Biomedical Engineering Reference
In-Depth Information
associated to the functional is
(
κ
g
−∇
g
,
n
)
n
=
0
(5.8)
where
κ
is the mean curvature of the evolving surface and
n
is its outer uni-
tary normal vector. So, the evolution of the surface is driven by the associated
gradient descent equation
S
t
=
(
κ
g
−∇
g
,
n
)
n
(5.9)
Following this equation, the surface evolves toward the minimum of the
functional, which is achieved at the edges of the image. The resulting steady-
state surface is a model of the object of interest. The level set method [29] is used
to track its motion, allowing topological changes in the surface and avoiding
numerical instabilities. Basically, the level set method consists in embedding
the evolving surface in a manifold one dimension higher than
S
, and implicitly
represented by a function
φ
. The surface
S
can be reconstructed as the level set
zero of
φ
. If the manifold evolves following the equation
∇
φ
|∇
φ
|
φ
t
=
g
·
di
v
|∇
φ
|−∇
g
,
∇
φ
(5.10)
then the evolution of the zero level set of
φ
is equivalent to the evolution of
S
driven by Eq. (5.9).
5.2.4.2
Introduction of Region-Based Information
in the GAC Model
The GAR model [19] combines the classical GAC model [27] with region-based
statistical information incorporated into the classical energy functional. There-
fore, in places where the gradient is weak, regional information drives the evo-
lution of the surface thus being more robust than GAC.
A surface
S
provides a partition of the space in three regions: inside (
in
),
outside (
out
), and the boundary itself (
S
). Considering the surface evolving in
the domain of a three-dimensional image, the surface provides the following
partition at each time:
(
t
)
=
in
(
t
)
∪
out
(
t
)
∪
S
(
t
)
.
(5.11)
