Biomedical Engineering Reference
In-Depth Information
In this higher-dimensional formulation, topological changes can be efficiently
implemented. Numerical schemes are stable, and the model is general in the sense
that the same formulation holds for 2D and 3D, as well as for merges and splits.
Besides, the surface geometry is easily computed. For example, the front normal
and curvature are given by:
∇
,
G
(
x, t
)
n
=
∇
G
(
x, t
)
,
K
=
∇·
(15)
∇
G
(
x, t
)
respectively, where the gradient and the divergent (
∇·
) are computed with respect
to
x
.
The initialization of the model through Eq. (14) is computationally expensive
and not efficient if we have more than one front to initialize [4].
The
narrow-band
technique is much more appropriate for this case. The key
idea of this technique comes from the observation that the front can be moved
by updating the level set function at a small set of points in the neighborhood of
the zero set instead of updating it at all the points on the domain (see [8, 30] for
details).
To implement this scheme, we need to pre-set a distance ∆
d
to define the
narrow band. The front can move inside the narrow band until it collides with
the narrow-band frontiers. Then, the function
G
should be reinitialized by treating
the current zero set configuration as the initial one.
Also, this method can be made cheaper by observing that the grid points that
do not belong to the narrow band can be treated as sign holders [8], following the
same idea of the characteristic function defined in Equation (2). Thus, the result
of steps (1)-(5) (Section 5) can be used to initialize the Level Sets model if the
narrow-band extension technique is applied.
At first, the user interaction procedure can also be implemented by using the
narrow-band approach. From the entropy condition, manually burnt nodes remain
burnt and consequently the split components will not merge again. However,
we should be careful to rebuild the narrow band consistently after the manual
intervention. We believe that this is the main challenge with this direction.
Offsets can also be implemented through Level Sets in a manner that resembles
the application of level sets for grid generation [36]. For example, in the case of
convex initial shape,
F
K
, where
K
is given in Eq. (15), is effective. For a
non-convex initial body, points with high curvature can move against the front. A
possible solution is to include a threshold value that ensures that the front always
moves outward:
F
=
−
K, F
threshold
). However, for highly non-convex
bodies, like the ones generated by steps (1)-(5), the speed function is not simple.
Then, sorting of domain decomposition would be required [36, 30].
= min (
−
These are
further directions for our work, as dicussed next.
