TOPMESH

A Tool for Extracting Topological

Information from Non-manifold Objects

Leila De Floriani, Laura Papaleo

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

Annie Hui

Department of Computer Science, University of Maryland, U.S.A.

Keywords: Shape Modelling, Shape Understanding, Structural Representation, Non-manifold Geometry.

Abstract: We present TopMesh, a tool for extracting topological information from non-manifold three-dimensional

objects with parts of non-uniform dimensions. The boundary of such objects is discretized as a mesh of

triangles and of dangling edges, representing one-dimensional parts of the object. The geometrical and

topological information extracted include the number of elements in the mesh, the number of non-manifold

singularities and the Betti numbers, which characterize the topology of an object independently of the

discretization of its boundary. TopMesh also computes a decomposition of the mesh into connected parts of

uniform dimension, into edge-connected components formed by triangles, and into oriented edge-connected

sub-meshes. We describe the functionalities of TopMesh and the algorithms implementing them.

1 INTRODUCTION

A 3D object is most commonly described through a

discretization of its boundary into a mesh consisting

of triangles and of dangling edges, which do not

bound any triangle. We call these meshes, known in

algebraic topology as simplicial meshes, triangle-

segment meshes. They are common in a variety of

application domains, including computer graphics,

solid modeling, finite element analysis, virtual

reality, animation and scientific visualization.

Triangle-segment meshes are specifically well suited

for describing complex non-manifold objects.

Informally, a manifold 3D object is an object in

which the neighbourhood of every point on its

boundary is topologically equivalent to a disk, or to

a half disk. If the boundary of the object is

triangulated, the manifold condition translates into

the fact that the set of triangles incident at any vertex

form a disk, or a half disk. Objects, that do not fulfil

this property at one or more points, are called non-

manifold, and if they also contain parts of different

dimensionalities, are called non-regular.

While there exist tools to extract geometric and

topological information from manifold shapes, much

less work exists on extracting such information from

non-manifold ones. TopMesh is a tool for the

extraction of topological information about a 3D

object based on an analysis of its combinatorial

representation. The basic idea, here, is translating

topological properties in the continuum case into

combinatorial properties of the discrete

representation of the object as a triangle-segment

mesh.

Triangle-segment meshes arise from the

idealization of manifold triangular meshes obtained

by the discretization of CAD models for finite

element analysis.

Features of TopMesh are related to the extraction

of non-manifold singularities, parts of the object

boundary with different degrees of connectivity, and

with different dimensions. Moreover, TopMesh

allows computing a topological signature for a 3D

object, given by the so-called the Betti numbers.

Betti numbers provide a characterization of the

topology of the object independent of the

discretization of its boundary, which is particularly

useful in shape analysis and retrieval. Betti numbers

are topological quantities corresponding to the

De Floriani L., Papaleo L. and Hui A. (2010).

TOPMESH - A Tool for Extracting Topological Information from Non-manifold Objects.

In Proceedings of the International Conference on Computer Graphics Theory and Applications, pages 21-29

DOI: 10.5220/0002819400210029

 SciTePress

connectivity of the object and to number of through-

holes or handles, and of hollow cavities in the

object.

TopMesh is implemented based on a library for

encoding and manipulating triangle-segment

meshes, the Triangle-Segment (TS) Library, freely

available in (AIM@SHAPE, 2007).

The reminder of this paper is organized as

follows. In Section 2, we provide the mathematical

background with definitions and intuitive

descriptions. In Section 3, we discuss related work,

while in Section 4 we briefly describe the TS data

structure. Section 5 is devoted to the description of

the main functionalities of TopMesh, and in Section

6 we focus on the extraction of the Betti numbers. In

Section 7, we presents results obtained on specific

non-manifold datasets. Finally, in Section 8 we draw

some concluding remarks.

2 BACKGROUND NOTIONS

In this Section, we introduce fundamental definitions

as basis for our work.

We call a triangle-segment mesh any two-

dimensional mesh discretizing the boundary of a

non-manifold and non-regular three-dimensional

object. Thus, a triangle-segment mesh is a two-

dimensional simplicial complex embedded in the

three dimensional Euclidean space (Agoston, 2005).

