A Compact Representation for Topological Decompositions

of Non-manifold Shapes

David Canino and Leila De Floriani

Department of Computer Science, University of Genova, Genova, Italy

Keywords:

Geometric Modeling, Topological Data Structures, Simplicial Complexes, Non-manifold Shapes, Decompo-

sition, Structural Representation.

Abstract:

Simplicial complexes are extensively used for discretizing digital shapes in several applications. A structural

description of a non-manifold shape can be obtained by decomposing the input shape into a collection of

meaningful components with a simpler topology. Here, we consider a unique and dimension-independent

decomposition of a non-manifold shape into nearly manifold components, known as the Manifold-Connected

(MC-) decomposition. We present the Compact Manifold-Connected (MC-) graph, an efﬁcient graph-based

representation for the MC-decomposition, which can be combined with any topological data structure for

encoding the underlying components. We present the main properties of this representation as well as algo-

rithms for its generation. We also show that this representation is more compact than several topological data

structures, which do not explicitly describe the non-manifold structure of a shape.

1 INTRODUCTION

Modeling digital shapes requires efﬁcient representa-

tions, which integrate geometric, topological, and se-

mantic aspects. A digital shape is often discretized

by any simplicial complex. A very common rep-

resentation of this latter is provided by topological

data structures, which encode a subset of simplices,

and the connectivity information among such sim-

plices. In the literature, there is a large amount of

research about these representations (De Floriani and

Hui, 2005). In any case, they describe only combina-

torial information of shapes, and do not expose their

relevant components and their connectivity.

In order to overcome these limitations, a shape

may be decomposed into meaningful components,

which are easily distinguished from the remaining

part of the object so as to reduce the complexity of

a shape. The resulting representation highlights com-

ponents of a shape, and their connectivity, namely it

is a structural model. Here, geometric details are ab-

stracted, and this model provides a high-level descrip-

tion of a shape. Hence, this model is a suitable basis

for semantic annotation and reasoning.

Most structural models are deﬁned for manifold

shapes (Shamir, 2008). Informally, a manifold is a

subset of the Euclidean space such that the neighbor-

hood of each point is homeomorphic to an open ball.

Non-manifold shapes do not satisfy this property at

one or more points, which are called the non-manifold

singularities. Non-manifold shapes arise in several

applications, for instance, as the result of the ideal-

ization process during numerical simulations (Thakur

et al., 2009). There is also an increasing interest for

modeling non-manifold shapes, discretized through

abstract simplicial complexes, which are not neces-

sarily embedded in any Euclidean space, like the

Vietoris-Rips complexes (Gromov, 1987).

The most natural decomposition of any non-

manifold shape consists of cutting this latter along

its non-manifold singularities without modifying the

manifold parts. The resulting representation high-

lights manifold components and their connectivity,

and non-manifold singularities are exposed explicitly.

However, this decomposition is possible only for 2-

complexes. In three or higher dimensions, it may in-

troduce artiﬁcial ”cuts” in the input shape, and cre-

ate almost manifold components (De Floriani et al.,

2003). In six or higher dimensions, this decomposi-

tion is not feasible, since the class of manifolds is not

decidable (Nabutovsky, 1996).

Here, we describe a dimension-independent ex-

tension of the Manifold-Connected (MC-) decompo-

sition, introduced in (Hui and De Floriani, 2007) for

2D and 3D shapes. This latter is a structural model for

abstract simplicial complexes, which are decomposed

100

Canino D. and De Floriani L..

A Compact Representation for Topological Decompositions of Non-manifold Shapes.

DOI: 10.5220/0004294501000107

In Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information

Visualization Theory and Applications (GRAPP-2013), pages 100-107

ISBN: 978-989-8565-46-4

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

as a collection of Manifold-Connected (MC-) com-

ponents, which form a decidable superclass of man-

ifolds. These components may contain non-manifold

singularities, and thus they cannot be represented by

topological data structures speciﬁc for manifolds.

In this paper, we propose a two-level graph-based

representation for the MC-decomposition, which we

call the Compact Manifold-Connected (MC-) graph.

Here, the lower level consists of a topological data

structure, capable to represent non-manifold shapes.

