Biomedical Engineering Reference
In-Depth Information
physics define the underlying principle of how a geometrical shape can vary over
space and time. An active contour model permits an arbitrary shape to evolve
to a meaningful shape guided by the image properties and constrained by the
physical laws. The physical laws provide the desired intuitive nature to the evolving
shape. In particular, for the snake, the points does not evolve independently but are
constrained by the motion of the two nearest points on either side, thus confining its
degrees of freedom, bringing an elastic model into the structure. Thus, it evolves
from the elastic theory paradigm, generally in a Lagrangian dynamics setting. It
stems from the theory of an elastic string deforming naturally to applied forces
and constraints defined by various sources.
Guided by the physical laws, the model is driven to deform toward a lower-
energy or equilibrium state monotonically. The local image statistics should be
formulated within the deformable paradigm in such a way that the model is guided
to delineate the desired anatomic structure. The optimization theory blends these
two different forms of constraints within the same framework. The local im-
age statistics-based features thus need to be defined within the framework of this
physics-based geometric model, such that the “equilibrium state” is achieved only
when the anatomic structure is delineated.
Definition of this physics-based model that governs the deformation property
of the string is the main essence that makes the deformable model an attractive
proposition to capture the local statistics of the image globally. A deformable
model, and in particular an active contour model, by definition optimally integrates
similar salient features within the geometric model.
The active contour model, or snake, proposed by Kass et al. [10] is an elastic
contour that deforms under the guidance of attributed geometric and image prop-
erties. This phenomenon of deformation, as guided by physical laws, is defined in
terms of an energy minimization framework. By definition, it is minimization of
the total energy over the entire shape, defined by
E
snake
(
τ
)=
E
int
(
τ
)+
E
ext
(
τ
)
(1)
where
E
snake
(
τ
) is the total energy of the contour
, composed of the internal
energy
E
int
(
τ
) and external energy
E
ext
(
τ
). Internal energy is defined by the
physical constraints that describe the degrees of freedom of the contour
τ
τ
(
s
), and
the external energy is defined by the image properties and other user constraints
(e.g., landmark).
As defined previously, the physical constraints of the active contour model
have their origin in the physics of an elastic body, which is described in the first
term of the functional in Eq. (1). The internal energy term can be expressed as
α
(
s
)
2
ds,
1
2
∂
2
τ
(
s
)
∂s
2
∂
τ
(
s
)
∂s
E
int
(
τ
)=
+
β
(
s
)
(2)
0