The basic entities in a triangle-segment mesh are

triangles, edges that do not bound any triangles, that

we call wire-edges, and vertices. The collection of

all the vertices and edges of a triangle-segment mesh

 form the 1- skeleton of the mesh.

The star of a vertex v is defined as the set of

triangles and edges that are incident at v. The link of

v is defined as the set of edges and vertices in the

star of v which are not incident in v (Agoston, 2005).

Figure 1(a) and Figure 1 (b) show, respectively, the

star and the link of a vertex. In a similar way, the

star of an edge e is the set of triangles that are

incident at e. The link of e is the set of vertices of

the triangles in the star of e which are not extreme

vertices e.

Figure 1 (c) and Figure 1 (d) show,

respectively, the star and the link of an edge.

A triangle-segment is connected when its 1-

skeleton is a connected graph. A triangle-segment is

edge-connected when, for every pair of triangles t

and t

of , there is a path in  composed of an

alternating sequence of triangles and edges such that

any edge in the path is shared by two triangles

preceding and following it. A triangle-segment mesh

without wire-edges is just a triangle mesh. If it

contains at least one edge that is bounding exactly

one triangle, it is called a triangle mesh with

boundary.

e

(a) (b) (c) (d)

Figure 1: The star and the link of a vertex and an edge: (a)

the star of vertex v; (b) the link of vertex v ; (c) the star of

edge e; (d) the link of edge e.

A vertex v in a triangle-segment mesh is a

manifold vertex if its link consists of one or two

vertices, i.e., v belongs only to one or two wire-

edges, or if the link of v consists of a closed or open

chains of edges, i.e., v is shared by a fan of triangles

forming a full disk (as in the example

Figure 1(a)) or

a half disk (in this latter case v is a boundary vertex).

An edge is a manifold edge if its link consists of one

or two vertices (as in the example

Figure 1(c)).

A vertex, or an edge which does not satisfy the

above properties is called a non-manifold vertex or

non-manifold edge, respectively. Examples of non-

manifold vertices and edges are shown in

Figure 2, as

their relative links.

e

(a) (b) (c) (d)

Figure 2: (a) The star of non-manifold vertex v consists of

three components, one of which is a wire-edge and two of

which consist of triangles; (b) The link of v for the

example in (a); (c) The star of non-manifold edge e

consists of three triangles; (d) The link of e for the

example in (c) consists of three vertices.

The fundamental characterization of a three-

dimensional object is given by its Betti numbers β

and β

(Agoston, 2005). Specifically, β

is the

number of connected components, β

is the number

of 1-cycles, and β

is the number of 2-cycles.

Intuitively, the 1-cycles define the independent

tunnels, or handles, while the 2-cycles are parts of

the object which enclose empty space (voids). A 2-

cycle C in a triangle-segment mesh  can be viewed

as an edge-connected component of  without

boundary, such that the three-dimensional region

enclosed by C is connected, that is, every two points

in the region can be joined by a curve which does

not intersect any simplex of .

Figure 3(a) shows an

example of an object with one 1-cycle and one 2-

cycle (the cube), while

Figure 3(b) shows an example

GRAPP 2010 - International Conference on Computer Graphics Theory and Applications

of an object with two 2-cycles (the two cubes

sharing a face). The Betti numbers are topological

invariants and are independent of the way the object

boundary is discretized.

On the other hand, the quantity β

-β

+β

, which

is called the Euler characteristic of the object, is

related by Euler Poincare’s formula

n – e + f = β

– β

+ β

(1)

where n is the number of vertices, e the number

of edges, and f the number of triangles in the

triangle-segment mesh discretizing the object

boundary.

(a) (b)

Figure 3: (a) An object formed by one 2-cycle and one 1-

cycle; for this object: β

=β

=1); (b) an object with two

2-cycles, each of which is the interior of a parallelepiped:

for this object: β

=1 β

=0 and β

=2).

3 RELATED WORK

In this Section, we review some related work on

topological segmentation of non-manifold shapes

and on the computation of topological invariants.

