Biomedical Engineering Reference
In-Depth Information
inability to respond to topological variations. The topology of the structure of
interest must be known a priori since traditional snakes models being parametric
representations are incapable of topological transformations without additional
intervention. Samadani [58] used a heuristic technique based on deformation en-
ergies to split and merge active contours. More recently, Malladi et al. [54] and
Caselles et al. [51] independently developed a topology-independent active con-
tour scheme based on themodeling of propagating fronts with curvature-dependent
speeds, where the propagating front is viewed as an evolving level set of some im-
plicitly defined function.
5.
TOPOLOGICAL SNAKE
Topological adaptivity requires the multiple instances of the model to be dy-
namically created or destroyed, or can seamlessly split or merge as the object to
be segmented changes its topology. Much research has been dedicated to this area
[59-61]. The main principle behind each of these approaches involves using the
grid information to establish a relation between the parametric curve and the pixel
domain. Conversion to and from the traditional snakes model formulation requires
the ability to discard or impose the grid within the framework at any time. The
grid needs to provide a simple and effective means to extend the geometric and
topological adaptability of snakes. McInerney and Terzopoulos [59] developed a
parametric snakes model that has the power of an implicit formulation by using
a superposed simplicial grid to reparameterize the model during the deformation
process. Of all the approaches, we will detail this approach since it is the one most
widely used. As previously mentioned, the idea is to incorporate the traditional
snakes model within the framework of simplicial domain decomposition.
The grid of discrete cells used to approximate the snake model is an example
of space partitioning by simplicial decomposition. There are two main types
of domain decomposition methods: non-simplicial and simplicial. Most non-
simplicial methods employ a regular tessellation of space. The marching cubes
algorithm is an example of this type of method. These methods are fast and
easy to implement, but they cannot be used to represent surfaces or contours
unambiguously without the use of a disambiguation scheme.
Simplicial methods, on the other hand, are theoretically sound because they
rely on classical results from algebraic topology. In simplicial decomposition,
space is partitioned into cells defined by open simplices, where an
n
-simplex is
the simplest geometrical object of dimension
n
. A simplicial cell decomposition
is also called a
triangulation
.
The simplest triangulation of Euclidean space
R
n
is a Coxeter-Freudenthal
triangulation (Figure 22a). It is constructed by subdividing space using a uniform
cubic grid, and the triangulation is obtained by subdividing each cube into
n
!
simplices. Simplicial decompositions provide an unambiguous framework for
