Biomedical Engineering Reference
In-Depth Information
the problem of volume segmentation. For more details on level set methods
see [7, 31].
There are two different approaches to defining a deformable surface from
a level set of a volumetric function as described in Eq. (8.1). Either one can
think of φ ( s )asa static function and change the isovalue k ( t ) or alternatively
fix k and let the volumetric function dynamically change in time, i.e. φ ( s , t ).
Thus, we can mathematically express the static and dynamic models respecti-
vely as
φ ( s ) = k ( t ) ,
(8.2a)
φ ( s , t ) = k .
(8.2b)
To transform these definitions into partial differential equations which can be
solved by standard numerical techniques, we differentiate both sides of Eq. (8.2)
with respect to time t , and apply the chain rule:
φ ( s ) d s
dk ( t )
dt ,
dt =
(8.3a)
∂φ ( s , t )
t +∇ φ ( s , t ) ·
d s
dt = 0 .
(8.3b)
The static equation (8.3a) defines a boundary value problem for the time-
independent volumetric function φ . This static level set approach has been
solved [32, 33] using “Fast Marching Methods.” However, it inherently has some
serious limitations following the simple definition in Eq. (8.2a). Since φ is a func-
tion (i.e. single-valued), isosurfaces cannot self-intersect over time, i.e. shapes
defined in the static model are strictly expanding or contracting over time. How-
ever, the dynamic level set approach of eq. (8.3b) is much more flexible and shall
serve as the basis of the segmentation scheme in this chapter. Equation (8.3b)
is sometimes referred to as a “Hamilton-Jacobi-type” equation and defines an
initial value problem for the time-dependent φ . Throughout the remainder of this
chapter we shall, for simplicity, refer to this dynamical approach as the level set
method, and not consider the static alternative.
Thus, to summarize the essence of the (dynamic) level set approach, let
d s / dt be the movement of a point on a surface as it deforms, such that it can be
expressed in terms of the position of s U and the geometry of the surface at
that point, which is, in turn, a differential expression of the implicit function, φ .
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