INVARIANT CATEGORISATION OF POLYGONAL OBJECTS

USING MULTI-RESOLUTION SIGNATURES

Roberto Lam

Instituto Superior de Engenharia, Universidade do Algarve, Campus da Penha, Faro, Portugal

J. M. Hans du Buf

ISR-Vision Laboratory, FCT, Universidade do Algarve, Campus de Gambelas, Faro, Portugal

Keywords:

3D Shape matching, Volumetric models, Manifold meshes.

Abstract:

With the increasing use of 3D objects and models, mining of 3D databases is becoming an important issue.

However, 3D object recognition is very time consuming because of variations due to position, rotation, size

and mesh resolution. A fast categorisation can be used to discard non-similar objects, such that only few

objects need to be compared in full detail. We present a simple method for characterising 3D objects with

the goal of performing a fast similarity search in a set of polygonal mesh models. The method constructs,

for each object, two sets of multi-scale signatures: (a) the progression of deformation due to iterative mesh

smoothing and, similarly, (b) the inﬂuence of mesh dilation and erosion using a sphere with increasing radius.

The signatures are invariant to 3D translation, rotation and scaling, also to mesh resolution because of proper

normalisation. The method was validated on a set of 31 complex objects, each object being represented with

three mesh resolutions. The results were measured in terms of Euclidian distance for ranking all objects, with

an overall average ranking rate of 1.29.

1 INTRODUCTION AND

RELATED WORK

One might say that technological developments will

lead us towards using increasingly complex illustra-

tions, i.e., moving from 2D to 3D space. There are

digital scanners which produce 3D models of real ob-

jects. CAD software can also produce 3D models,

from complex pieces of machinery with lots of cor-

ners and edges to smooth sculptures. Very complex

protein structures play an important role in pharma-

cology and related medical areas. Many actors in the

World Wide Web have started to incorporate 3D mod-

els in sites and home pages. As a consequence of

this trend, there is a strong interest in methods for 3D

similarity analysis (Bustos et al., 2005; Tangelder and

Veltkamp, 2007). Similarity analysis is a fast way to

discard many irrelevant objects from a database, i.e.,

before precise object recognition (matching) which

may be very time consuming because of all variations

that may occur: different position (object origin), ro-

tation, size and also mesh resolution.

Similarity analysis does not require precise shape

comparisions, global nor local. Instead, this approach

is based on computing sets of features (FV or feature

vector) of a query object and comparing its FV with

all FVs of known objects in a database. The FVs can

be obtained by a variety of methods, from very sim-

ple ones (bounding box, area-volume ratio, eccentric-

ity) to very complex ones (curvature distribution of

sliced volume, spherical harmonics, 3D Fourier coef-

ﬁcients) (Saupe and Vranic, 2001; Pang et al., 2006;

Sijbers and Dyck, 2002). We mention two approaches

which are related to our own approach. Assfalg et

al. (2006) projected a 3D object onto 2D curvature

maps. This is preceded by smoothing and simpliﬁca-

tion of the polygonal mesh, and ﬁnal retrieval is based

on comparing the 2D curvature maps. Chuang et al.

(1991) and Suzuki (2007) used the fractal dimension

for characterising 3D objects.

The intrinsic nature of the objects may pose

some constraints, and some methods may be more

168

Lam R. and M. Hans du Buf J. (2009).

INVARIANT CATEGORISATION OF POLYGONAL OBJECTS USING MULTI-RESOLUTION SIGNATURES.

In Proceedings of the International Conference on Knowledge Discovery and Information Retrieval, pages 168-173

DOI: 10.5220/0002301401680173

 SciTePress

suitable—and faster—for the extraction of FVs than

others. For example, methods based on spherical har-

monics and 3D Fourier coefﬁcients are not suitable

for concave objects (non-star-shaped), whereas other

methods have problems with open (non-closed) ob-

jects. Some limitations can be solved by combin-

ing two or more methods. However, since many ob-

jects can yield very similar FVs by applying only one

method, i.e., mathematically possibly an inﬁnite num-

ber of objects, normally several methods are com-

bined to achieve the best results.