In our experiments we have used the Incidence Sim-

plicial (IS) data structure (De Floriani et al., 2010),

and the Generalized Indexed data structure with Ad-

jacencies (IA

∗

) (Canino et al., 2011). The upper level

consists of MC-components, expressed through ref-

erences to simplices in the underlying data structure,

and acts in the same way as a spatial index (Samet,

2006). Then, we present an algorithm for extract-

ing the compact MC-graph from any topological data

structure. We also show that the compact MC-graph,

if combined with the IS and the IA

∗

data struc-

tures, is more compact than a widely used data struc-

ture, namely the Incidence Graph (IG) (Edelsbrun-

ner, 1987), which does not expose the structure of a

shape explicitly. Finally, we also show that the com-

pact MC-graph, combined with the IA

∗

data structure,

is more cost effective than the Initial Quasi-Manifold

(IQM-) decomposition (De Floriani et al., 2003), and

than the Double Level Decomposition (DLD) data

structure (Hui et al., 2006), which contains the IQM-

decomposition of simplicial 3-complexes, embedded

in the Euclidean space E

The remainder of this paper is organized as fol-

lows. In Section 2, we present some background no-

tions, while, in Section 3, we brieﬂy discuss related

work. In Section 4, we deﬁne the MC-decomposition

for a non-manifold shape in arbitrary dimensions. In

Section 5, we introduce and analyze our compact MC-

graph, while, in Section 6, we present some experi-

mental results. Finally, in Section 7, we draw con-

cluding remarks and discuss future work.

2 BACKGROUND NOTIONS

In this section, we introduce background notions,

which we will use throughout the paper.

A Euclidean p-simplex σ is the convex hull of a

set V

of p +1 afﬁne points in the Euclidean space E

with p ≤ n, and p = dim(σ) is the dimension of σ. A

k-simplex σ

, with k ≤ p, generated by k + 1 vertices

in V

, is a k-face of σ. Here, σ and σ

are mutually

incident. A ﬁnite collection Σ of Euclidean simplices

is a Euclidean simplicial complex if (i) all the faces of

each simplex are in Σ, and (ii) for each pair of sim-

plices σ and σ

in Σ, either σ ∩ σ

0, or σ ∩ σ

in Σ. The maximum dimension d of the simplices in

Σ is the dimension of Σ, which will thus be called a

simplicial d-complex. The domain of a simplicial d-

complex Σ embedded in E

, with d ≤ n, is the subset

of E

spanned by all the simplices of Σ. A Euclidean

simplicial complex Σ is the geometric realization of

an abstract simplicial complex, which represents the

combinatorial structure of Σ.

The combinatorial boundary b(σ) of a simplex σ

consists of all the faces of σ. The star St(σ) is formed

by all the simplices incident at σ. If St(σ) =

0, then

σ is a top simplex. Top d-simplices are called maxi-

mal. A simplicial d-complex Σ is regular if all the top

simplices are maximal. The link Lk(σ) of a simplex

σ consists of all the faces of the simplices in St(σ),

which are not incident at σ. Two k-simplices are ad-

jacent if they share a (k − 1)-face, while two vertices

are adjacent if they are connected by a common edge.

An h-path is any sequence of (h + 1)-simplices

(σ

)

i=0

such that two consecutive simplices σ

i−1

and

are adjacent. Two simplices σ

and σ

are h-

connected if an h-path (σ

)

i=0

connects σ

and σ

A subcomplex Σ

of any simplicial complex Σ is h-

connected if any two simplices in Σ

are h-connected.

A (d − 1)-simplex σ in a regular simplicial d-

complex is combinatorial manifold, if at most two d-

simplices are incident at σ. A k-simplex σ, such that

Lk(σ) is homeomorphic to the (d − k)-sphere, is com-

binatorial manifold, otherwise σ is a non-manifold

singularity. A simplicial d-complex, such that all ver-

tices are combinatorial manifold, is a combinatorial

d-manifold (De Floriani et al., 2003). Note that there

are some algorithmically non-recognizable combina-

torial d-manifolds for d ≥ 6 (Nabutovsky, 1996), thus

the class of combinatorial d-manifolds is not always

decidable. A regular (d − 1)-connected simplicial d-

complex Σ, such that the star of each (d − 1)-simplex

consists of at most two d-simplices, is a combinato-

rial pseudo-manifold.

3 RELATED WORK

In this section, we brieﬂy discuss some structural

models, speciﬁc for non-manifold shapes.