In the literature, several techniques have been

proposed for segmenting the boundary of a 3D

manifold shape, see (Shamir, 2006) for a survey.

Such techniques try to decompose an object into

meaningful components, i.e., components which can

be perceptually distinguished from the remaining

part of the object. Some methods have been

developed in CAD/CAM for extracting the so-called

form features, like protrusions, depressions or

through-holes, which produce a boundary-based

decomposition of a 3D object guided by geometric,

topological and semantic criteria, see (Shah, 1991)

for a survey.

Much less work has been done on decomposition

of non-manifold shapes. A common approach to

represent a non-manifold shape consists of

decomposing it into manifold components. Some

techniques have been proposed in the literature for

decomposing the boundary of regular non-manifold

3D shapes (called r-sets), i.e., for non-manifold

shapes which do not contain dangling faces or edges,

which are described by their boundary. In

(Falcidieno and Ratto, 1992), the idea of cutting a

two-dimensional non-manifold shape into manifold

pieces is exploited to develop compression

algorithms. In (Delfinado and Edelsbrunner, 1995;

Desaulnier and Stewart, 1992), a representation

scheme based on the decomposition of an r-set into

its manifold parts. In (Rossignac and Cardoze,

1999), a decomposition algorithm for non-manifold

shapes is presented which minimizes the number of

duplications introduced by the decomposition

process. In (Pesco et al., 2004) the authors propose a

decomposition of a 2D cell complex based on a

combinatorial stratification of the complex, inspired

by Whitney stratification and a set of topological

operators for manipulating it.

In (Delfinado and Edelsbrunner, 1995) an

incremental algorithm is proposed for computing the

ranks of the homology groups (Betti numbers) for

simplicial 3-complexes embeddable in the three-

dimensional Euclidean space R

The basic idea is to build a complex by adding

one simplex at a time, and at each step the Betti

numbers are updated to reflect the changes in the

homology groups.

The authors discuss implementations of the

algorithm that run in time O(n(n)) and O(n), where

n is the number of simplexes in the complex.

(Dey and Guha, 1998) present an approach to

compute the generators of the homology groups for

compacted triangulated 3-manifolds in R

. Then,

they show that this approach can be applied to

arbitrary simplicial complexes in R

, after thickening

the complex to produce a 3-manifold homotopic to

it. Finally, a classical algebraic approach to the

computation of homology groups is discussed in

(Gueziec et al., 1998).

4 THE TS LIBRARY

We recall that a triangle-segment mesh is encoded as

a Triangle-Segment (TS) data structure. The TS data

structure is the non-manifold extension of the

indexed data structure with adjacencies, which is the

most common representation for triangle meshes. It

is compact and scalable and it supports efficient

navigation algorithms (De Floriani et al., 2004). The

TS data structure encodes the triangles, wire-edges

and vertices of the mesh. The wire-edges and the

triangles are represented by encoding relations wire-

edge-vertex and triangle-vertex, respectively.

 For each wire edge e, wire-edge-vertex relation

associates the two extreme vertices with edge e;

 For each triangle t, the triangle-vertex relation

associates triangle t with its three vertices (see

TOPMESH - A Tool for Extracting Topological Information from Non-manifold Objects

Figure 4 (a)).

The relation among a triangle t and its edge-

adjacent triangles may involve an arbitrary number

of triangles along each edge of t, since an edge can

be non-manifold.

For compactness, the TS data structure stores, for

each edge e of a triangle t, the two triangles that are

immediately preceding and succeeding t around e,

when the triangles incident at e are sorted in counter-

clockwise order around e, as shown in

Figure 4 (b)).

We call such relation the partial triangle-triangle-at-

edge relation The counter-clockwise order of an

edge e is defined by considering one of the two

possible orientations of e and by applying the right-

hand rule with the thumb pointing consistently with

vu, if e = (v, u) and the orientation vu is chosen

for e. If edge e is an interior manifold edge, the

predecessor and the successor are identical and

consist of the only other triangle t′ which share e

with t (see

Figure 4 (c)). When there is no triangle

adjacent to t along e, the predecessor and the

successor of t are both t.

