A 3D Object Retrieval System Based on Multi-Resolution Reeb Graph

Ding-Yun Chen and Ming Ouhyoung Department of Computer Science and Information Engineering,

National Taiwan University, Taipei, Taiwan E-mail: {dynamic, ming}@cmlab.csie.ntu.edu.tw

Abstract This paper proposes a 3D model retrieval

system which extended the work of Hilaga in 2001.

We apply the pre-processing stage to 3D models in

practical use. The demo system with over 450 3D

models from the Net is on the web page:

http://3dsite.dhs.org/~dynamic, and can also be used

in PocketPC with wireless LAN card. There are 445

various models in our database.

1. Introduction In this decade, multimedia data, which usually

doesn’t need any text to represent, grow rapidly.

Content-based retrieval for multimedia data becomes

more and more important. In order to communicate

with people’s information, the MPEG group aims to

create MPEG-7 international standard, also known as

“Multimedia Content Description Interface”, for the

description of the multimedia data, including image,

video, audio, 2D shape and 3D object [8]. 3D object

retrieval research is active now, because the

technique of 3D modeling and digitizing tools is on a

progressive improvement.

In the last few years, several articles have been

devoted to the study of 3D object retrieval. Cyr and

Kimia [5] proposed an aspect-graph approach. They

generate a set of 2D silhouette for each 3D object, and

then measure the similarity between two views by 2D

shape similarity metrics. Kolonias et al. [3] proposed

aspect ratio, a binary 3D shape mask and set of paths

outlining the shape of the 3D object for matching.

Paquet and Rioux [7] presented an approach for 3D

models retrieval using the distribution of moment,

normal, cord, color, material and texture. Zhang and

Chen [2] propose a 3D model retrieval system using

volume-surface ratio, aspect ratio, moment invariants

and Fourier transformation coefficients. Elad et al. [6]

apply relevant feedback to 3D object retrieval, which

uses moments as features. Osada et al. [4] propose and

analyze a method for computing shape signatures for

arbitrary 3D polygonal models. Hilaga et al. [1]

propose a technique in which similarity between

polyhedral models is quickly, accurately, and

automatically calculated by comparing the skeletal and

topological structure. The structure decomposes 3D

model to a one-dimensional graph structure. The graph

is invariant to translation, rotation and scaling, robust

against connectivity changes, and resistant against

noise, certain changes due to deformation.

Our system is based on the research of Hilaga,

which is one of the best ideas among the previous

works. Fig. 1 shows the flow chart of our system

when querying by a 3D object. The last two stages

are the same with Hilaga, please refer to [1]. Chapter

2 details the pre-processing stage in our system.

2. Pre-Processing of 3D Object This stage is designed for accurately and fast

getting the search key. There are four steps in this

stage: merging vertices, merging parts, re-sampling

and adding short-cut edges. The first two steps solve

practical problems when many models are used. The

last two steps are modified from the approach of

Hilaga [1].

Fig. 1 The flow chart of our system.

The same vertices may be shared in duplicate at

adjacent triangles for 3D models. The first step is to

merge these vertices. The way we used is to sort the

vertices according to the coordinate, and then check

the adjacent vertices. The time complexity is O(nlgn),

since we use the heapsort algorithm.

3D models are saved as different parts in many

cases for easy editing and animation. The approach

we used is to construct the search key by the whole

model rather than some separated parts. Therefore,

we add edges to connect different parts. The

approach is similar to the first step. The coordinate of

each vertex is quantized first, and an edge is created

to connect the same quantized value for the separated

parts. The time complexity is also O(nlgn).

For more accurate calculation of the search key,

each triangle has to be re-sampled. Each triangle

splits into smaller triangles until all edges are less

than a threshold. To speedup re-sampling time in all

cases, each triangle is split at once. There are three

cases for all triangles. In the first case, there is only

one edge in triangle larger than threshold, just split

into two triangles. In the second case, if two edges in

a triangle are larger than threshold, the two edges are

averagely split into many segments. Re-sample by

connecting the new vertices from the two edges. In

the third case, if all three edges in triangle are larger

than threshold, select the two largest edges and split

into many triangles first, then each triangle

recursively is split using the second case.