The Selective Geometric Complexes (Rossignac

and O’Connor, 1989) exploit decompositions of non-

manifold shapes, encoded through the IG data struc-

ture (Edelsbrunner, 1987), whose cells can be either

open or not connected. In (Desaulniers and Stewart,

1992) the authors propose a decomposition of non-

manifold shapes into regular parts, providing interest-

ACompactRepresentationforTopologicalDecompositionsofNon-manifoldShapes

101

ing topological information. In (Falcidieno and Ratto,

1992) the authors discuss the identiﬁcation of form

features in simplicial shapes, decomposing them into

regular parts. In (Gueziec et al., 1998) the authors

propose a decomposition-based technique to convert

non-manifold shapes into manifolds without address-

ing geometric aspects. In (Rossignac and Cardoze,

1999) the authors improve over this technique by

taking in account also geometry: the key idea is to

split and duplicate non-manifold singularities in or-

der to avoid self-intersections. A further extension

to volumetric shapes is proposed in (Attene et al.,

2009), which applies local updates on the neighbor-

hood of a non-manifold singularity. In (Pesco et al.,

2004) the authors propose a non-unique combinato-

rial stratiﬁcation of a cell 2-complex on which they

deﬁne editing operators. The Initial Quasi-Manifold

(IQM-) decomposition (De Floriani et al., 2003) is a

unique and dimension-independent decomposition of

abstract simplicial d-complexes. A simplicial shape

is decomposed into IQM-components, which are reg-

ular simplicial d-complexes such that any pair of d-

simplices in the star of each vertex is connected by

a (d − 1)-path such that two consecutive d-simplices

share a (d − 1)-simplex. A graph-based data structure

for representing the IQM decomposition of any shape

embedded in the Euclidean space E

is the Double-

Level Decomposition (DLD) data structure (Hui et al.,

2006). In this decomposition, each IQM-component

is encoded through the Indexed data structure with

Adjacencies (IA) (Paoluzzi et al., 1993).

4 THE MC-DECOMPOSITION

In this section, we describe a dimension-indepen-

dent extension of the Manifold-Connected (MC-) de-

composition, initially deﬁned in (Hui and De Floriani,

2007) only for 2D and 3D shapes. This decomposi-

tion is deﬁned on abstract simplicial complexes.

In a regular simplicial d-complex Σ, two d-

simplices σ and σ

are said to be manifold-connected

if and only if there exists a (d − 1)-path joining σ and

such that two consecutive d-simplices share a man-

ifold (d − 1)-simplex. A (d − 1)-simplex in Σ is man-

ifold if its star contains at most two d-simplices. We

call a (d − 1)-path with such properties as a manifold

(d − 1)-path. Figure 1(a) shows a manifold 1-path (in

yellow) between two triangles σ and σ

(in red) in a

regular simplicial 2-complex, namely a torus. A reg-

ular d-complex Σ, such that any pair of d-simplices is

manifold-connected, is a Manifold-Connected comp-

lex (MC-complex) of dimension d. It is clear that any

manifold is also an MC-complex, like torus in Fig-

ure 1(a), but the reverse is not true. In fact, an MC-

complex may contain non-manifold singularities, like

non-manifold edge e in MC-complex of dimension

3 in Figure 1(b). This shows that the class of MC-

complexes is a superclass of combinatorial manifolds.

(a) (b)

Figure 1: (a) A manifold 1-path (in yellow) connecting two

triangles σ and σ

(in red) in a regular simplicial 2-complex,

which is also an MC-complex of dimension 2. (b) An ex-

ample of MC-complex of dimension 3, which is pinched at

a non-manifold edge e. All the tetrahedra incident at e are

also highlighted (in green).

The MC-decomposition is deﬁned as the decom-

position of any regular simplicial d-complex X into

a collection of MC-complexes of dimension d, called

the MC-components of X , which can be described in

terms of top d-simplices. Manifold-connectivity re-

lation, restricted to top d-simplices in X , deﬁnes an

equivalence relation on X . MC-components of X are

the equivalence classes of top d-simplices with re-

spect to the manifold-connectivity relation. Specif-

ically, any top d-simplex σ, called the representa-

tive simplex of X , is equivalent to all the top d-

simplices in X , which are reachable from σ through

a manifold (d − 1)-path. The collection of all the

MC-components in X form the Manifold-Connected

(MC-) decomposition of X . This decomposition is

unique, since any top d-simplex σ in X belongs to

only one MC-component. Several MC-components

in X may have a common intersection, formed by

non-manifold singularities, i.e., a subcomplex of X

of dimension lower than d. Figure 2(a) shows the

MC-decomposition of a regular simplicial 2-complex,

which is formed by three MC-components of dimen-

sion 2, respectively, in red, purple, and green. These

MC-components are connected by two non-manifold

vertices and a non-manifold edge.

A non-regular simplicial d-complex Σ is uniquely

(a) (b)

Figure 2: (a) Three MC-components of dimension 2 for a

regular simplicial 2-complex, respectively, in red, purple,

and green. (b) MC-components of dimension 2 (in green)

and 3 (in purple) for a non-regular simplicial 3-complex.

