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A SIMPLE AND EFFICIENT APPROACH FOR 3D MESH
APPROXIMATE CONVEX DECOMPOSITION
Khaled Mamou and Faouzi Ghorbel
Groupe de recherche GRIFT, Laboratoire CRISTAL, Ecole Nationale des Sciences de l’Informatique
University of Manouba, 2010 Manouba-Tunisia
khaled mamou@yahoo.fr, faouzi.ghorbel@ensi.rnu.tn
ABSTRACT
This paper presents an original approach for 3D mesh approx-
imate convex decomposition. The proposed algorithm com-
putes a hierarchical segmentation of the mesh triangles by ap-
plying a set of topological decimation operations to its dual
graph. The decimation strategy is guided by a cost function
describing the concavity and the shape of the detected clus-
ters. The generated segmentation is finally exploited to con-
struct a faithful approximation of the original mesh by a set
of convex surfaces. This new representation is particularly
adapted for collision detection. The experimental evaluation
we conducted shows that the proposed technique efficiently
decomposes a concave 3D mesh into a small set (with respect
to the number of its facets) of nearly convex surfaces. Fur-
thermore, it automatically detects the anatomical structure of
the analyzed 3D models, which makes it an ideal candidate
for skeleton extraction and patterns recognition applications.
Index Terms—Approximate convex decomposition, 3D
mesh, collision detection, hierarchical segmentation
1. INTRODUCTION
Collision detection is essential for realistic physical interac-
tions in video games, computer animation and physically-
based modeling. In order to ensure real-time interactivity with
the player/user, video game and 3D modeling software de-
velopers usually approximate the 3D models composing the
scene (e.g. animated characters, static objects...) by a set of
simple convex shapes such as ellipsoids, capsules or convex-
hulls. In practice, these simple shapes provide poor approxi-
mations for concave surfaces and generate false collision de-
tections.
In this article, we present a simple and efficient approach
to decompose a 3D mesh into a set of nearly convex surfaces.
This decomposition is directly exploited to compute a faithful
approximation of the original 3D mesh, particularly adapted
to collision detection.
The remainder of this paper is structured as follows.
Section 2 introduces the approximate convex decomposition
problem. The proposed segmentation technique is described
in Section 3 and its performances are evaluated in Section 4.
Finally, Section 5 concludes the paper and suggests possibili-
ties for future work.
2. APPROXIMATE CONVEX DECOMPOSITION
Let Sbe a 3D mesh defined in 3,ξ={v1,v
2, ..., vV}the
set of its vertices (Vrepresents the number of vertices) and
θ={t1,t
2, ..., tT}the set of its triangles (Trepresents the
number of triangles).
Computing an exact convex decomposition of Sconsists
in partitioning it into a minimal set of convex sub-surfaces.
In [1], the authors prove that computing such decomposition
is an NP-hard problem and propose different heuristics to re-
solve it. In [2, 3], Lien et al. point out the fact that the pro-
posed algorithms are non-practical since they produce a high
number of clusters. In order to provide a tractable solution,
they propose to relax the exact convexity constraint and con-
sider instead the problem of computing an approximate con-
vex decomposition of S. Here, for a fixed parameter ,the
goal is to determin a partition Π={π1,π
2, ..., πK}of θwith
a minimal number of clusters Kand verifying that each clus-
ter has concavity lower than .
To the best of our knowledge, the approximate convex de-
composition problem was only addressed by [2, 3]. This later
technique exploits a divide-and-conquer strategy, which con-
sists in iteratively dividing the mesh until the concavity of
each sub-part is lower than the threshold . Here, at each step
i,thevertexv∗
iwith the highest concavity is selected and the
cluster to which it belongs is divided into two sub-clusters by
considering a bisection plane incident to v∗
i. The main limita-
tion of this appraoch is related to the choice of the ”best” cut
plane, which requires a sophisticated analysis of the model
features [2, 3]. Moreover, considering only plane-based bi-
sections is in practice too restrictive and may lead to poor
decompositions (cf. Figure 3).
