A Robust 3D Shape Descriptor based on the Electrical Charge

Distribution

Fattah Alizadeh and Alistair Sutherland

School of Computing, Dublin City University, Dublin, Ireland

Keywords: 3D Model Retrieval, Charge Density Distribution, Shape Descriptor, Non-rigid Objects.

Abstract: Defining a robust shape descriptor is an enormous challenge in the 3D model retrieval domain. Therefore,

great deals of research have been conducted to propose new shape descriptors which meet the retrieving

criteria. This paper proposes a new shape descriptor based on the distribution of electrical charge which

holds valuable characteristics such as insensitivity to translation, sale and rotation, robustness to noise as

well as simplification operation. After extracting the canonical form representation of the models, they are

treated as surfaces placed in a free space and charge Q is distributed over them. Following to calculating the

amount of charge on each face of the model, a set of concentric spheres enclose the model and the total

amount of distributed charge between the adjacent spheres on the model’s surface generates the Charge

Distribution Descriptor (CDD). A beneficial two-phase description using the number of Charged-Dense

Patches for each model is utilized to boost the discrimination power of the system. The strength of our

approach is verified using experiments on the McGill dataset. The results demonstrate higher ability of our

system compared to other well-known approaches.

1 INTRODUCTION

Recent growth in the computer technology has

resulted in an increasing number of 3D models. 3D

scanners and cameras, 3D modelling software,

mobile phones and etc. are among the new

technologies which speed up the creation of these

models. Nowadays, thousands of models are

available in the domain-specific datasets. In

addition, the rapid developments of the internet have

hooked more attractions for retrieving 3D models

based on their contents. On the other hand, due to

the higher complexity of the models, annotating and

retrieving these models using text-based retrieving

systems is a non-trivial task. Consequently,

researches have drawn a particular attention to the

proposing new shape descriptors by which the

models can be searched, indexed and retrieved in a

beneficial manner.

During last decade, several shape descriptors for

model retrieval have been introduced and some of

them have a good retrieval quality (Kazhdan et al.,

2003); (Chen et al., 2003); (Lian et al., 2010). But

defining a robust shape descriptor to enhance the

retrieval quality especially for non-rigid objects and

partial matching is still a challenging area. A typical

constructive shape descriptor should be invariant to

the linear transformations such as the translation,

scale and rotation. Moreover, robustness to noise,

model deformations and simplifications are some

other characteristics which result in boosting the

retrieval ability.

In this paper, we propose a histogram-based

shape descriptor based on the distribution of

electrical charge which describes non-rigid objects

effectively. It is insensitive to the linear

transformations and some other modifications such

as noise and simplification. A two-phase describing

framework is utilized in order to defining models in

a more distinguishable manner.

We organize the rest of the paper as follows:

section 2 mainly dedicated to give a brief summary

of the related works. The proposed approach is

mainly discussed in the chapter 3. Experimental

results are presented in section 4 and finally in

section 5 we discuss our conclusion and the future

works.

2 RELATED WORKS

Research on the 3D model retrieval started less than

213

Alizadeh F. and Sutherland A..

A Robust 3D Shape Descriptor based on the Electrical Charge Distribution.

DOI: 10.5220/0004295502130218

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2013), pages 213-218

ISBN: 978-989-8565-47-1

Copyright

c

2013 SCITEPRESS (Science and Technology Publications, Lda.)

2 decades ago. Since then, dozens of techniques

have been proposed most of which use shape

descriptors to represent the models in an informative

way. Based on the information used, they can be

classified into 4 main categories namely statistic-

based (Histogram-Based), Transform-based, Graph

based and view-based approaches. A beneficial

survey about the aforementioned approaches can be

found in the works by Bustos et al and Tangelder

and Veltkamp (Bustos et al., 2005); (Tangelder et

al., 2008).

Since our proposed descriptor lies in the first

category, in the sequel we provide a brief review of

available methods of statistic-base category.

2.1 Histogram-based Descriptors

In these approaches, a histogram which accumulates

the numerical values of a specific property is used to

represent the model features. Shape Distribution,

Shape Histogram, Extended Gaussian Images (EGI)

and Electrostatic Fields are only to name a few of

these techniques.

