Biomedical Engineering Reference
In-Depth Information
to be tuned in order to guarantee the success of the process. Examples of these
parameters include, but are not limited to, the iteration step, speed terms, time
step, etc.
In [7] a more efficient 3D segmentation technique was proposed. In that work,
where almost no parameter setting is required, surface evolution is controlled by
current probabilistic region information. Probability density functions for the
object and the background are estimated using SEM. The resulted level set model
is based on these density functions. However, the proposed model works only for
bimodal images, and this may be too restrictive for many applications.
In [9] we proposed a novel and robust level set-based segmentation technique.
A statistical model of regions is explicitly embedded into partial differential equa-
tions describing the evolution of the level sets. SEM is used to estimate the initial
values of class parameters. The probability density function for each region is
modeled by a Gaussian with adaptive parameters. These parameters and the prior
probability of each region are automatically re-estimated at each iteration of the
process. The designed level set model depends on these density functions. The
region information over the image is also taken into account. Our proposed model
differs from that in [7] due to its suitability for multimodal images and due to
adaptive estimation of the probability density functions. Our experiments in 3D
segmentation of MR images demonstrate the accuracy of the algorithm.
4.2.4. Curve/surface modeling by level sets
Within the level set formalism [37], the evolving surface is a propagating front
embedded as the zero level of a higher-dimensional scalar function
φ
(
x, t
). This
hypercurve/surface is usually defined as a signed distance function: positive inside
the region enclosed by the evolving interface, negative outside, and zero on the
boundary of the enclosed region. As we have seen earlier, the continuous change
of
φ
can be described by the following PDE:
∂φ
(
x, t
)
∂t
+
F
∇
φ
(
x, t
)
=0
,
(7)
where
F
is a scalar velocity function depending on the local geometric properties
(e.g., local curvature) of the front and on the external parameters related to the
input data, e.g., image gradient. The function
φ
deforms iteratively according to
F
, and the position of the 2D/3D front is given at each iteration step by the equation
φ
(
x, t
)=0.
The design of the velocity function
F
plays a major role in the evolutionary
process. In our formulation, we have chosen the formulation given by (5), that is,
F
=
ν
−
κ
, where the local curvature
κ
of the front is defined in the 3D case as
