Biomedical Engineering Reference
In-Depth Information
8.2 Level Set Surface Models
When considering deformable models for segmenting 3D volume data, one is
faced with a choice from a variety of surface representations, including triangle
meshes [19, 20], superquadrics [21-23], and many others [18, 24-29]. Another
option is an implicit level set model, which specifies the surface as a
level set
of a scalar volumetric function,
φ
:
U
→
IR, where
U
⊂
IR
3
is the range of the
surface model. Thus, a surface
S
is
S
= {
s
|
φ
(
s
)
=
k
}
,
(8.1)
with an isovalue
k
. In other words,
S
is the set of points
s
in IR
3
that composes
the
k
th isosurface of
φ
. The embedding
φ
can be specified as a regular sampling
on a rectilinear grid.
Our overall scheme for segmentation is largely based on the ideas of Osher
et al.
[30] that model propagating surfaces with (time-varying) curvature-
dependent speeds. The surfaces are viewed as a specific level set of a higher
dimensional function
φ
—hence the name level set methods. These methods
provide the mathematical and numerical mechanisms for computing surface
deformations as isovalues of
φ
by solving a partial differential equation on the
3D grid. That is, the level set formulation provides a set of numerical methods
that describes how to manipulate the grayscale values in a volume, so that the
isosurfaces of
φ
move in a prescribed manner (shown in Fig. 8.1). This chapter
does not present a comprehensive review of level set methods, but merely
introduces the basic concepts and demonstrates how they may be applied to
(a)
(b)
Figure 8.1: (a) Level set models represent curves and surfaces implicitly using
grayscale images. For example, an ellipse is represented as the level set of an
image shown here. (b) To change the shape of the ellipse we modify the grayscale
values of the image by solving a PDE.