(a) (b)

Figure 4: Topological information encoded in the TS data

structure for a triangle t (a), for an edge e (b) (c) and for a

vertex v (d) (e).

To simplify the task of navigating the mesh

around a vertex, the TS data structure encodes for

each vertex v (see

Figure 4 (d-e))

(i) the vertex-triangle relation, which

associates with v one triangle for each edge-

connected component in the star of v;

(ii) the vertex-wire-edge relation which

associates with v all the wire edges incident at

vertex v.

All the topological relations among the entities in

the TS data structure (vertices, wire-edges and

triangles) can be extracted in time linear in the

number of entities involved in the relations (De

Floriani et al., 2004). The source code of the TS

data structure library is freely available in an online

tool repository (AIM@SHAPE, 2007).

5 BASIC FUNCTIONALITIES

We present here the basic functionalities of

TopMesh, and describe the algorithms for extracting

the topological information on which it is based.

TopMesh extracts the following topological

information from an object whose boundary is

discretized as a triangle-segment mesh:

 non-manifold singularities: non-manifold

vertices, non-manifold edges and wire-edges;

 components with various degrees of

connectivity, namely connected components,

wire-webs, which are maximal connected

components formed by the wire-edges, and edge-

connected components, which describe the parts

of the object which do not contain non-manifold

vertices;

In the following subsections we show how,

relying on the TS data structure, TopMesh extracts

the information listed above. In Section 6, we

describe the computation of the Betti numbers, and

specifically, we propose an algorithm for computing

the number of 2-cycles.

5.1 Extracting Non-manifold

Singularities

We consider a triangle-segment mesh encoded in a

TS data structure. Wire-edges are explicitly encoded

in the TS and thus their retrieval and counting is

trivial.

A non-manifold vertex v is identified by

counting the number of connected components in the

link of v. This is performed by considering the

vertex-triangle and vertex-wire-edge relations at v

encoded in the TS data structure: v is manifold if

only one of the two relations is not empty, and

(i) if the vertex-triangle relation is not empty,

and it consists only one triangle; or

(ii) if the vertex-wire-edge relation is not

empty, and it consists of either one or two wire-

edges.

Otherwise vertex v is non-manifold.

A non-manifold edge e is detected by

considering a triangle t incident at e and checking

whether t has a predecessor and successor in the

triangle-triangle-at-edge relation of t at e in the TS

data structure, which are different. Otherwise edge e

is manifold.

5.2 Extracting Connected Components

and Wire-webs

Connected components of a triangle-segment mesh

GRAPP 2010 - International Conference on Computer Graphics Theory and Applications

 are identified by applying a connected component

labeling algorithm to . This is performed as a

breadth-first traversal in the TS data structure. The

traversal of each connected component starts at an

arbitrary unvisited vertex v of , and visits all

triangles in the vertex-triangle relation encoded in

the TS data structure, and all the wire-edges in the

vertex-wire-edge relation at v. Then, all the vertices

which are bounding such triangles and wire-edges

are considered, and if they are not visited, they are

inserted in a queue Q.

The traversal continues by extracting the first

vertex in the queue and considering it as current

vertex. A connected component is completely

traversed when the queue is empty. By repeating this

traversal process until no unvisited vertex is left, all

the connected components are retrieved.

Wire-webs are connected components formed

only of wire-edges. To detect and count such

components, a traversal similar to the one described

above is performed, but, at each vertex v, only the

incident wire-edges are considered.

The edge-connected components are computed

by considering the sub-mesh ′ obtained from the

original one  by eliminating all the wire-webs. The

edge-connected components correspond to

connected components of the dual graph

representing ′. In the dual graph, the nodes

correspond to the triangles of ′ and the arcs to the

edges shared by two or more triangles.

Edge-connected components are extracted from

the TS data structure starting from an arbitrary

triangle t of ′. For each edge e of t, all the triangles

incident at e are extracted, marked as visited and

inserted in a queue Q. Then, the first triangle in Q is

extracted and the traversal is repeated from such

triangle.

