Biomedical Engineering Reference
In-Depth Information
where
d
is the distance from
x
to the surface
S
(
x
,
t
=
0) and the signal indicates
if the point is interior (
−
) or exterior (
+
) to the initial front.
Finite difference schemes, based on a uniform grid, can be used to solve
Eq. (7.19). The same entropy condition of T-surfaces (
once a grid node is burnt
it stays burnt
) is incorporated in order to drive the model to the desired solution
(in fact, T-surfaces was inspired on the level sets model [50]).
In this higher dimensional formulation, topological changes can be efficiently
implemented. Numerical schemes are stable, and the model is general in the
sense that the same formulation holds for 2D and 3D, as well as for merge and
splits. Besides, the surface geometry is easily computed. For example, the front
normal and curvature are given by:
∇
G
(
x
,
t
)
∇
G
(
x
,
t
)
n
=∇
G
(
x
,
t
)
,
(7.21)
K
=∇·
,
respectively, where the gradient and the divergent (
∇·
) are computed with re-
spect to
x
.
7.3 Initializing Traditional Deformable
Models
In the area of deformable models, the definition of the initial estimation (see
Eq. (7.9)) from which we can start the model evolution (the initialization step)
is a difficult and important task. Problems associated with fitting the model to
data could be reduced if a better start point for the search were available. In this
section, we show a set of methods used to find the initial curve (or surface).
We start with methods that use image statistics and morphological tech-
niques, and later we present modern approaches, such as neural nets.
7.3.1 Region-Based Approaches
The simplest way to initializing deformable models is through a preprocessing
step in which the structures of interest are enhanced.
This can be done by image statistics extracted by image histograms or pattern
recognition techniques [69] (see [39] for a recent review). These statistics can
be represented by a mean
µ
and variance
σ
of the image field
I
or any other field
