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Stationary Formulation: The Boundary Value PDE View
The central idea of
level set methods
is to track a propagating interface by embedding it as the zero
level set of a higherdimensional function.
This is an initial value partial differential equation; the
initial front gamma(t=0) provides initial data as the zero level
set of the level set function phi, and tracking the evolution of
phi is identified with the evolution of the interface under its normal
speed F.
In the case where the front is moving with a speed F which is always
positive, front propagation problems can be recast as a
stationary boundary value problem. In the special case where this
speed depends only on position, the resulting Eikonal equation is
a static HamiltonJacobi equation with a long history; it has
applications in such areas as geometric optics, seismology, and
electromagnetics.
More precisely, given a interface initially at gamma(t=0), let
T(x,y) be the time T at which the interface crosses the point (x,y).
Then a static HamiltonJacobi equation of the form