2 OVERVIEW OF OUR

APPROACH

We use a set of 31 models, each one represented with

four different mesh resolutions. The models were se-

lected from the (AIM@SHAPE, 2008) database. This

database has high-deﬁnition objects which can be

converted to other mesh resolutions by means of one

parameter between 9.9 (max mesh size) and 5.5 (min

mesh size). The models were downloaded in PLY

format and only “watertight” ones—closed, without

gaps and with regular meshes—were selected. Fig-

ure 1 shows a few examples, and Table 1 a list of all

objects with their mesh resolutions: the ﬁrst three res-

olutions are used for creating the training set FVs, the

fourth one as test object for similarity search.

In order to obtain invariance to translation and

scale (size), the models were normalised to the uni-

tary sphere (radius 1.0) after the origin of the models

was translated to the center of the sphere. Rotation in-

variance is achieved by the fact that our FV is global

to the model as proven in (Vrani´c, 2004). Invariance

to mesh resolution is obtained by proper feature nor-

malisations, which will be explained below. We ap-

ply two different but complementary methods in order

to generate two kinds of features for object retrieval.

These are based on mesh smoothing (Section 2.1) and

on dilation and erosion (Section 2.2).

2.1 Mesh Smoothing

Mesh smoothing serves to reduce noise, for example

for decreasing the mesh size by re-triangulation of

planar areas. (Glendinning and Herbert, 2003) used

smoothing of principal components for shape classi-

ﬁcation in 2D. Here, the idea is related to iterative

and adaptive (nonlinear) mesh smoothing in 3D, i.e.,

smoothing in quasi-planar regions but not at sharp

edges (Lam et al., 2001). However, here we simply

apply the linear version which will smooth the mesh

Table 1: All 31 models with their mesh resolutions, the last

resolution was used in similarity search.

N Model Resolutions

1 Blade 6.5; 7.5; 9.9; 8.0

2 Bimba 6.0; 8.5; 9.5; 8.0

3 Block 5.0; 6.5; 8.0; 8.5

4 Bunny 6.5; 7.5; 9.9; 8.0

5 Cow 6.0; 6.4; 9.9; 7.1

6 Cow2 6.0; 7.5; 9.9; 8.9

7 DancingChildren 6.0; 7.5; 9.9; 6.8

8 Dragon 6.0; 8.0; 9.5; 7.7

9 Duck 6.0; 7.5; 9.9; 6.7

10 Eros 6.0; 7.5; 9.9; 6.5

11 Fish 6.0; 7.5; 8.0; 8.0

12 FishA 6.0; 7.5; 9.9; 7.0

13 GreekSculpture 6.5; 7.0; 7.7; 8.5

14 IsidoreHorse 6.0; 7.5; 9.9; 7.0

15 Mouse 6.0; 7.5; 9.9; 7.8

16 Pulley 6.0; 7.5; 9.9; 7.0

17 Torso 6.0; 7.5; 9.9; 7.7

18 CamelA 6.0; 7.5; 9.9; 7.8

19 Carter 6.0; 8.5; 9.5; 7.3

20 Chair 6.5; 7.5; 9.9; 6.9

21 Dancer 6.0; 7.5; 99; 7.7

22 Dente 6.0; 7.5; 9.9; 7.0

23 Elk 6.0; 7.5; 9.9; 7.9

24 Grayloc 6.0; 7.5; 9.9; 7.8

25 Horse 6.0; 7.5; 9.9; 8.0

26 Kitten 6.0; 7.5; 9.9; 7.3

27 Lion-dog 6.0; 7.5; 9.9; 8.0

28 Neptune 6.0; 8.0; 9.5; 7.6

29 Ramesses 6.0; 7.5; 9.9; 8.0

30 Rocker 6.0; 7.5; 9.9; 7.1

31 Squirrel 6.0; 7.5; 9.9; 7.2

at all vertices: it starts by eliminating very sharp ob-

ject details like in- and protruding dents and bumps,

and then, after more iterations, less sharp details. The

sum of the displacements of all vertices, in combi-

nation with the contraction ratio of the surface area,

generates a quadratic function which can characterise

the model quite well.