The Shape Distribution descriptor (Osada et al.,

2001) contains a set of functions based on geometric

measurements (e.g., angles, distances, areas, and

volumes) using some random points on the surface

of the 3D model. The accuracy of the appropriate

histograms could be altered by changing the number

of random points. Even though D2, one of their

functions, had better retrieval quality than the other

functions, generally speaking, none of the functions

have enough ability for describing 3D models. This

work was extended later by Ohbuchi et al (Ohbuchi

et al., 2003) by using quasi-random sequence of

numbers instead of pseudo-random sequences.

The shape histogram proposed by Ankerst et al

(Ankerst et al., 1999) has been evaluated in the

context of molecular biology and reached good

accuracy and performance. They decomposed the

3D models using one of these three techniques: Shell

model, Bin model and spider-web or combined

model. Their technique is not invariant to rigid

transforms and so they had to do pose-normalization

as a pre-processing step. Also, since the approach

proceeds with voxel data, 3D objects represented by

polygonal meshes need to be voxelised prior to

descriptor extraction.

The Extended Gaussian Image (EGI) is a

spherical histogram in which bins accumulate the

count of the spherical angles of the surface normal

per triangle, usually weighted by triangle area

(Zhang, et al., 2006). It is a histogram that records

the variation of surface area with surface orientation.

Later some extensions of the original EGI;

Complex-EGI and Volumetric-EGI were introduced

to enhance the original EGI especially for

differentiate between convex and non-convex shapes

without any pose normalization (Kang and Ikeuchi,

1997); (Horn, 1984).

Paquet et al (Paquet and Rioux, 1997); (Paquet et

al., 2000) exploited both the geometric features and

photometric properties such as cord, angle, colour,

reflection and texture. Their techniques are easy to

implement but since they only consider the global

property of the model, their proposed approach is

not very discriminative about objects details.

Recently, Mademlis et al employed electrostatic

fields to 3D model retrieval (Mademlis et al., 2008).

They considered the complete voxelised 3D model

as a distribution of electric charge. Changing control

parameters of descriptors enabled them to extract 24

histograms for each 3D model. Despite of robustness

with respect to object’s degeneracies and native

invariance under rotation and translation, their

descriptor is sensitive to non-rigid transforms.

Some other techniques have been proposed to

use histograms for 3D model retrieval such as

utilizing the Probability Distance Function (Akgul et

al., 2009) and distance function by the 3D Poisson

equation (Pan et al., 2011) which in addition to good

retrieval ability, they are robust to shape

perturbation and noise.

The main advantage of histogram-based

approach is their simplicity of implementation.

Almost all of the aforementioned methods are very

straightforward to implement and understand. And if

they are combined with the other methods as a pre-

processing step or active filter they can improve

their retrieval performance.

3 PROPOSED APPROACH

Our motivation for proposing the Charge

Distribution Descriptor (CDD) comes from a famous

fact in physics-electricity which says: “the electric

charges on the surface of conductor tend to

accumulate at the sharp convex areas and disappear

at the sharp concavity areas”.

We treat the 3D model as a conductor placed in a

free space (the space with no electric charge). Then,

a predefined electrical charge Q is distributed on the

surface of the 3D models. The amount of distributed

charges over each face of the model becomes the

descriptor of that face. Figure 1 illustrates the 6 steps

of our proposed approach.

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214

Figure 1: The proposed retrieval system.

To computing the charge distribution on the

triangular faces of 3D models we employed the

Finite Element Method (FEM) technique proposed

by Wu and Levin (Wu and Levin, 1997). Using the

Gauss’s law and conservation-of-charge fact, they

were able to calculate the charge distribution density

on the any arbitrary surface.

Since the charge distribution is calculated

regardless of coordinate systems, it is invariant to

the translation and rotation transformations but it is

not constant during resizing the models. So, we use

the amount of distributed charge (instead of charge

density) on the surface of each triangle. It is

insensitive to scale transformation and simply is

calculated via the underneath formula:

Chrgamnt

i

=chrgDns

i

*TriArea

i

, i=1,...,m (1)

where TriArea

i

and ChrgDns

i

are the area and

charge density of face i respectively and m is the

number of faces on the surface of each model.

Figure 2 shows four different coloured models

based on their charge distribution; the redder areas

indicate the surfaces holding more electrical charge.

As displayed in this figure, the sharper points

located in the convex areas have more electrical

charge than the other parts and vice versa.