In order to overcome such limitations, this paper intro-
duces a novel hierarchical segmentation approach for 3D
mesh approximate convex decomposition, described in the
next section.
3501978-1-4244-5654-3/09/$26.00 ©2009 IEEE ICIP 2009
3. PROPOSED APPROACH
The proposed hierarchical segmentation approach proceeds as
follows. First, the dual graph of the mesh is computed. Then
its vertices are iteratively clustered by successively applying
topological decimation operations, while minimizing a cost
function related to the concavity and the aspect ratio of the
produced segmentation clusters.
Let’s first recall the definition of the dual graph of the
mesh.
3.1. Dual graph
The dual graph S∗associated to the mesh Sis defined as fol-
lows:
•each vertex of S∗corresponds to a triangle of S,
•two vertices of S∗are neighbours (i.e., connected by an
edge of the dual graph) if and only if their correspond-
ing triangles in Sshare an edge.
3.2. Decimation operator
Once the dual graph S∗is computed, the algorithm starts
the decimation stage which consists in successively apply-
ing halfe-edge collapse [4] decimation operations. Each half-
edge collapse operation applied to an edge (v, w), denoted
hecol(v, w), merges the two vertices vand w.Thevertexw
is removed and all its incident edges are connected to v(cf.
Figure 1).
Fig. 1.Half-edge collapse decimation operation.
Let A(v)be the list of the ancestors of the vertex v.Ini-
tially, A(v)is empty. At each operation hecol(v, w)applied
to the vertex v, the list A(v)is updated as follows :
A(v)←A(v)∪A(w)∪{w}.(1)
3.3. Simplification strategy
The decimation process described in the previous section is
guided by a cost function describing the concavity and the
aspect ratio [5] of the surface S(v, w)resulting from the uni-
fication of the vertices vand wand their ancestors:
S(v, w)=A(v)∪A(w)∪{w, v}.(2)
As in [5], we define the aspect ratio Eshape (v, w)of the
surface S(v, w)as follows:
Eshape (v, w)= ρ2(S(v, w))
4π×σ(S(v, w)) ,(3)
where ρ(S(v, w)) and σ(S(v, w)) are respectively the perime-
ter and the area of S(v, w).
The cost Eshape(v, w)was introduced in order to favor
the generation of compact clusters. In the case of a disk this
cost equals one. The more irregular a surface is the highier its
aspect ratio.
The decimation cost E(v, w)associated to the edge (v, w)
is given by:
E(v, w)=C(S(v, w))
D+αEshape(v, w),(4)
where
•C(S(v, w)) is the concavity of S(v, w)(cf. Section
3.4),
•Dis a normalization factor equal to the diagonal of the
bounding box of S,
•αis a parameter controlling the contribution of the
shape factor Eshape(v, w)with respect to the concavity
cost (cf. Section 3.5).
At each step of the decimation process, the hecol oper-
ation with the lowest decimation cost is applied and a new
partition Π(n)={πn
1,π
n
2,π
n
3, ..., πn
K(n)}is computed as fol-
lows:
∀k∈{1, ..., K (n)},π
n
k=pn
k∪A(pn
k),(5)
where (pn
k)k∈{1,...,K(n)}represent the vertices of the dual
graph S∗obtained after nhalf-edge collapse operations. This
process is iterated until all the edges of S∗generating clusters
with concavities lower than are decimated.
3.4. Concavity measure
As discussed in [2, 3], there is no consensus in the literature
on a quantitative concavity measure. In this work, we define
the concavity C(S)of a 3D mesh S, as follows (cf. Figure 2):
C(S) = arg max
M∈SM−P(M),(6)
where P(M)represents the projection of the point Mon
the convex-hull CH(S)[6] of S, with respect to the half-ray
with origin Mand direction normal to the surface Sat M.
Let us note that the concavity of a convex surface is zero.