GRAPP2013-InternationalConferenceonComputerGraphicsTheoryandApplications

102

decomposed into maximal regular subcomplexes Σ

i.e., collections of top p-simplices in Σ, with 0 <

p ≤ d. The Manifold-Connected (MC-) decomposi-

tion of Σ consists of the MC-decompositions of all

the subcomplexes Σ

. Note that each top p-simplex

in Σ belongs to only one subcomplex Σ

, thus the

MC-decomposition of Σ is unique. Hence, Σ is de-

composed into MC-complexes of dimension p, for

0 < p ≤ d, which may have a common intersec-

tion, formed by non-manifold simplices. Figure 2(b)

shows the MC-decomposition of a non-regular simp-

licial 3-complex, formed by several MC-components

of dimension 2 (in green) and 3 (in purple), which are

connected by chains of non-manifold edges.

5 THE COMPACT MC-GRAPH

The MC-decomposition for an abstract simplicial d-

complex Σ can be represented through a two-level

data structure. At the lower level, we use any topo-

logical data structure M

for non-manifold shapes to

encode Σ. On the contrary, the upper level encodes

the connectivity among MC-components through a

graph-based data structure. An MC-component C in

Σ corresponds to a node, containing a reference to the

representative top simplex of C , while any arc corre-

sponds to the common intersection S of several MC-

components, and contains references to non-manifold

simplices in Σ. The resulting representation acts as a

sort of spatial index (Samet, 2006), imposed on any

topological data structure M

. Note that if Σ is mani-

fold, then its MC-decomposition consists of only one

MC-component. Hence, only a reference to one top

simplex in Σ is stored.

Thus, the main difference among several graph-

based representations of the MC-decomposition de-

pends on the encoding of hyperarcs, since nodes are

described uniquely. The ﬁrst graph-based represen-

tation of the MC-decomposition, satisfying these de-

sign choices, has been proposed in (Boltcheva et al.,

2011), but it may become verbose due to the presence

of cliques (Canino, 2012).

The Exploded Manifold-Connected (MC-) graph,

introduced in (Canino and De Floriani, 2011), is a

hypergraph G

= (N

, A

), such that each hyperarc

corresponds to a non-manifold simplex σ, and con-

nects n

nodes, corresponding to the MC-components

incident at σ. We denote the collection of non-

manifold singularities in Σ as Σ

. Let n

and a

be,

respectively, the number of nodes and arcs in G

, then

the storage cost S

of the exploded MC-graph, ex-

pressed in terms of references, is equal to:

= n

∑

a∈A

(1 + n

) = n

+ a

∑

σ∈Σ

(1)

Figure 3(b) shows the exploded MC-graph, which

represents the MC-decomposition of non-regular 2D

shape in Figure 3(a). In this case, there are three

hyperarcs, related, respectively to non-manifold ver-

tices v

and v

, and to non-manifold edge e

. For in-

stance, hyperarc in red relates non-manifold vertex v

and MC-components C

, C

and C

, which are in-

cident at v

. The exploded MC-graph is robust with

respect the presence of cliques (Canino, 2012). How-

ever, a subset of MC-components may be duplicated

and related to several hyperarcs, increasing its stor-

age cost (Canino, 2012). For instance, in Figure 3(b),

MC-components C

, C

, and C

are connected twice

by hyperarcs related to non-manifold vertices v

and

. Thus, the number of redundancies must be re-

duced in order to deﬁne a compact representation of

the MC-decomposition.

Here, we propose the Compact Manifold-Connec-

ted (MC-) graph, which overcomes all the drawbacks

of the exploded MC-graph. The key idea consists of

grouping together all the hyperarcs of the exploded

MC-graph, which are related to the same subcomplex

of non-manifold singularities, in order to reduce the

storage cost. The compact MC-graph is a hypergraph

= (N

, A

), such that any node corresponds to

one MC-component C , while any hyperarc describes

the maximal set of non-manifold singularities, shared

by MC-components C

, . .. , C

. Thus, any hyperarc

a = (C

, . . . , C

) satisﬁes the following properties:

i) intersection Σ

i=1,...,k

, is a subcomplex of

Σ, not necessarily connected;

ii) there is no MC-component C

6= C

, for all i =

1, . . . , k, such that Σ



i=1,...,k



∩ C

For each node, corresponding to one MC-component

C, one reference to the representative simplex of C is

stored. For any hyperarc a, k

= k references to MC-

components C

, . . . , C

are stored, plus s

references

to non-manifold singularities in Σ

. Let n

and a

be, respectively, the number of nodes and hyperarcs

in G

, then the storage cost S

of the compact MC-

graph, expressed in terms of references, is equal to:

= n

∑

a∈A

+ s

) (2)

It is clear that the exploded and the compact MC-

graph have the same number of nodes, thus n

= n

Figure 3(c) shows the compact MC-Graph, which

represents the MC-decomposition of non-regular 2D

shape in Figure 3(a). The resulting hypergraph is mo-

ACompactRepresentationforTopologicalDecompositionsofNon-manifoldShapes

103

(a) (b) (c)

