Biomedical Engineering Reference
In-Depth Information
easily with open and closed surfaces. The topology of the particle-based surface
can be modified during the triangulation step. However, this has the disadvan-
tages of being expensive (
O
(
N
) log
N
) where
N
is the number of particles) and
that it may be difficult or cumbersome to find good initial seed particle sites,
especially automatically [50].
A more general approach to incorporate topological changes in the paramet-
ric snake models is the T-snakes model [47-50]. The method embeds the snake
model within a framework defined by a simplicial domain decomposition, using
classical results in the field of numerical continuation methods [1]. The resulting
model has the power of an implicit one without the need for a higher dimen-
sional formulation [46]. Besides, it can be efficiently extended to 3D, generating
the T-surfaces model [49].
The sensitivity to the initialization is a very common problem for deformable
models. The use of simulated annealing for minimization was proposed in [62].
Despite the global optimization properties, the use of this technique is limited
to both its computational complexity and memory requirements.
Levine
et al.
[44] applied hierarchical filtering methods, as well as a contin-
uation method based on a discrete scale-space representation. At first, a scale-
space scheme is used at a coarse scale to get closer to the global energy mini-
mum represented by the desired contour. In further steps, the optimal valley or
contour is sought at increasingly finer scales.
These methods address the nonconvexity problem but not the adverse effects
of the internal normal force. This force is a contraction force which makes the
curve collapse into a point if the external field is not strong enough. In Cohen [18]
and Gang
et al.
[79] this problem is addressed by the addition of another internal
force term to reduce the adverse effects of the contraction force. In both works
the number of parameters is increased if compared with the original model and
there are some trade-offs between efficiency and performance.
Another way to remove the undesired contraction force of the original snake
model is to use the concept of invariance, which is well known in the field of
computer vision [26, 36]. This concept has been applied to closed contours,
and consists in designing an internal smoothing energy, biased toward some
prior shape, which has the property of being invariant to scale, rotation, and
translation. In these models, the snake has no tendency to expand or contract,
but it tends to acquire a natural shape.
An example of a technique, which applies invariance concepts, is the dual
active contour (dual ACM) [37]. This approach basically consists of one contour
