Biomedical Engineering Reference
In-Depth Information
images. Furthermore, there are applications not commonly thought of as moving
interface problems, like optimal path planning and construction of the shortest
geodesic paths on surfaces, which can be recast as front propagation problems
with significant advantages [39].
Consider a boundary, either a curve in 2D or a surface in 3D, separating one
region from another. Assume that this curve/surface moves in a direction normal
to itself with a known speed function
F
. The goal is to track the motion of this
interface as it evolves. Here, the motions of the interface in its tangential directions
are ignored. At a specific moment, the speed function
F
(
L, G, I
) describes the
motion of the interface in the normal direction.
1. Speed factor
L
depends on local geometric information (e.g., curvature
and normal direction).
2. Speed factor
G
depends on the shape and position of the front (e.g., inte-
grals along the front, heat diffusion).
3. Speed factor
I
does not depend on the shape of the front (e.g., an underlying
fluid velocity that passively transports the front).
Langragian techniques are based on parameterizing the contour according to
some sampling strategy and then evolving each element according to the speed
function. While such a technique can be very efficient, it suffers from various lim-
itations, like deciding on the sampling strategy, estimating the internal geometric
properties of the curve, and changing its topology, addressing problems in higher
dimensions [40].
The level set method was initially proposed to track a moving front by Osher
and Sethian in 1988 [41] and was widely applied across various imaging domains
in the late 1990s. They can be used to efficiently address the problem of curve
or surface propagation in an implicit manner. The central idea is represent the
evolving contour using a signed function, which is in a higher-dimensional space,
where its zero level corresponds to the actual contour. Then, according to the
motion equation of the contour, one can easily derive a similar flow for the implicit
surface such that when applied to the zero level it will reflect the propagation of
the contour. Basically, there are two kinds of formulations regarding the level set
method: boundary value formulation and initial value formulation.
2.2. Boundary Value Formulation
Assume for the moment that
F>
0, and the front is moving all the way
outward. We can define the arrival time
T
(
x, y
) of the front as it crosses each
(
x, y
). Based on the fact that distance = rate
×
time, we have the following
expression in the 1D situation:
1=
F
dT
dx
.
(1)