Figure 3: (a) The MC-decomposition of a non-regular 2D shape is formed by four MC-components, connected through non-

manifold edge e

and vertices v

and v

. Some arcs (in red) in the corresponding (b) exploded and (c) compact MC-graphs.

re compact than the corresponding exploded MC-gra-

ph, shown in Figure 3(b). Here, hyperarc (C

, C

)

(in red) is related to non-manifold singularities v

and e

, while other hyperarc (C

, C

) is related

only to non-manifold vertex v

, since C

is not inci-

dent at v

and e

The compact MC-graph, if combined with any

topological data structure M

, can be computed in

two steps. In the ﬁrst step, all the MC-components

are retrieved as described in (Boltcheva et al., 2011).

Recall that the identiﬁcation of MC-components is al-

ways deﬁned and completely dimension-independent.

We encode the result of the ﬁrst step as an array L

such that, for each non-manifold singularity σ, any

location L[σ] contains l

labels of MC-components

incident at σ. Note that this allows deﬁning the ex-

ploded MC-graph immediately. In the second step, all

the hyperarcs in the compact MC-graph are retrieved

by analyzing each location L[σ] through an auxiliary

array B, as follows:

1. If L[σ] contains only one label C , create a tuple

(σ, C ) in B, describing a self-loop related to σ.

Otherwise, sort L[σ] in increasing order as a list

, and create a tuple (σ,

) in B.

2. Sort all the tuples in B with respect to the lexico-

graphic order of lists of labels. Tuples related to

the same subset of labels are stored in consecutive

locations of B.

3. Create and complete a new hyperarc for each sub-

set of MC-components, identiﬁed at Step 2.

The identiﬁcation of MC-components depends on

the topological data structure M

, used in combina-

tion with the compact MC-graph. The IA

∗

data struc-

ture (Canino et al., 2011) offers the best support for

these operations (Canino, 2012). On the contrary, re-

trieving the hyperarcs does not depend on M

. The

time complexity of Step 1 is O(l

logl

) for each non-

manifold simplex σ, since L[σ] is sorted. In Step 2, s

locations of B are sorted, one for each non-singularity

σ in Σ (i.e., in Σ

), thus the time complexity of this

step is O(s

logs

). Step 3 can performed in O(1).

Hence, the time complexity required for computing

hyperarcs in the compact MC-graph is linear in:

logs

∑

σ∈Σ

logl

(3)

6 EXPERIMENTAL RESULTS

In our experiments, we have combined the compact

MC-graph with all the data structures within the Man-

grove Topological Data Structure (Mangrove TDS)

framework (Canino, 2012). This latter is a framework

for the fast prototyping of any topological data struc-

ture, representing simplicial complexes without re-

strictions. A C++ implementation of this framework

is contained in the Mangrove TDS Library (Canino

and De Floriani, 2012), released in public domain.

Here, we focus our attention only on the Incidence

Simplicial (IS) data structure (De Floriani et al., 2010)

and the Generalized Indexed data structure with Ad-

jacencies (IA

∗

) (Canino et al., 2011), which offer the

best performance with respect to queries efﬁciency

and storage cost (Canino, 2012). Note that the IS data

structure represents all the simplices explicitly, while

the IA

∗

data structure offers a more compact represen-

tation by encoding only vertices and top simplices.

In Table 1, we provide experimental comparisons

on our graph-based data structures for a subset of sim-

plicial shapes, freely available from (Hui and De Flo-

riani, 2009). First, we compare the storage costs S

and S

of the exploded and the compact MC-graph,

respectively. As expected, our results in Table 1 show

that the compact MC-graph is more compact than the

exploded MC-graph: S

≈ 2 ×S

, on average. This is

due to the clustering policy of MC-components, con-

nected by hyperarcs of the compact MC-graph, as dis-

cussed in Section 5. Note that the number of hyper-

arcs a

in the compact MC-graph is smaller than the

number a

of hyperarcs in the exploded MC-graph.