An edge-connected component is completely

traversed when Q is empty. By repeating this

traversal process until no unvisited triangle is left, all

the edge-connected components are found.

It can be easily seen that all the above algorithms

have a time complexity which is linear in the size of

the mesh.

6 COMPUTING BETTI

NUMBERS

In this Section, we show how to compute a

topological signature of a 3D object based on its

Betti numbers β

, β

and β

. Recall that β

is the

number of connected components, β

is the number

of 1-cycles, and β

is the number of 2-cycles.

Given a 3D object represented as a triangle-

segment mesh encoded with the TS data structure, β

can be computed as the total number of the

connected components, β

needs to be computed by

extracting oriented sub-meshes of the edge-

connected components which enclose voids. Finally,

is obtained from Euler-Poincare’s formula by

computing the number of vertices, edges and

triangles in the mesh. Thus, the main step, here, is to

compute β

In the following subsections, we first provide

basic definitions and successively we present the

algorithm for computing the number of 2-cycles.

6.1 Oriented and Folded Surfaces

The algorithm for computing the Betti number β

a triangle-segment mesh  is based on the concepts

of triangle sides, oriented surfaces and folded

surfaces.

A triangle t has two triangle sides (with opposite

normals) defined by the two possible orderings of its

vertices. Each triangle side induces an orientation on

each of its edges. In

Figure 5 (a) an example is shown

where the edge connecting vertices a and b has

orientation ab on one triangle side, and ba on

the opposite side. Two triangle sides are adjacent at

their common edge e if one is reachable from the

other by crossing e.

For each edge-connected component  in the

mesh, we can extract connected collections of

triangle sides for the triangles in , that we call

oriented surfaces. An oriented surface is a maximal

edge-connected set of triangles sides, such that, for

each pair of triangle sides s′, s′′ in the set sharing an

edge e, s′ is the successor of s′′ according to one

orientation of e, and s′′ is the successor of s′

according to the opposite orientation of e. In other

words, the two triangle sides that belong to two

different triangles, which share a common edge,

induce an opposite orientation on their common

edge (see

Figure 5 (b)).

To clarify the definition of oriented surface, let

us consider the example shown in

Figure 5 (c).

Imagine an ant walking on the upper side of triangle

t (that is T

It reaches the triangle side U

by crossing edge

(a, b). It reaches the opposite side of t (T

) by

crossing edge (a, c), because (a, c) is a boundary

edge. An oriented surface is basically the surface

formed by all the triangle sides that are reachable by

a walking ant.

TOPMESH - A Tool for Extracting Topological Information from Non-manifold Objects

An oriented surface may enclose a void. An

oriented surface S with boundary, such that for each

triangle side s that belongs to S the other side s′′ of

the same triangle also belongs to S, cannot enclose

any void.

a

b

a

b

c

a

b

c

a

(a) (b)

a

b

c



U



2



(c)

Figure 5: (a) Example of the two possible triangle sides for

a triangle t (b) example of adjacency between two triangle

sides. (c) example for an oriented surface. An oriented

surface is basically the surface formed by all the triangle

sides that are reachable (via adjacency) by a walking ant.

We call such oriented surface a folded surface. If

an edge-connected component  consists of only one

oriented surface, it must be a folded surface.  may

consist of two surfaces, as it is always the case if it

contains only manifold edges.

For instance, if  is a triangulated sphere, it

consists of two oriented surfaces, each of which

corresponding to an orientation of the surface of the

sphere. If  consists of two triangulated cubes

sharing a face, then there are three oriented surfaces

associated with it, namely the two cubes with

normals pointing towards the inside and the oriented

surface consisting of the union of the two cubes with

the normals pointing towards the external empty

space (see

Figure 6).

It can be easily seen that, if we consider an edge-

connected component  of the original mesh, there

exists exactly one oriented surface in the set of

oriented surfaces associated with it, that we denote

out

, such that the normals to the triangle sides

forming it point towards the outside space.

out

defines a cycle of triangles which can be

expressed as the linear combination of the other

oriented surfaces associated with . Thus, unlike the

other oriented surfaces in , S

out

does not define a 2-

cycle, since it contains the remaining oriented

surfaces.