Figure 2: Four coloured models from the McGill dataset;

the redder parts specify the denser faces.

3.1 Concentric CDD

In order to describe each model j, the N

s

concentric

spheres are drawn on the centre-of-mass of the

model. The radii of the spheres monotonically

increase to enclose the model entirely. The range of

radii should meet the following criteria:

min(d

j

) < R

k

< max(d

j

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

Here d

j

is the distance of face from center-of-mass

and m is the number of faces on each model. The

sum of charge amount in each layer between two

adjacent spheres is assigned to each layer and the

N

s

-1 dimensional feature vector describes the whole

3D model. Figure 3 shows three different sample

models and their corresponding concentric spheres.

(Here Ns=4).

Figure 3: Three different models and their corresponding

concentric spheres.

It is important to note that the deformation of models

has significant effect on the amount of charge

distributed on the model surface located between

two adjacent spheres. To overcome this problem, we

use the canonical form

1

representation by which

non-rigid shape similarity problem can be mapped

into an easier problem of rigid similarity. To this

end, we utilize the Least-Square technique with the

CAMCOF algorithm. Since both of SAMCOF

algorithm and the geodesic distance extraction are

time consuming tasks, we first simplify all of the

models so that they have 2000 faces using the

MeshLab (MeshLab1.1.0, 2008). Later on in this

paper we show that the proposed descriptor is robust

to model simplification (see figure 6).

After simplification, geodesic distance extraction

and canonical form computation, the amount of

charge on each face of the simplified models is

computed. Figure 4 displays the results for three

different poses of a spectacles model; although the

poses are deferent but their canonical form

presentations and distribution of charge are quite

similar.

1

The canonical form is a bending-invariant representation in

which the geodesic distances are approximated by the Euclidian

ones (Elad-Elbaz and Kimmel, 2003).

OriginalModels

ModelSimplification

Canonical

Form

Extraction

ChargeDistribution

(DescriptionPhase1)

Concentric

Spheres

DensePatch

Extraction

(DescriptionPhase2)

Histogram

Generationand

Comparison

RetrievedModels

ARobust3DShapeDescriptorbasedontheElectricalChargeDistribution

215

Figure 4: (a): Three different poses of a spectacle models,

(b): corresponding canonical forms and (c): distribution of

electrical charge.

3.2 Two-Phase Descriptor

In order to boost the retrieval quality, a two-phase

shape description framework is leveraged. To this

end, we extract the number of High-Density-Patches

(HDP) on the surface of each charged model and

utilize them to calculate the final dissimilarity

between the pairs of models. Each HDP includes a

local maxima point (a surface with higher electrical

charge than its neighbours) and a set of adjacent

faces on the model surface which have the charge

density more than a pre-defined threshold

τ. The

threshold τ is experimentally selected as shown in

Equation (3). Figure 5 shows some extracted dense

patches on the models based on the density

distribution:

τ = 0.3*max(ChrgDns

i

) , i=1,2,…,m (3)

Here m is the number of faces for each model and

chrgDns

i

is the charge density for face i.

Figure 5: Extracted dense patches on the surface of three

different models.

Testing our retrieval ability using the effect of the

HDP numbers for each model, we concluded that,

since the numbers of HDPs for most of the

articulated models are not constant, retrieving the

similar models based on the number of HDPs leads

to the lower ability of finding the similar models. To

solve this problem and balance the effect of HDP

numbers during the matching phase we assigned a

weight λ to each HDP which is defined using the

formula (4):

max( ) min( )

max(# ) min(# )

ChrgDis ChrgDis

HDP HDP

(4)

After calculating the dissimilarity measure between

each pair of models using the original CDD

descriptor, the weight factor λ is applied to extract

the final dissimilarity between two models i and j as

follows:

(, ) (, )

##*

ij

Dis i

j

chr

g

Dis i

j

HDP HDP

(5)

Where ChrgDis(.,.) is the dissimilarity measure

based on original CDD and #HDP

i

is the number of

High-Density-Patches on the model i.

4 EXPERIMENTAL RESULTS

We have tested our approach on the McGill dataset,

which is publicly available on the internet. It

consists of 458 models classified into 19 different

classes. (256 articulated models in 10 classes and

202 non-articulated models in 9 classes). Beside of

the retrieval quality of our descriptor, the robustness

to the simplification and noise are studied.

