Biomedical Engineering Reference
In-Depth Information
Figure 2.1:
Geometric representations of deformable surfaces.
functional for active contour models is expressed as:
1
1
1
C
(
s
)
C
(
s
)
ds
−
λ
∇
I
(
C
(
s
))
2
2
E
(
C
)
=
α
ds
+
β
ds
(2.1)
0
0
0
where (
α,β,λ
) are positive parameters. The first two terms control the rigidity
and elasticity of the contour (defining the internal energy of the deformable
object) while the last term attracts the model to high-gradient locations in the
image
I
(defining the external energy of the model).
Active contour segmentation via minimization of the energy functional in
Eq. (2.1) is typically implemented with a parametric framework in which the de-
formable model is explicitly formulated as a parameterized contour on a regular
spatial grid, tracking its point positions in a Lagrangian framework [11].
In their original paper from 1988 [7], Osher and Sethian introduced the con-
cept of geometric deformable models, which provide an implicit formulation
of the deformable contour in a level set framework. To introduce the concept
of the level set framework we focus on the boundary-value problem of a close