Note that, if  consists of one folded surface, this

latter does not define a 2-cycle as well. Since there is

exactly one oriented surface for each edge-

connected component which does not define a 2-

cycle, we have that the number of 2-cycles in the

model is equal to the total number of oriented

surface minus the number of 1-connected

components.

(a)

(

)

(

)



Figure 6: (a) Two triangulated cubes sharing a face; (b) the

oriented surface with normals pointing towards the

external space; (c) one of the two oriented surfaces with

normals pointing towards the enclosed void.

6.2 Algorithm

The oriented surfaces in an edge-connected

component  can be computed by defining a

di-graph G

on the collection of the triangle sides of

, that we call the oriented adjacency di-graph of .

In G

, the nodes are the triangle sides, while there is

a directed arc from a node s′ to a node s′′ if and only

if the two sides s′ and s′′ share an edge e, and side s′′

is the successor of s′ at e. Note that graph G

may

contain parallel arcs oriented in the opposite

direction since two sides of the same triangle can be

the successors of each other at a boundary edge.

Figure 7 (a)-(c) show three examples of oriented

adjacency di-graphs for (a) three triangles sharing a

non-manifold edge, (b) two triangles sharing a

manifold edge and (c) a triangle with a boundary

edge.

Each strongly connected sub-graph of the

oriented adjacency di-graph G

defines an oriented

surface. A strongly-connected component of graph

is any maximal subgraph H

of G

such that any

node s′′ in H

can be reached from any other node s′

of H

through a directed path. If we consider all the

triangle sides corresponding to the nodes in the

strongly-connected component H

, they form a

maximal edge-connected set of triangle sides such

that any pair of triangle sides s′ and s′′ are connected

through a path on the complex formed by edge-

adjacent triangles which are consistently oriented.

Thus, they form an oriented surface.

GRAPP 2010 - International Conference on Computer Graphics Theory and Applications

We do not actually build the oriented adjacency

di-graph, but we simulate it on the TS data structure.

To this aim, we label each triangle t in the TS data

structure with two bit flags.

(a) (b) (c)

Figure 7: (a) Three triangles sharing a non-manifold edge

e and the corresponding di-graph; (b) Two triangles

sharing a manifold edge e and the di-graph at e; (c) A

triangle with a boundary edge e and the di-graph at e.

The first bit flag corresponds to the side of t

defined by the order of the vertices of t as the they

are encoded in the triangle-vertex relation, the

second one to the opposite side of t. A bit flag is

equal to 0 if the corresponding side has not been

visited, it is equal to 1 otherwise. The traversal of

the directed arcs emanating from a node s′ in the

oriented adjacency di-graph corresponds to finding

the successors of triangle side s′ in the TS data

structure along its three edges which have an

orientation consistent with s′. Let us consider an

edge e of s′ and let s′′ be the successor of s′ in the

oriented surface (and, thus, (s′, s′′) is an arc in G

We consider the triangle t corresponding to s′. Let

the orientation of e = (v, u) in the TS data structure

be vu. We consider relation triangle-triangle-at-

edge of t at edge e, which returns the pair (f

, f

where f

is the successor of t in counter-clockwise

and f

is the successor of t in clock-wise order,

according to the orientation of e.

Now, s′′ will be a side of f

chosen in such a way

that s′′ induces an opposite orientation on edge e

with respect to the one induced by s′. Note that, if e

is a boundary edge, than f

=t and s′′ is the other

side of t with respect to s′.

Also, since the algorithm performs a visit on the

di-graph G

and the number of nodes in G

proportional to the number of triangles and edges in

the input mesh, the algorithm complexity is linear in

the size of the mesh.

7 DISCUSSION

TopMesh is a command line tool written in C++. It

has been tested on several data sets on a dual-core

2.66GHz CPU, 1GB RAM WinXP platform.

In this Section, we show two examples of results

of the application of TopMesh on non-manifold

objects.

