Biomedical Engineering Reference
In-Depth Information
polygonal surfaces whose interior may contain the desired objects. If a polygonal surface
involves more than one object, the resolution is increased in that specific region, and the
marching procedure is applied again. Next, we apply T-Surfaces to improve the result. If
the obtained topology remains incorrect, we enable the user to modify the topology by an
interactive method based on the T-Surfaces framework. Finally, we discuss the utility of
diffusion methods and implicit deformable models for our approach.
1.
INTRODUCTION
Deformable Models, which include the popular
snake models
[1] and de-
formable surfaces [2, 3], are well-known techniques for boundary extraction and
tracking in 2D/3D images. Basically, these models can be classified into three
categories: parametric, geodesic snakes, and implicit models. The relationships
between these categories have been demonstrated in several works [4, 5].
Parametric Deformable Models consist of a curve (or surface) that can dy-
namically conform to object shapes in response to internal (elastic) and external
(image and constraint) forces [6]. In geodesic snakes formulations, the key idea is
to construct the evolution of a contour as a geodesic computation. A special metric
is proposed (based on the gradient of the image field) to let the state of minimal
energy correspond to the desired boundary. This approach allows addressing the
parameterization dependence of parametric snake models and can be extended to
three dimensions through the theory of minimal surfaces [7, 5]. Implicit models,
such as the formulation used in [8], consist of embedding the snake as the zero
level set of a higher-dimensional function and to solve the corresponding equation
of motion. Such methodologies are best suited for the recovery of objects with
unknown topologies.
When considering the three mentioned categories, two aspects are funda-
mental within the context of the present work: user interaction and topological
changes. Parametric models are more suitable for user interaction than the others
because they use neither the higher-dimensional formulations of Level Sets nor
globally defined features, like the metric in the geodesic approach. However, for
most parametric methods the topology of the structures of interest must be known
in advance since the mathematical model cannot deal with topological changes
without adding extra machinery [9, 10].
Recently, McInerney and Terzopoulos [11, 9, 10] proposed the T-Snakes/
T-Surfaces model to add topological capabilities (
splits and merges
) to a para-
metric model. The resulting method has the power of an implicit approach without
requiring a higher-dimensional formulation.
The basic idea is to embed a discrete deformable model within the framework
of a simplicial domain decomposition (
triangulation
) of the image domain. In this
framework, the reparameterization is based on the projection of the curve/surface
over the triangulation and on a
Characteristic Function
, which distinguishes the
interior grid nodes of the (closed) curve/surface from the exterior ones. The set
