Biomedical Engineering Reference
In-Depth Information
shape modeling, and visual tracking. The first kind of active contour models
are represented explicitly as parameterized contours (i.e., curves or surfaces) in
a Lagrangian framework, which are called parametric active contours. Due to
the special properties of the level set method, active contours can be represented
implicitly as level sets function that evolve according to an Eulerian formulation,
which are called geometric active contours. Although it involves only a simple
difference in the representation for an active contour, the level set method-based
geometric active contours have many advantages over parametric active contours.
1. They are completely intrinsic and therefore are independent of the para-
meterization of the evolving contour. In fact, the model is generally not
parameterized until evolution of the level set function is complete. Thus,
there is no need to add or remove nodes from an initial parameterization
or adjust the spacing of the nodes as in parametric models.
2. The intrinsic geometric properties of the contour such as the unit normal
vector and the curvature can be easily computed from the level set function,
in contrast to the parametric case, where inaccuracies in the calculations
of normals and curvature result from the discrete nature of the contour
parameterization.
3. The propagating contour can automatically change topology in geometric
models (e.g., merge or split) without requiring an elaborate mechanism to
handle such changes, as in parametric models.
4. The resulting contours do not contain self-intersections, which are com-
putationally costly to prevent in parametric deformable models.
Since its introduction, the concept of active contour for image segmentation
defined in a level set framework has motivated the development of several families
of methods, which include: the front-evolving geometric model, geodesic active
contours, and region-based level set active contours.
4.1. Front-Evolving Geometric Models of Active Contours
Front-evolving geometric models of active contours (Caselles et al. [48] and
Malladi et al. [49]) are based on the theory of curve evolution, implemented via
level set algorithms. They can automatically handle changes in topology. Hence,
without resorting to dedicated contour tracking, unknown numbers of multiple ob-
jects can be detected simultaneously. Evolving the curve
C
in the normal direction
with speed
F
amounts to solving the following differential equation:
∂
∂t
=
|∇
Φ
|
u
0
|
)(
div
(
∇
Φ
|∇
Φ
|
g
(
|∇
)+
γ
)
,
(8)
