On the Strategy to Follow for Skeleton Pruning

Sequential, Parallel and Hybrid Approaches

Maria Frucci, Gabriella Sanniti di Baja, Carlo Arcelli and Luigi P. Cordella

Institute of Cybernetics “E.Caianiello”, CNR, Via Campi Flegrei 34, Pozzuoli (Naples), Italy

Keywords: Shape Representation, Skeleton, Pruning.

Abstract: Pruning is an important step in a skeletonization process and a number of pruning criteria have been

suggested in the literature. However, the modality to be followed when checking the pruning criterion is not

generally described in detail. In our opinion, two main pruning modalities can be envisaged and in this

paper we discuss their impact on the performance of pruning. Moreover, we introduce a third modality,

which we regard as able to provide a more satisfactory pruning performance.

1 INTRODUCTION

The skeleton of an object is one of the most popular

medial representations and is useful for shape

analysis. A number of papers exist in the literature

proposing skeletonization methodologies or dealing

with the use of the skeleton for practical

applications, e.g., see (Siddiqi and Pizer, 2008).

However, a factor limiting the use of the skeleton for

applications is its sensitivity to deformations along

the boundary of the object. In fact, even negligible

noise along the boundary may cause spurious

branches in the skeleton. Thus, proper techniques

able to remove scarcely significant skeleton

branches, or to prevent their creation, are of interest.

In general, skeleton branches are expected in

correspondence with regions of the object that are

perceived as individually meaningful. In particular,

peripheral branches are expected in correspondence

with limbs and smooth boundary convexities.

However, the structure of the skeleton may be very

complex, especially if continuous skeletonization

methodologies, such as those based on the Voronoi

diagram (Ogniewicz and Kubler, 1995), are used.

Thus, a one-to-one correspondence between skeleton

branches and object regions may not be satisfied.

Pruning is aimed at removing scarcely significant

peripheral branches so that the previous

correspondence can be established.

The most commonly employed criteria to

evaluate the significance of skeleton branches and

accordingly perform pruning were extensively

discussed in (Shaked and Bruckstein, 1998) and deal

with propagation velocity, maximal thickness, radius

function, axis arc length, and the boundary/axis

length ratio. More recently, new pruning

methodologies have been suggested in (Bai et al.,

2007) and in (Shen et al., 2011) dealing with contour

partitioning via discrete curve evolution and with

bending potential ratio, where pruning can be

accomplished during a post-processing phase, or can

be integrated into the skeleton computation process.

Generally, any significance criterion aims at

establishing a strict relation between a skeleton

branch and the relevance of the object part the

branch represents. Branch removal caused by

pruning modifies the skeleton in such a way that the

object it represents turns out to differ from the

original object for a smoother boundary or for the

number of protrusions. A pruning process is

adequate if these differences are negligible in the

framework of the specific application.

An aspect that has not received enough attention

in the literature is the modality that is followed when

pruning is seen as a post-processing phase, after the

skeleton has been computed. Another aspect that is

not taken into account is that, due to branch removal,

branches that were internal in the original skeleton

are likely to be transformed into peripheral branches.

Since the new peripheral branches may be not

significant, pruning may need to be iterated. Thus,

the problem of establishing how many times pruning

can be iterated has to be faced to avoid that the

structure of the skeleton be excessively simplified.

In this paper we discuss the performance of

pruning methods by taking into account both the

263

Frucci M., Sanniti di Baja G., Arcelli C. and P. Cordella L. (2013).

On the Strategy to Follow for Skeleton Pruning - Sequential, Parallel and Hybrid Approaches.

In Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods, pages 263-266

DOI: 10.5220/0004193302630266

 SciTePress

above aspects. We do not introduce new criteria to

evaluate the significance of skeleton branches, but

investigate how the application of a given pruning

criterion may condition the obtained results.

Reference to a specific criterion will be done only to

exemplify the different modalities of pruning.

2 PRELIMINARY NOTIONS

The skeleton S of a digital object P is a subset of P

consisting of the union of curves symmetrically

placed within P and characterized by the same

topology of P. Each point of S (a pixel in 2D and a

voxel in 3D) is labeled with the value of its distance

from the complement of P. A point p of S is an end

point if it has only one neighbor in S, a normal point

if it has two neighbors in S, and a branch point if it

has more than two neighbors in S.

A skeleton branch is a curve of S entirely

consisting of normal points, except for the two

extremes of the curve that are end points or branch

points. The only case in which a skeleton branch can

be delimited by two end points is when the skeleton

consists of a single branch. Such a simple case of

skeleton structure is not of interest in the framework

of pruning. A skeleton branch delimited by one end

point and one branch point is a peripheral branch. A

skeleton branch delimited by two branch points is an

internal branch.

In an ideal skeleton, branches should correspond

to perceptually meaningful object regions, while the

skeleton S of P includes a generally much larger

number of branches, some of which originating from

noisy convexities along the boundary of P.

Therefore, pruning can be done to remove scarcely

significant branches, so favoring the similarity

between S and the ideal skeleton.

Pruning consists in removing, partially or totally,

skeleton branches from S. It should be accomplished

exclusively on peripheral branches to guarantee that

the pruned skeleton is characterized by the same

topology as the original skeleton and, hence, the

same topology as P. Due to the linear structure of S,

pruning can be efficiently implemented by resorting

to skeleton tracing, starting from the end points.