If V

,i = 1, N, is the object’s vertex list with as-

sociated coordinates (x

), the triangle list T(V)

can be used to determine the vertices at a distance of

one, i.e., all direct neighbour vertices connected to V

by only one triangle edge. If all neighbour vertices

of V

are V

i, j

, j = 1,n, the centroid of the neighbour-

hood is obtained by

= 1/n

∑

j=1

i, j

. Each vertex

is moved to

, with displacement

= ||V

−

||.

The total displacement is D =

∑

i=1

. The en-

tire procedure is repeated 10 times, because we are

mainly interested in the deformation of the object at

INVARIANT CATEGORISATION OF POLYGONAL OBJECTS USING MULTI-RESOLUTION SIGNATURES

169

Figure 1: Examples of models: Squirrel (top), IsidoreHorse

(middle) and GreekSculpture (bottom). Low resolutions at

left and high ones at right.

the start, when there still are many object details,

and more iterations do not add useful information

anymore. Hence, displacements are accumulated by

∑

m=1

with m = 1...10. In order to obtain

invariance to mesh size, in each iteration m the dis-

placement D

is corrected using

:= D

· N

· S

, (1)

with N the total number of vertices, NP

the number

of participating vertices (in non-planar regions which

contributed to the displacement), S

the surface of the

object (sum of all triangles) after the smoothing step,

and A

the ﬁnal, maximum accumulated displace-

ment after all 10 iterations. Then the curve of each

object and each mesh resolution is further normalised

by the total contraction ratio deﬁned by S

(ﬁnal

surface and original surface), and the three curves (10

data points) are averaged over the three mesh resolu-

tions. In the last step, the averaged A

is least-squares

approximated by a quadratic polynomial in order to

reduce 10 parameters to 3. Figure 3 shows represen-

tative examples of curves A

. It should be stressed

that, in contrast to the second method as described be-

low, no re-triangulation of the mesh of the object after

each iteration is done, i.e., the number of vertices—

and triangles—remains the same. Figure 2 shows a

model and the inﬂuence of mesh smoothing.

2.2 Mesh Dilation and Erosion

The second method is based on the estimation of the

3D fractal dimension by applying a sphere as struc-

tural element with increasing radius. A sphere is ap-

plied to the model, its origin being placed at each

vertex. This yields two surfaces: the dilated surface

grows and the eroded one shrinks as a function of

sphere radius, both showing less object detail. In-

stead of computing the fractal dimension, we com-

pute the volume between the two surfaces as a func-

tion of sphere radius in order to obtain characteristic

curves, like the ones described in the previous sec-

tion, for characterising the objects. The growth (in-

creasing radius) of the sphere is related to the mesh

resolution of the model. Therefore, before comput-

ing the volumes, we eliminate vertices which are in-

side the neighbourhooddeﬁned by the sphere with the

radius used in the dilation-erosion process. If ∆L is

the difference between the maximum and minimum

edge lengths of a model, i.e., ∆L = L

max

− L

min

, then

∆R = 0.05∆L. Hence, the radius at iteration m is

= m∆R, which results in volumes V

(the volume

of the original model) andV

(the volume between di-

lated and eroded models at iteration m). For obtaining

invariance to mesh size, we apply

:= V

; m = 1,2,... (2)

As can be seen in Fig. 4, the dilation-erosion curves

are quite similar, for different mesh resolutions, at the

start of the process, but then start to diverge when the

radius becomes too big and noise is introduced to the

model. Therefore we averaged the V

of the three

mesh resolutions and only included in the FV two pa-

rameters: V

and V

. Figure 5 shows the Bimba model

with erosion and dilation.

3 RESULTS AND DISCUSSION

The 31 models listed in Table 1 were used, each

with four mesh resolutions. As explained before,

the ﬁrst three mesh resolutions were used for con-

structing the FV of the model, and the last one was

used for testing. Each model was characterised by

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170

Figure 2: Mesh smoothing applied to IsidoreHorse. From

top to bottom: original and smoothed meshes after 3, 6 and

10 iterations.