4.1 Robustness against Simplification

and Noise

As mentioned before, the introduced shape

descriptor is invariant to the linear transformations.

But we verify the robustness of it against some

geometry operations. We use the pictorial

presentation of models to show the effect of

transformations in the distribution of electrical

charge on the surface of the models. Figure 6

illustrates that the CDD descriptor proposed in this

paper is remains stable after the simplification and

noise; The original models in figure 6-(a) are

simplified from 20K faces into 3K in 6-(b). In

addition, a random noise is applied to the boundary

of original models in figures 6-(c).

Comparing the distribution of the electrical

charge for all of these modified models in figures 6

and 7 supports our claim that the CDD descriptor is

invariant to the aforementioned transformations; the

CDD descriptor’s histogram of these modified

models has small variations but they are still quite

similar. The reason for insensitivity to noise and

simplification can be explained as follows: as Wu

a)

b)

c)

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216

Figure 6: The original models and some modifications.

(a): original models, (b): simplified models, and (c): noisy

models.

and Levin charge on each face is contributed to by

all other faces. So, the small boundary changes

which are caused by noise and simplification have

almost no meaningful effects on the density. It is a

great advantage of our approach compared to the

curvature-based approaches (e.g. mean-curvature

and curvature-index); they are considerably affected

by any surface perturbations.

4.2 Algorithm Parameters

We tested several different options for number of

concentric spheres to enclose and describe the

models and observed some evaluation factors for

each one. The evaluation factors such as Nearest

Neighbour (NN), First Tier (FT), Second Tier (ST),

E-Measure, and Discounted Cumulative Gain

(DCG) in the following table shows that 20 spheres

is the best choice for the sphere counts.

4.3 Retrieval Ability

In order to verify the ability of our shape descriptor

the Precision-Recall plot is employed to compare

our system with 6 other well-known approaches.

These approaches are MDS-CM-BOF (Lian et al.,

2010), D2 (Osada et al., 2001), G2 (Mahmoudi and

Sapiro 2009), GSMD (Papadakis et al., 2007), SHD

(Kazhdan et al., 2003) and LFD (Chen et al. 2003).

As mentioned in (Lian et al., 2011), the MDS-CM-

BOF descriptor is one the state-of-the-art approaches

which showed the great ability in the SHREC’11

contest. Furthermore, The LFD had the best quality

comparing to the other 12 descriptors in (Shilane et

al., 2004).

As depicted in figure 8, thanks to specific

matching scheme (the Clock Matching scheme), the

MDS-CM-BOF descriptor is the best one, and our

approach ranked second. The figure shows that our

approach provides the higher retrieving quality than

other 5 approaches by far.

Figure 7: The CDD histograms of models in figure 6.

Figure 8: The Precision-Recall plot for our and 6 different

other methods.

Table 1: Evaluation factors for different number of

concentric spheres in the Concentric-CDD method.

Sphere

Count

NN FT ST E DCG

5

0.7212 0.3414 0.4710 0.4012 0.6337

10

0.8563 0.4940 0.6359 0.4719 0.8172

20

0.8812 0.6052 0.7744 0.5019 0.8461

50

0.7375 0.4803 0.6433 0.4735 0.7968

200

0.4063 0.2641 0.3805 0.2879 0.5868

5 CONCLUSIONS

A robust shape descriptor introduced in this paper

describes the 3D models based on the distributions

0

0,1

0,2

0,3

0,4

0,5

12345678910

OccurrenceProbability

BinNumber

NormalDolphin

NoisyDolphin

SimplifiedDolphin

NormalFourleg

NoisyFourleg

SimplifiedFourleg

0

0,2

0,4

0,6

0,8

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Precision

Recall

D2

SHD

G2

LFD

GSMD

MDS‐CM‐BF

Concentric‐CCD

a)

b)

c)

ARobust3DShapeDescriptorbasedontheElectricalChargeDistribution

217

of electrical charge over the triangular faces of each

model. In addition to the distribution of charge, a

beneficial two-phase description mechanism is also

utilized in order to describe models in a more

distinguishing manner; the number of High-Density-

Patches on each model enabled us to boost the

retrieval quality. Experimental results show that the

proposed descriptor is invariant to the linear

transformations as well as some geometry

operations. In the next step of our work, we try to

adjust our descriptor to support partial matching.

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