Intuitively, the more concave a surface the ”farther” it is from
its convex-hull. The definition (6) extends directly to open
meshes once oriented normals are provided for each vertex.
3.5. Choice of the parameter α
The clusters detected during the early stages of the algorithm
are composed of a low number of adjacent triangles with a
concavity almost equal to zero. Therfore, the decimation cost
Eis dominated by the aspect ratio cost Eshape ,whichfavors
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Fig. 2. Concavity measure for a 3D mesh.
the generation of compact surfaces. This behavior is progres-
sively inverted during the decimation process since the clus-
ters are becoming more and more concave. In order to ensure
that the cost (α.Eshape )has no influence on the choice of the
last decimation operations, we have set the parameter αas
follows:
α=
10 ×D.(7)
This choice garanty, for disk-shaped clusters, that the cost
(α.Eshape)is ten times lower than the concavity-related cost
C(S(v,w))
D.
4. EXPERIMENTAL RESULTS
In order to validate our approach, we have first compared it
segmentation results to those of the ACD technique1intro-
duced in [2, 3]. Figure 3 presents the convex-hulls gener-
ated by our approach to those of [2, 3]. The reported results
show that our approach provides better convex approxima-
tions while detecting a lower number of clusters.
Figure 4 presents the segmentation results and the convex-
hull approximations generated by our approach for diffrent
3D meshes. Let us note first that, for all the models, the gen-
erated decompositions ensures a concavity lower than while
generating a small number of clusters, which provides an ef-
ficient representation for collision detection. Moreover, the
proposed technique successfully detects the convex parts (e.g.
the four spheres of Figure 4.p) and the anatomical structure of
the analyzed 3D models.
5. CONCLUSION
We have presented a hierarchical segmentation approach for
approximate convex decomposition of 3D meshes. The gen-
erated segmentations are exploited to construct faithful ap-
proximations of the original mesh by a set of convex surfaces.
This new representation is particularly adapted for collision
detection. The experimental evaluation we conducted shows
that the proposed technique efficiently decomposes a concave
3D model into a small set of nearly convex surfaces, while
automatically detecting its anatomical structure which makes
1We have considered the only public implementation of the algo-
rithm avaible at http://codesuppository.blogspot.com/2006/04/approximate-
convex-decomposition.html
it an ideal candidate for skeleton extraction and pattern recog-
nition applications.
(a) T= 5804 (b) K=28 (c) K=23
(d) T= 39689 (e) K=12 (f) K=12
(g) T= 19563 (h) K=26 (i) K=22
Fig. 3. Comparative evaluation: (a,d,g) original meshes,
(b,e,h) convex-hulls generated by [2, 3] and (c,f,i) convex-
hulls generated by our approach (Knumber of clusters and T
number of triangles).
6. REFERENCES
[1] B. Chazelle, D.P. Dobkin, N. Shouraboura, and A. Tal,
“Strategies for polyhedral surface decomposition: an ex-
perimental study,” in Symposium on Computational Ge-
ometry, 1995, pp. 297–305.
[2] J.M. Lien and N.M. Amato, “Approximate convex de-
composition,” in Symposium on Computational Geome-
try, 2004, pp. 457–458.
[3] J.M. Lien and N.M. Amato, “Approximate convex de-
composition of polyhedra and its applications,” Com-
puter Aided Geometric Design, pp. 503–522, 2008.
[4] H. Hoppe, “Progressive meshes,” in International
Conference on Computer Graphics and Interactive Tech-
niques, 1996, pp. 99–108.
[5] A. K. Jain, “Fundamentals of digital image processing,”
Prentice-Hall International, 1989.
[6] F.P. Preparata and S.J. Hong, “Convex hulls of finite sets
of points in two and three dimensions,” Communication
of the ACM, vol. 20(2), pp. 87–93, 1977.
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Fig. 4. Segmentation results and generated convex-hulls (=0.03 ×D,Dlength of the bounding box diagonal, Knumber of
clusters and Tnumber of triangles).
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