Biomedical Engineering Reference
In-Depth Information
4.2.4 Reinitialization
As noted in the introduction to this chapter, there are two means by which the
level set method can be kept stable for arbitrary speed functions. For nearly all
applications of the level set method, one of these techniques must be used. One
method involves using velocity extensions, and the other uses reinitialization.
Both methods are frequently used, and there is disagreement as to which method
is preferred. Recent advances in the level set method have resulted in either
method producing good results. For balance, both methods are presented, with
reinitialization treated here and velocity extensions to follow in Section 4.2.5.
Reinitialization was first introduced in [19], where it was observed that the
only part of the level set function which is of interest is the portion immediately
around the zero level set. While initially, the level set function can be constructed
to be the signed distance function to the interface, most speed functions,
F
, will
not preserve this property over time. This can lead to instability, and ultimately
failure of the method. Reinitialization is, therefore, a process where the level set
function is reconstructed to be the signed distance function.
Let
φ
be the level set function, and let
˜
φ
be the desired reconstructed level
set function, then
˜
φ
solves
˜
φ
−
1
(0)
=
φ
−
1
(0)
,
(4.29)
˜
φ
=
1
.
(4.30)
∇
This pair of equations is precisely the type of problem the fast marching method
is designed to solve, with
F
≡
1 in Eq. 4.21. Furthermore, the function
φ
can be
used to initialize the fast marching method, as described in Section 4.2.3. The
solution
˜
φ
of Eqs. 4.29 and 4.30 is now called reinitialized.
Early implementations of reinitialization suffered from accuracy, particularly
in regions of high curvature. When the interface was reinitialized, there was
significant error in the computed solution in Eq. 4.29. This was primarily due to
the low-order accurate methods used for interpolating
φ
. More recent methods,
such as the one presented in Section 4.2.3, significantly reduced this error, as
illustrated in Fig. 4.8.
It has been observed recently [100] that for the specific application of reini-
tialization, it is not necessary to use the heap sort method. In fact, the same
results can be achieved by simply taking a first-in-first-out strategy for the order
of the grid points. In other words, instead of maintaining the binary tree and