3 PRUNING APPROACHES

To decide whether the peripheral branch currently

traced should be pruned, suitable criteria are

necessary to evaluate the perceptual relevance of the

object region mapped into that skeleton branch. The

pruning criterion can be checked on the whole

peripheral branch, so as to establish whether the

entire branch should be removed or should be kept

in S. Alternatively, while tracing the branch pruning

can remove skeleton points one after the other as far

as the pruning criterion is satisfied, leading to a

possibly partial skeleton branch removal.

Let N be the number of end points (hence of

peripheral branches) of S. During the same iteration

of pruning, all peripheral branches are analyzed.

Total removal of a peripheral branch may cause a

point of S, classified as branch point in the initial

skeleton, to be transformed into a normal point.

Thus, two different pruning modalities (here called

parallel and sequential modalities) can be

considered. If the parallel modality is followed,

points classified as branch points in the initial

skeleton maintain their status until all peripheral

branches have been examined and possibly pruned.

If the sequential modality is followed, the branch

point status is updated as soon as a branch point is

reached by pruning. Thus, during an iteration of

pruning according to the parallel modality, all

peripheral branches can be removed at most up to

the branch points delimiting them in the initial S. In

the sequential modality, a peripheral branch B

delimited by the branch point b

in the initial S, may

be pruned until a branch point b

, more internal than

in the initial S. This happens if the status of b

has

been transformed from the status of branch point to

that of normal point before analyzing the branch B,

due to removal of other already examined peripheral

branches meeting into b

Once the N peripheral branches have been

examined and those satisfying the pruning criterion

have been removed, the skeleton structure results to

be modified. Let M be the number of end points

characterizing the pruned skeleton. Obviously, it is

MN. Some of the M peripheral branches can

originate from end points already existing in the

initial skeleton. The delimiting branch points of

these peripheral branches may differ from those

delimiting the corresponding branches in the initial

skeleton. Some of the M peripheral branches can

originate from new end points that in the initial

skeleton were classified as branch points. A second

iteration of pruning can then be accomplished by

considering the M peripheral branches. Pruning can

be iterated producing at each iteration a skeleton

with a structure simpler than that of the skeleton

obtained at the previous iteration.

To our knowledge, no discussion has been done

in the literature regarding both the modality

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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

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representative power of the skeleton is prevented. As

an example, see Figure 2, where the initial skeleton

and the results of pruning after one iteration, when

following the parallel approach, the sequential

approach (with two different inspection orders) and

the hybrid approach are shown from left to right.

The pruning criterion involving the ratio R and the

value =0.4 for the threshold have been used.

Figure 2: From left to right, the initial skeleton and the

pruned skeletons obtained by following parallel,

sequential and hybrid approaches.

Once the current iteration is completed, end

points and branch points in the pruned skeleton are

identified and a new iteration can be accomplished.

Actually, pruning iterations are accomplished until

the pruning criterion is not satisfied by any

peripheral branch, so leading to a pruned skeleton

that, in the limits of the adopted threshold, has

simple structure and adequate representative power.

Figure 3: Initial skeletons (odd lines) and skeletons pruned

by following the hybrid approach (even lines).

We have checked the above pruning criterion on

a number of digital objects by following the parallel,

sequential and hybrid modalities. Sometimes the

results obtained by the hybrid approach were equal

to those obtained by the sequential method in a

given branch inspection order; sometimes the hybrid

and the parallel approaches provided the same

results. Sometimes the results provided by the three

approaches were all different. Though the

differences in the results are not so large to clearly

show the supremacy of one of the three modalities,

we think that the hybrid approach should be

preferred since it is less affected by drawbacks.

A few examples of the performance of pruning

involving the ratio R with =0.4, and accomplished

by the hybrid approach are shown in Figure 3.

5 CONCLUSIONS

In this paper we have discussed the performance of

skeleton pruning accomplished by following parallel

and sequential modalities. We have also introduced

a hybrid modality that, in our opinion, overcomes

the drawbacks of the parallel and sequential

approaches.

We have used a simple pruning criterion since

we are interested in showing that the selected

modality has an impact on the results. Our future

work will deal with checking whether the hybrid

approach can still be seen as preferable with respect

to the sequential and parallel approaches even if

using more sophisticated pruning criteria.

REFERENCES

Siddiqi, K., Pizer, S. M., (Eds.), 2008. Medial

Representations: Mathematics, Algorithms and

Applications. Springer, Berlin.

Ogniewicz, R. L., Kubler, O., 1995. Hierarchic Voronoi

Skeletons. Pattern Recognition 28(3), 343-359.

Shaked, D., Bruckstein, A. M., 1998. Pruning medial axes.

Computer Vision and Image Understanding 69(2),

156-169.

Bai, X., Latecki, L. J., Liu, W-Y., 2007. Skeleton pruning

by contour partitioning with discrete curve evolution.

IEEE Trans. PAMI 29(3), 449-462.

Shen, W., Bai, X., Hu, R., Wang, H., Latecki, L.J., 2011.

Skeleton growing and pruning with bending potential

ratio. Pattern Recognition 44, 196-209.

Pfaltz, J. L., Rosenfeld, A., 1967. Computer representation

of planar regions by their skeletons. Comm. ACM

10(2), 119-125.

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