function to the whole narrow band area. A simple way to do this is to extrapolate
the value on the zero level set to the nearby grid points. Many approaches for such
extensions have been discussed in [4].
When the curve evolves to the boundary of the narrow band, the narrow band
should be reinitialized. If the point difference between sequential time steps is
smaller than the designed threshold in the narrow band, the propagating speed will
be set to zero. When the narrow band is not reinitialized within a certain time, this
means that curve evolvement has stopped. So the iteration ends, and the final zero
level curve is the boundary of the desired object.