For instance, for the 2D shape ”Tower” in Table 1,

= 1.4k and a

= 165, while S

≈ 2.8 × S

We also compare the total storage costs S

and

∗

of the compact MC-graph, combined with the

IS and the IA

∗

data structures, respectively, and the

storage cost S

of the Incidence Graph (IG) (Edels-

brunner, 1987). This latter represents all the sim-

GRAPP2013-InternationalConferenceonComputerGraphicsTheoryandApplications

104

Table 1: Statistics on the MC-decomposition and the corresponding exploded and compact MC-graphs of some simplicial

2- and 3-complexes, freely available from (Hui and De Floriani, 2009). Here, column s

shows the number of top simplices

in the input shape, n

is the number of nodes, while a

and a

are, respectively, the number of arcs in the exploded and in

the compact MC-graphs. Columns S

and S

contain the storage costs of our graph-based representations, expressed as the

number of references. In addition, columns S

∗

, S

and S

show, respectively, the storage costs of the IA

∗

, IS, and IG data

structures. Finally, S

∗

and S

denote the storage costs of the compact MC-graph, combined, respectively, with the IA

∗

and

the IS data structures. In other words, S

∗

= S

+ S

IA∗

and S

= S

+ S

Shape s

∗

Carter 8k 45 641 48 3.8k 1.2k 52k 75k 53.2k 76.2k 95k

Chandelier 8.1k 130 616 96 2.6k 1k 120k 174k 121k 175k 220k

Pinched Pie 2.3k 120 1.4k 192 4.8k 1.9k 17k 20.5k 18.9k 22.4k 25k

Tower 19k 169 1.4k 165 5.9k 2.1k 124k 175k 126.1k 177.1k 221k

Chime 376 27 29 28 133 127 3.2k 8.5k 3.3k 8.6k 12k

Flasks 4k 8 76 6 300 98 29k 75k 29.1k 75.1k 104k

Teapot 13k 2.9k 1.2k 1k 10.4k 10.1k 85k 163k 95.1k 173.1k 220k

Wheel 1.2k 115 136 88 675 563 9.9k 23.7k 10k 24k 33.4k

Table 2: Statistics on storage costs of the compact MC-graph, combined with the IA

∗

data structure, and of the IQM-

decomposition (De Floriani et al., 2003), for several versions of simplicial shape ”Sierpinski”, embedded in the Euclidean

space E

, with 2 ≤ d ≤ 5. Here, columns s

and s

contain, respectively, the numbers of vertices and of d-simplices in the

input shape. Columns S

, S

∗

, S

∗

, and S

are the same as in Table 1. Finally, column S

IQM

contains the storage cost of the

IQM-decomposition.

Shape s

∗

IQM

Sierpinski 2D 88.5k 59k 324k 345k 669k 757k 1.1M

Sierpinski 3D 131k 65.5k 458k 524k 0.98M 1.1M 3.67M

Sierpinski 4D 195.3k 78.1k 664k 781k 1.44M 1.64M 11.6M

Sierpinski 5D 140k 46.6k 467k 559.6k 1M 1.16M 7.7M

plices explicitly, and is commonly used in the ap-

plications (Popovic and Hoppe, 1997). Our experi-

mental results in Table 1 (see columns S

, S

∗

, and

) show that the compact MC-graph is more cost

effective than the incidence graph. In fact, S

≈

1.12 × S

and S

≈ 1.45 × S

∗

, for 2D shapes, on

average, while, for 3D shapes, S

≈ 1.3 × S

and

≈ 3.2 × S

∗

. Columns S

∗

and S

in Table 2

show how the compact MC-graph, combined only

with the IA

∗

data structure, continues to be more cost

effective than the incidence graph in higher dimen-

sions. For instance, S

≈ 14.8 × S

∗

for simplicial

4-complexes. This result is very interesting, since the

IG data structure, unlike the compact MC-graph, does

not explicitly describe the structure of a shape, and

does not allow for an efﬁcient identiﬁcation of non-

manifold singularities (Canino, 2012).

Finally, we have also compared our compact MC-

graph, combined with the IA

∗

data structure, and

the IQM-decomposition (De Floriani et al., 2003) for

simplicial d-complexes, embedded in the Euclidean

space E

. For the sake of simplicity, we focus our at-

tention on Sierpinski d-shape, embedded in E

, which

is formed by s

vertices and s

d-simplices. Figure 4

shows a simpliﬁed 2D version of this shape. Here,

non-manifold singularities occur only at vertices, and

only d + 1 vertices are manifold. Any non-manifold

vertex v is shared by two d-simplices, and each of

them is both a MC- and an IQM-component.

v v

Figure 4: Simpliﬁed 2D versions of the Sierpinski shape,

embedded in the Euclidean space E

. In this shape, any

non-manifold vertex v is shared by two triangles (in black).

The most relevant difference between these two

representations consists of the different encoding for

the connectivity of subcomponents. Each node of

the IQM data structure contains one d-simplex, en-

coded as an IA data structure (Paoluzzi et al., 1993),

thus it requires (2d + 3)s

references. Each arc corre-

sponds to a non-manifold vertex v, and connects two

nodes, one for each d-simplex incident at v. In ad-

dition, for each arc, two copies of v are stored in or-

der to guarantee that any d-simplex is a valid IQM-

component. Hence, the storage cost S

IQM

of the IQM

data structure is S

IQM

= (2d + 3)s

+ 4(s

− d − 1).

