Biomedical Engineering Reference
In-Depth Information
of interest. On the other hand, the internal constraints control the smoothness and
continuity of the model.
It is well known that there are two main branches of deformable models ac-
cording to parametrization of the model. The first, the parametric one, is based on
the classical Newtonian mechanics equations that govern the elasticity and stretch-
ing of the deformable model. In this branch, the curve defining the deformable
model is explicitly parameterized. This fact means that the resulting model is usu-
ally restricted to a single object. In this sense, some parametric models [9] try to
solve this drawback by re-parameterizing the model at each step of the evolution.
The other branch relies on the theory of geodesic curves and level sets [2, 10].
In this formulation the snake is defined using an implicit parametrization. The
deformation process is defined by a changing Riemannian surface that minimizes
the length of the level set curve under the constraints of the image features. The
main advantage of this formulation is that it can naturally deal with topological
changes during snake evolution.
In this latter branch, the equation that governs the evolution of the deformable
model is divided into two terms. The first is the normal component of the gradient
of a potential defined by the image features. The role of this term is to rule
the convergence to contours. The second depends on the snake curvature, and
endows the snake with a means to deform at null gradient regions while ensuring
regularity. However, the role of the curvature has a major impact in the numeric
scheme: on one hand, it restricts the maximum speed of the evolution and, on the
other, interferes with convergence in concave areas. The usual way to overcome
this latter issue is to add a constant motion term, the balloon force [3], that pushes
the snake into concave regions.
In general, deformablemodels were originally designed to be used in a contour
space — an image defining the contours of the regions of interest. However,
contours are not always available, especially in textured or complex images. Due
to this problem, region-based schemes were introduced. They aim at finding a
partition of the image such that the descriptors of each of the regions conform
to a given “homogeneity” criterion. In this sense, the force guiding the snake
is derived from the competition of the descriptors. Several authors address this
issue: Ronfard [11] set the velocity function proportional to the difference of
simple statistical features. Zhu [12] and Paragios and Deriche [5] defined the
region evolution as a quotient of probabilities corresponding to different regions.
In Yezzi et al. [13] the difference of mean gray levels inside and outside the
evolving front at each iteration defines the motion of the deformable model. Along
the same lines, Besson et al. [7] propose a difference of simple statistics, variance,
and covariance matrix, inside and outside the evolving curve. Chakraborty et al.
[14] consider an evolution using a Fourier parametrization over the original image
and previously classified image regions. Probably the most notable technique for
complex images is the one proposed by Paragios and Deriche [5] and Samson
et al. [15] based on supervised learning of the features of the regions of interest.
