Biomedical Engineering Reference
In-Depth Information
where
κ
is the curvature of Γ,
n
its inward unit normal, and
<, >
stands
for the scalar product of two vectors.
We can give the following interpretation to each of the terms involved in
the above formula. The term
<
g,n>n
is a vector field defined on the
curve pointing to the region of interest that attracts the snake to the object
boundary. Since its computation essentially relies on image edges, from
a vector flow point of view, it can be considered as a
Static Vector Field
locally defining the target object. The curvature term,
g
∇
κn
, influences
different aspects of the snake evolution. On one hand, it defines its motion
when it is located far away from the object boundaries. Since it depends
on the evolving snake, it acts as a
Dynamic Vector Field
in the convergence
process. On the other hand, it serves as a curve-regularizing term, ensuring
continuity of the final segmenting snake in a similar fashion [16] as the
membrane term of parametric snakes does. Finally, it gives to the process
a smooth behavior and ensures continuity during the deformation, in the
sense that it prevents shock formation [17]. However, incorporating the
curvature term into the convergence scheme has some disadvantages. First,
it is difficult to facilitate snake convergence to concave areas. Second,
guidance through the curvature is extremely slow, so in spite of giving
regularity to the evolution equation, it hinders the numerical scheme since
the time increment is bounded by the second-order term [18].
·
The main problem of (1) is that convergence to the object of interest relies
on the properties of the external field. Even considering a regularization
[19] of the external force, concave regions such that the unit tangent turns
around more than
π
degrees between consecutive inflexion points of the
object contour, cannot be reached [20]. In order to increase convergence to
concavities and to speed up the evolution, a constant balloon force velocity
term,
V
0
, corresponding to area minimization is added:
∂
∂t
=(
g
·
κ
+
V
0
−
<
∇
g,n>
)
·
n.
(2)
Notice that, in order to ensure that the scheme will stop at the boundary
of interest, an equilibrium between the constant shrinking velocity,
V
0
,
and the static vector field,
g
, must be achieved. One easily realizes that,
should this condition be satisfied, incorporating the curvature term into the
convergence scheme constitutes a significant drawback.
V
0
must overpass
the magnitude of
κ
to enter into concave regions but, at the same time,
it should be kept under
min
∇
(minimum taken on the curve to
detect!) to guarantee nontrivial steady states. This dichotomy motivates
bounding the scope of
V
0
to a given image region [7]:
|∇
g,n
|