On the contrary, in the compact MC-graph, each sim-

plex is stored only once in the underlying IA

∗

data

ACompactRepresentationforTopologicalDecompositionsofNon-manifoldShapes

105

structure, while, for each arc, references to two MC-

components and to a non-manifold vertex are stored.

Hence, the storage cost of the compact MC-graph

is equal to S

∗

= (d + 2)s

+ 5s

− 4(d + 1). Ex-

perimental results in Table 2 (see columns S

∗

and

IQM

) show that the compact MC-graph tends to be

more cost effective than the IQM data structure for

any dimension. For instance, S

IQM

≈ 1.14 × S

∗

and

IQM

≈ 1.16 ×S

∗

for, respectively, simplicial 4- and

5-complexes. Note that these representations coincide

when encoding manifolds, since they are equivalent to

the IA data structure, plus one additional reference to

a top simplex in the input shape.

In addition, our experimental results in Table 2

(see columns S

IQM

and S

) show that also the IQM

data structure is more compact than the incidence

graph in any dimension, e.g., S

≈ 8 ×S

IQM

for sim-

plicial 4-complexes.

7 CONCLUDING REMARKS

We have presented a structural model for non-ma-

nifold shapes, which are decomposed into a collection

of MC-components, a decidable superclass of mani-

folds of any dimension. We have designed and imple-

mented the Compact MC-graph, a graph-based repre-

sentation for the MC-decomposition (Hui and De Flo-

riani, 2007), which can be combined with any topo-

logical data structure representing non-manifolds.

We have combined the compact MC-graph with

all the topological data structures, which are currently

implemented in the Mangrove TDS Library (Canino

and De Floriani, 2012). Our tests show that the com-

pact MC-graph, if combined with the IS (De Flori-

ani et al., 2010) and the IA

∗

(Canino et al., 2011)

data structures, is more compact than the incidence

graph (Edelsbrunner, 1987), which is a widely used

data structure in several applications. However, this

latter, unlike our compact MC-graph, does not ex-

pose the structure of a shape explicitly, and does not

support the identiﬁcation of non-manifold singulari-

ties efﬁciently (Canino, 2012). Our tests also show

that the compact MC-graph is more cost effective than

the IQM-decomposition (De Floriani et al., 2003) and

than the DLD data structure (Hui et al., 2006), even

for high dimensions.

There is an increasing interest in quad and un-

structured hexahedral meshes in geometry process-

ing, animation, and numerical simulations. Some data

structures, speciﬁc for simplicial complexes, like the

IS and IA* data structures, can be easily extended to

deal with such shapes, since all the simplifying as-

sumptions, that make the two data structures com-

pact in the case of simplicial complexes, hold also

for quad and hexahedral meshes. Thus, also the MC-

decomposition can be extended to such meshes and

also to more general cell complexes .

Finally, we are designing new graph-based rep-

resentations for the IQM-decomposition. The prop-

erties of the IQM components may allow for a very

compact encoding. In fact, an IQM-component is al-

most manifold, thus it may be representable through

very compact data structures, speciﬁc for manifolds,

like (Gurung and Rossignac, 2009; Gurung et al.,

2011a; Gurung et al., 2011b), just to mention few.

ACKNOWLEDGEMENTS

This work has been partially supported by the Italian

Ministry of Education and Research under the PRIN

2009 program, and by the National Science Founda-

tion under grant number IIS-1116747.

REFERENCES

Attene, M., Giorgi, D., Ferri, M., and Falcidieno, B. (2009).

On Converting Sets of Tetrahedra to Combinatorial

and PL Manifolds. Comp.-Aid. Des., 26(8):850–864.

Boltcheva, D., Canino, D., Merino Aceituno, S., L

eon, J.-

C., De Floriani, L., and H

etroy, F. (2011). An Iterative

Algorithm for Homology Computation on Simplicial

Shapes. Comp.-Aid. Des., 43(11):1457–1467.

Canino, D. (2012). Tools for Modeling and Analysis of Non-

manifold Shapes. PhD thesis, Department of Compu-

ter Science, University of Genova, Genova, Italy.

Canino, D. and De Floriani, L. (2011). A Decomposition-

based Approach to Modeling and Understanding Ar-

bitrary Shapes. In Proc. of the EG Italy, pages 53–60.

Canino, D. and De Floriani, L. (2012). The Mangrove To-

pological Data Structure (Mangrove TDS) Library.

http://mangrovetds.sourceforge.net.

