Biomedical Engineering Reference
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the common face. The process of obtaining v from v is called pivoting . Let us
present the basic continuation algorithm [1].
PL generation algorithm:
Find a transverse triangle σ 0 ;
={ σ 0 } ;V ( σ 0 ) = set of vertices of σ 0 ;
while V ( σ ) =∅ for some σ
. get σ such that V ( σ ) =∅ ;
. get v V ( σ ) ;
. obtain σ from σ by pivoting v into v
. f σ is not transverse
.
then drop v from V ( σ ) ;
.
else
if σ then
.
.
drop v from V ( σ ) , v from V ( σ )
.
else
⇐= + σ ;
.
V ( σ ) ⇐= set of vertices of σ ;
.
.
drop v from V ( σ ) , v from V ( σ )
Differently from tetra-cubes, once the generation of a component is started,
the algorithm runs until it is completed. However, the algorithm needs a set of
seed simplices to be able to generate all the components of an isosurface. This
is an important point when comparing continuation and marching methods.
If we do not have guesses about seeds, every simplex should be visited. Thus,
the computational complexity of both methods is the same ( O ( N ) where N is
the number of simplices).
However, if we know in advance that the target boundary is connected,
we do not need to search inside a connected component. Consequently, the
computational cost is reduced if continuation methods are applied.
Based on this discussion about marching cubes and PL generation, we can
conclude that, if we do not have the topological and scale restrictions given in
Section 7.4, tetra-cubes is more appropriate to initialize the T-surfaces. In this
case, it is not worthwhile to attempt to reconstruct the surface into neighboring
simplices because all simplices should be visited to find surface patches.
However, for the T-surfaces reparameterization (steps (1)-(4) in
Section 7.2.3), the situation is different. Now, each connected component is
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