Biomedical Engineering Reference
In-Depth Information
dependence of parametric snake models and can be extended to 3D through the
theory of minimal surfaces [11, 57].
Implicit models, such as the formulation used in [46], consist of embedding
the snake as the zero level set of a higher dimensional function and to solve the
corresponding equation of motion. Such methodologies are best suited to the
recovery of objects with unknown topologies.
Parametric deformable models are more intuitive than the implicit and
geodesic ones. Their mathematical formulation makes it easier to integrate im-
age data, initial estimated, desired contour properties and knowledge-based
constraints, in a single extraction process [6].
However, parametric models also have their limitations. First, most of these
methods can only handle objects with simple topology. The topology of the
structures of interest must be known in advance since the mathematical model
cannot deal with topological changes without adding extra machinery [21-47].
Second, parametric snakes are too sensitive to their initial conditions due to the
nonconvexity of the energy functional and the contraction force which arises
from the internal energy term [37, 79]. Several works have been done to address
the mentioned limitations.
Topological restrictions can be addressed through a two-step approach:
firstly, a method of identifying the necessity of a topological operation (split
or merge) and secondly, a procedure of performing it. In [21] we found such
a methodology that can split a closed snake into two closed parts. This is ac-
complished by first constructing a histogram of the image force norm along the
snake to identify the appropriate region to cut it (region with weakest image
field). Next, the method identifies two points in this region to be the end points
of the segment which will cut the curve into two parts. The criterion to do this
is based on the direction of an area force used to make the contour fit concave
parts. This methodology has the disadvantages of not dealing with the contour
merges and its extension to the 3D case is very difficult.
In [65] another approach is presented. It seeds particles on the surface of an
object until their density on the surface is within some threshold value. Its com-
ponents are a dynamical particle system and an efficient triangulation scheme
which connects the particles into a continuous polygonal surface model consis-
tent with the particles configuration. Particles are oriented; that is, each one has
a position and a normal vector associated. The interparticle forces are used to
encourage neighboring oriented particles to lie in each other's tangent planes,
and therefore favor smooth surfaces. This technique has the advantage of dealing