7 parameters, 5 from the method described in Sec-

tion 2.1 (surface of original model after normalisation

to unit sphere; contraction ratio after 10 iterations;

3 coefﬁcients of the quadratic approximation of the

Figure 3: Characteristic curves from mesh smoothing of the

Bimba and IsidoreHorse models.

Figure 4: Characteristic curves from mesh dilation and ero-

sion of the Mouse and Squirrel models.

smoothing curves), and 2 from Section 2.2 (volume

of original model after normalisation to unit sphere;

volume between dilated-eroded surfaces after 2 itera-

tions). The FVs of the objects’ test resolutions were

compared with all FVs of the database, and the ob-

jects were sorted by using the Euclidean distance be-

tween the FVs. Table 2 lists the results, starting with

the object with the smallest distance, then the object

with the next smallest distance, and so forth, until the

ﬁfth object. Table 2 shows that in 26 of 31 cases

the correct object was ranked ﬁrst. Similar objects

(IsidoreHorse, 14; CamelA, 18; Horse, 25) were al-

ways ranked at position 1 to 3. The average ranking

rate R = (1/31)

∑

i=1

, where P

is the ranked posi-

tion of object i, is 1.29. This means that the majority

of objects is ranked at position 1 or 2, at least within

the ﬁrst 3 or 5 positions for narrowing a full object

comparison in a big and complex database. It should

INVARIANT CATEGORISATION OF POLYGONAL OBJECTS USING MULTI-RESOLUTION SIGNATURES

171

Figure 5: Mesh dilation and erosion applied to the Bimba

model. From top to bottom: original plus eroded and dilated

models after 4 iterations.

be stressed that, although 26 objects were ranked ﬁrst,

this does not mean that the correct object has been

identiﬁed. The most similar object may have been de-

tected, but in real conditions, i.e., with big and com-

plex databases like protein structures, the search has

been narrowedin order to save time for detailed object

comparisons. Future work involves improving further

the methods described in this paper, but in relation to

detailed object comparisons which may provide addi-

tional parameter models for a similarity search. Most

important is to improve the erosion-dilation method

such that identical curves are obtained for different

object resolutions at larger sphere radii, after which

the relative importance of the individual features and

feature sets can be studied.

Table 2: Results.

N Testing Model Ordered output

1 Blade 1-6-30-22-12

2 Bimba 2-26-31-22-15

3 Block 3-27-9-15-31

4 Bunny 4-7-23-14-25

5 Cow 12-5-6-17-11

6 Cow2 6-12-30-22-1

7 DancingChildren 7-4-23-14-10

8 Dragon 8-31-23-9-2

9 Duck 9-31-15-8-2

10 Eros 18-25-10-14-26

11 Fish 11-6-12-1-30

12 FishA 12-6-1-5-11

13 GreekSculpture 13-28-20-17-5

14 IsidoreHorse 14-25-18-10-26

15 Mouse 15-31-9-22-2

16 Pulley 16-24-19-3-9

17 Torso 17-5-12-20-6

18 CamelA 25-18-14-10-26

19 Carter 19-24-16-8-9

20 Chair 20-17-5-12-1

21 Dancer 11-12-5-6-21

22 Dente 22-26-30-2-6

23 Elk 23-7-4-14-8

24 Grayloc 24-19-16-8-9

25 Horse 25-14-18-10-26

26 Kitten 26-2-22-10-30

27 Lion-dog 27-3-9-15-17

28 Neptune 28-21-5-20-17

29 Ramesses 29-10-14-4-7

30 Rocker 22-30-26-2-6

31 Squirrel 31-15-9-2-22

ACKNOWLEDGEMENTS

We would like to thank the author of Binvox software,

Patrick Min. Binvox was used in order to compute the

volume of the models.

Research supported by the Portuguese Foundation

for Science and Technology (FCT), through the pluri-

annual funding of the Inst. for Systems and Robotics

through the POS Conhecimento Program (includes

FEDER funds), and by the FCT project SmartVision:

active vision for the blind (PTDC/EIA/73633/2006).

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172

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