Biomedical Engineering Reference
In-Depth Information
Chapter 4
Recent Advances in the Level Set Method
David Chopp 1
4.1 Introduction
The level set method was introduced in the groundbreaking paper by Osher
and Sethian in 1988 [85]. The ground work for this paper began with a paper
by Sethian on moving fronts [102]. The original application for this work was
to solve problems in flame propagation, where the flame speed was given as a
function of the local mean curvature of the propagating flame front. The work
in [85] combined two fundamental ideas together in a unique way, and formed
the basis for the level set method in wide use today.
The first fundamental idea was the choice of an implicit representation for
the moving interface. At first glance, this appears to be a completely unnatural
choice; it is more difficult to specifically locate the interface at any given time,
and, in its simplest form, requires an order of magnitude greater computational
cost. However, this approach also offers powerful geometric properties which
no other method can as easily provide, and can be extended to higher dimen-
sions with vastly greater ease. Specifically, the implicit representation allows for
changes in the topology of the interface to happen naturally without requiring
collision detection and interface reconstruction as required by Lagrangian-type
methods. Also, the evolution equation they derived for propagating the interface
can be written entirely in terms of the embedding function, so that (at least for
their application) the actual location of the interface at any given time need not
be determined.
1 Engineering Sciences and Applied Mathematics Department, Northwestern University.
201
 
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