Chapter 4 Vortex detection - Computer Graphics and Visualization

2.4. Deformable models
where ÜÖ is the radial component (in the direction Ö) of an external force, with an
associated weight, Ò is an internal force, with an associated weight, and ÑÔ is a
damping force which increases the stability and ensures termination of the iteration.
From these forces, the acceleration for the nodes is easily derived using the relation
Ñ, where is Ñ is a constant mass for all nodes. From the acceleration, the
velocity is derived, and using the velocity, the new position Ô of a node is determined
using a numerical integration:
Ô Ø ¡Ø Ô Ø Ú Ø ¡Ø (2.3)
where ¡Ø is the time interval between subsequent new positions.
There are several important differences between Discrete Dynamic Contours (DDC)
and previous models.
1. The first difference is that in DDCs, minimization of the energy is done for each
individual node, not globally for the entire contour, hence the name discrete.
2. The second difference is that displacement of the nodes is done only in the radial
direction, or the direction normal to the contour, filtering out any tangential displacement
components. This is done for two reasons: tangential displacements
do not contribute to deformation of the model, and tangential displacements
may cause local accumulation of nodes on the contour, which is not desirable.
3. The third important difference is that DDCs perform resampling of the curve
and add new nodes, which can make the final contour much larger than the one
originally defined.
In principle, deformable contours are limited to 2D images, or 2D slices of 3D
data sets. When 3D representations of objects are required, a frequently applied technique
connects contours from several stacked slices by filling the intermediate space
by polygonal tilings.
Our deformable surfaces, as described in Chapter 5 of this thesis, have in common
with the above methods that they are based on a minimization process. They
have in common with the Discrete Dynamic Contours that minimization is done per
node rather than for the entire surface. For the node displacement directions, we have
compared various mechanisms, including a mechanism similar to the above greedy
algorithm, and a mechanism which displaces only in the normal direction.
2.4.2 Deformable surfaces
Deformable surfaces are a logical extension of deformable curves: the 1D curves in 2D
space are extended to 2D surfaces defined in 3D space. Whereas deformable curves
consist of points Ü × on a parametric curve, deformable surfaces are either parametric
surfaces Ü Ù Ú , or unstructured triangle meshes, which are less straightforward to
parameterize.
One example of deformable surfaces consisting of unstructured triangular meshes
are the Geometrically Deformed Models (GDMs) described in [Miller, 1990; Miller
et al., 1991]. A GDM utilizes a cost function somewhat similar to the energy of active
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