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followed when pruning the skeleton, and the number
of iterations necessary to get a satisfactory result. In
this paper, we are interested in discussing both the
above aspects. Do parallel and sequential modalities
produce the same results? If not, is one of those
modalities preferable? How could we establish the
number of iterations of pruning that sufficiently well
simplify the structure of S while still producing a
satisfactory shape representation?
Obviously, the performance of pruning strongly
depends on the goodness of the pruning criterion.
However, independently of the selected criterion, we
think that the parallel and sequential modalities are
likely to produce different results. Since we are not
interested in judging the goodness of pruning
criteria, but only in showing that the selected
modality has an impact on the results, we may use a
very simple pruning criterion. To this aim, we note
that among the points of S, a crucial role is played
by the centers of maximal balls, CMBs, (Pfaltz and
Rosenfeld, 1967). In fact, the union of the balls
associated to the CMBs of S recovers the object.
Then, we use a simple pruning criterion based on the
ratio R between the number of CMBs in a peripheral
branch and the total number of points in the branch.
The rationale is that the larger is the percentage of
CMBs in a branch, the higher is the representative
power of that skeleton branch. The proper value of
the threshold for R should be fixed depending on
the problem at hand. In this paper, we set =0.4.
Since we are also interested in finding a way to
determine the proper number of iterations for
pruning, we consider pruning that either removes a
whole peripheral branch or keeps it in the skeleton.
In fact, in order pruning can be iterated, necessarily
some initial branch points have to be transformed
into new end points and this is not guaranteed when
partial skeleton branch removal is considered.
In our opinion, pruning in sequential modality is
likely to be more conservative as far as preserving
shape information is concerned. Its main drawback
is that the result is conditioned by the order in which
branches are examined. By changing the branch
inspection order, the delimiting branch point for the
currently traced peripheral branch may be more or
less internal in S. The order also conditions the
number of possible further iterations.
As for the parallel modality, the result is
obviously independent of the order in which
peripheral branches are examined. The main
problem occurs when all peripheral branches
meeting in common branch points are pruned and
pruning is iterated. In fact, some of the end points in
the pruned skeleton were branch points in the initial
skeleton, but the pruning criterion is checked only
for the branches that are peripheral at the current
iteration. Thus, the relevance of an object region
mapped into a subset of the initial skeleton, whose
branches are pruned at different iterations, is not
correctly evaluated. As a consequence, successive
iterations may cause an additive negative impact on
the representative power of the skeleton.
For illustrative purposes, let us refer to a 2D case
and consider the object in Figure 1 left, where the
skeleton is shown superimposed on the object. The
result after one iteration of pruning done according
to the parallel modality is shown in Figure 1 middle
left. The results obtained at the end of the first
iteration when following the sequential modality,
and by selecting a different order for tracing the
peripheral branches are shown in Figure 1 middle
right and Figure 1 right, respectively. We observe
that the obtained results are different
notwithstanding the same pruning criterion based on
the ratio R and the same value for the threshold
have been adopted.
Figure 1: From left to right, the initial skeleton and the
pruned skeletons obtained with different modalities.
4 HYBRID APPROACH
We think that a possible solution to the drawbacks of
sequential and parallel pruning modalities can be
obtained by following an hybrid approach that mixes
the sequential and parallel modalities in such a way
to take the benefits of both. We suggest that the
branch point status is not updated during the current
iteration. We also suggest that if all peripheral
branches meeting in a common branch point satisfy
the pruning criterion, the peripheral branch
characterized by the highest relevance is not
removed. By postponing branch point status
updating, we exploit the good feature of parallel
pruning that the result is not influenced by the order
in which branches are examined. By keeping in S
the most relevant branch, we exploit the positive
performance of sequential pruning. In fact, at the
end of each iteration some branches always exist
that originate from end points present in the initial
skeleton, so that the negative additive impact on the
OntheStrategytoFollowforSkeletonPruning-Sequential,ParallelandHybridApproaches
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