Canino, D., De Floriani, L., and Weiss, K. (2011). IA

∗

: an

Adjacency-based Representation for Non-Manifold

Simplicial Shapes in Arbitrary Dimensions. Comp.

& Graph., 35(3):747–753.

De Floriani, L. and Hui, A. (2005). Data Structures for Sim-

plicial Complexes: an Analysis and a Comparison. In

Proc. of the Symp. on Geom. Proc., pages 119–128.

De Floriani, L., Hui, A., Panozzo, D., and Canino, D.

(2010). A Dimension-independent Data Structure for

Simplicial Complexes. In Proc. of the 19th Int. Mes.

Round., pages 403–420.

De Floriani, L., Mesmoudi, M. M., Morando, F., and Puppo,

E. (2003). Decomposing Non-manifold Objects in Ar-

bitrary Dimension. Graph. Mod., 65(1/3):2–22.

Desaulniers, H. and Stewart, N. (1992). An Extension

of Manifold Boundary Representations to the r-sets.

ACM Trans. on Graph., 11(1):40–60.

GRAPP2013-InternationalConferenceonComputerGraphicsTheoryandApplications

106

Edelsbrunner, H. (1987). Algorithms in Combinatorial Ge-

ometry. Springer.

Falcidieno, B. and Ratto, O. (1992). Two-manifold Cell-de-

composition of r-sets. Comp. Graph. For., 11(3):391–

404.

Gromov, M. (1987). Hyperbolic Groups. Springer.

Gueziec, A., Taubin, G., Lazarus, F., and Horn, W. (1998).

Converting Sets of Polygons to Manifold Surfaces by

Cutting and Stitching. In Proc. of the IEEE Conf. on

Vis., pages 383–390.

Gurung, T., Laney, D., Lindstrom, P., and Rossignac, J.

(2011a). SQuad: a Compact Representation for Tri-

angle Meshes. Comp. Graph. For., 30(2):355–364.

Gurung, T., Luffel, M., Lindstrom, P., and Rossignac, J.

(2011b). LR: Compact Connectivity Representation

for Triangle Meshes. ACM Trans. on Graph., 30(4).

Gurung, T. and Rossignac, J. (2009). SOT: Compact Repre-

sentation for Tetrahedral Meshes. In Proc. of the ACM

Conf. on Sol. and Phys. Mod., pages 79–88.

Hui, A. and De Floriani, L. (2007). A Two-level Topo-

logical Decomposition for Non-Manifold Simplicial

Shapes. In Proc. of the ACM Conf. on Sol. and Phys.

Mod., pages 355–360.

Hui, A. and De Floriani, L. (2009). The Non-Manifold Me-

shes Repository. http://indy.disi.unige.it/nmcollection.

Hui, A., Vaczlavik, L., and De Floriani, L. (2006). A

Decomposition-based Representation for 3D Simpli-

cial Complexes. In Proc. of the Symp. on Geom. Proc.,

pages 101–110.

Nabutovsky, A. (1996). Geometry of the Space of Trian-

gulations of a Compact Manifold. Comm. in Math.

Phys., 181:303–330.

Paoluzzi, A., Bernardini, F., Cattani, C., and Ferrucci, V.

(1993). Dimension-Independent Modeling with Sim-

plicial Complexes. ACM Trans. on Graph., 12(1):56–

102.

Pesco, S., Tavares, G., and Lopes, H. (2004). A Stratiﬁ-

cation Approach for Modeling Two-dimensional Cell

Complexes. Comp. & Graph., 28:235–247.

Popovic, J. and Hoppe, H. (1997). Progressive Simplicial

Complexes. In Proc. of the ACM SIGGRAPH, pages

217–224.

Rossignac, J. and Cardoze, D. (1999). Matchmaker: man-

ifold BReps for Non-manifold R-sets. In Proc. of the

ACM Symp. on Sol. Mod. and Appl., pages 31–41.

ACM Press.

Rossignac, J. and O’Connor, M. (1989). A Dimension-

independent Model for Point-sets with Internal Struc-

tures and Incomplete Boundaries. In Geom. Mod. for

Prod. Eng., pages 145–180. North-Holland.

Samet, H. (2006). Foundations of Multidimensional and

Metric Data Structures. Morgan Kaufmann.

Shamir, A. (2008). A Survey on Mesh Segmentation Tech-

niques. Comp. Graph. For., 27(6):1539–1556.

Thakur, A., Banerjee, A. G., and Gupta, S. K. (2009). A

Survey of CAD Models Simpliﬁcation Techniques for

Physics-based Simulation Applications. Comp.-Aid.

Des., 41(2):65–80.

ACompactRepresentationforTopologicalDecompositionsofNon-manifoldShapes

107