Figure 8(a) shows a table fan. It has a wire-web

component (see Figure 8 (b)) and an edge-connected

component (see

Figure 8 (c)). Figure 8 (d) shows the

table fan connectivity at non-manifold edges. The

geometrical and topological data extracted by

TopMesh for this model are summarized in Table 1.

TopMesh also extracts the sub-meshes of the input

meshes forming connected components, wire-webs

and edge-connected components and also the

oriented surfaces generated in the process of

computing β

(a) (b)

Figure 8: A non-manifold model representing a table fan

(a). This model has a single wire-web in (b) and a single

edge-connected component in (c). In (d) a zoom showing

the non-manifold edges.

Figure 9 (a) shows a non-manifold model

representing a chandelier. It has a wire-web

component (see Figure 9 (b)) and 10 different edge-

connected components (two of them are shown in

Figure 9 (c)-(d)). Figure 9 (e) shows the chandelier

connectivity at a non-manifold edge., The

topological data extracted by TopMesh are

summarized in Table 2.

TOPMESH - A Tool for Extracting Topological Information from Non-manifold Objects

Table 1: Topological information extracted by TopMesh

from a non-manifold triangle-segment mesh representing a

table fan.

Property Value

Number of triangles 380

Number of Vertices 325

Number of edges 846

Number of wire-edges 272

Number of non-manifold vertices 145

Number of non-manifold edges 4

Number of wire-webs 1

Number of connected components (

)

Number of edge-connected components 2

Number of 1-cycles (

)

143

Number of 2-cycles (

)

(a)

 (b) (c)

 (d) (e)

Figure 9: A non-manifold model representing a chandelier,

which has a one wire-web (b) and 10 different edge-

connected components - two of them in (c) and (d). In (e)

chandelier connectivity at a non-manifold edge.

Table 2: Topological information extracted by TopMesh

from a non-manifold triangle-segment mesh representing a

chandelier.

Property Value

Number of triangles 18304

Number of Vertices 9242

Number of edges 27600

Number of wire-edges 136

Number of non-manifold vertices 80

Number of non-manifold edges 264

Number of wire-webs 1

Number of connected components (

)

Number of edge-connected components 10

Number of 1-cycles (

)

Number of 2-cycles (

)

A web page presenting TopMesh and showing

additional results, has been prepared and it is

available at the following URL: http://ggg.disi.

unige.it/topmesh/. On the web page the reader can

eventually find additional material and images of the

results.

8 CONCLUDING REMARKS

We addressed the problem of extracting topological

characteristics from non-manifold 3D shapes

containing parts of different dimensions, and

discretized as simplicial complexes.

We presented TopMesh as a tool for the

extraction of topological information by analysing a

3D object represented as a triangle-segment mesh

encoded in the TS data structure. We showed how,

by using the relations of the TS, we are able to

extract topological information of the input mesh

and we provived an algorithm for computing the

topological 3D object signature via the Betti

numbers.

Actually, TopMesh is being integrated as a core-

module in a Semantic Web system (Papaleo et al.,

2007; De Floriani et al., 2007), for inspecting 3D

shapes and for structuring and annotating such

shapes according to ontology-driven metadata. This

system has been designed to support researchers in

reasoning on digital shapes and in improving their

knowledge about multimedia data available on the

web.

As a further activity we want to extend the tool

for performing a semantic-oriented decomposition of

GRAPP 2010 - International Conference on Computer Graphics Theory and Applications

a shape. This decomposition could be applied, for

example, for classifying form features in simplicial

shapes obtained as idealization of CAD models in

the product analysis phase. Also, the decomposition

can be the basis for a semantic-based annotation of

complex 3D objects. We plan, also, to extend

TopMesh in order to manage large non-manifold

meshes using spatial indexing techniques.

ACKNOWLEDGEMENTS

This work has been partially supported by the

MIUR-FIRB project SHALOM under contract

number RBIN04HWR8, by the National Science

Foundation under grant CCF-0541032, and by the

MIUR-PRIN project on Multi-resolution modeling

of scalar fields and digital shapes. The authors

would like to thank May Huang for the support in

the implementation

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TOPMESH - A Tool for Extracting Topological Information from Non-manifold Objects