The fourth step is adding some edges called

"short-cut edges" to models. The purpose of short-cut

edges is used for accurately calculating the search

key. The short-cut edges make the distance of two

vertices in adjacent triangles to be straightforward

through the 3D surface, as shown in Fig. 2 (a). There

are two cases of triangles. In the first case, if a

triangle have two same vertices with other triangle.

Take Fig. 2 (b) as an example, triangle 1 and 2 share

vertex c and d, and then get the angle ace, which

is sum of acd and ecd. If the angle is less than

180 degree, add a short-cut edge between vertex a

and e. Alternative can start from the angle ade.

The angle is less than 180 degree means that edge ae

will inside the polygon aced. The distance of

short-cut edge ae can be calculated by the following

formula. Given the length of two edges (x, y) and

their angle (w), then the length of third edge is:

)cos(222 wyxyx ⋅⋅⋅−+ .

(a) (b)

Fig. 2 The short-cut edges make the distance of two vertices in adjacent triangles to be straightforward

through the 3D surface.

In the second case, if some triangle have only

one the same vertex with this triangle, and if there is

a triangle adjacent to the two triangles. For example,

triangle 1 and 3 share vertex d, and triangle 2

adjacent to them. Then get the angle edb, which is

2

1

3

e

d c

a b

Input a 3D model

Merge vertices

Merge different parts

Re-sample 3D model

Add short-cut edges

Geodesic distance

Create MRG

Compare models

Show results

Stage 1: Pre-processing is used for accurately and fast calculating the search key (Chapter 2)

Stage 2: Calculate the search key MRG for the 3D model

Stage 3: Compare the MRG to all models in the database

sum of edc, cda and adb. As similar to case

one, if the angle is less than 180 degree, add a

short-cut edge between vertex b and e.

3. Experimental Results The system is implemented in C language, and

is compiled using gcc in Linux. The system is on the

World Wide Web by using a simple php code. The

web page is http://3dsite.dhs.org/~dynamic. The user

can upload a 3D model and the system will calculate

the MRG for the model, and then compare it with all

models in the database. Alternatively, the user can

select one model in the database. The system

compares it with all other models and the results

show thumb pictures of top similar models. There are

445 various models, downloaded from [10] and [11],

in the database. The average time of comparing two

models is about 0.08 second in a PC with Pentium III

800MHz CPU. Fig. 3 shows results of 3D object

retrieval in our system. In addition, our retrieval

system can be used in PocketPC with wireless LAN

card, as shown in Fig. 4.

4. Discussion and Future Works While the system can work well in many cases,

there exist considerable improvements as to the

following problems: sub-graph matching, partial

matching, MRG doesn't always represent the skeletal

structure, different density of vertices, some vertices

inside the 3D object. To overcome the problems, we

plan to use the hierarchical medial axis to match the 3D

models. The 3D medial axis algorithm based on radial

basis function (RBF) is proposed in our group [9].

5. Conclusion A 3D object retrieval system, which improves

the practical use of Hilaga’s research [1], is proposed.

The pre-processing stage is applied for accurately

and fast getting the search key.

Reference

[1] M. Hilaga, Y. Shinagawa, T. Kohmura and T. L.

Kunii, “Topology Matching for Fully Automatic

Similarity Estimation of 3D Shapes”, ACM

SIGGRAPH, pp. 203-212, Aug. 2001.

[2] C. Zhang and T. Chen, “Efficient Feature

Extraction for 2D/3D Objects in Mesh

Representation”, IEEE International

Conference on Image Processing, Oct. 2001.

[3] I. Kolonias, D. Tzovaras, S. Malassiotis and M.

G. Strintzis, “Fast Content-Based Search of

VRML Models Based on Shape Descriptors”,

IEEE International Conference on Image

Processing, Oct. 2001.

[4] R. Osada, T. Funkhouser, B. Chazelle and D.

Dobkin “Matching 3D Models with Shape

Distributions”, Workshop on Shape-Based

Retrieval and Analysis of 3D Models, Oct. 2001.

[5] C. M. Cyr and B. B. Kimia, “3D Object

Recognition Using Shape Similiarity-Based

Aspect Graph”, International Conference on

Computer Vision, 2001.

[6] M. Elad, A. Tal and S. Ar, “Content Based

Retrieval of VRML Objects – A Iterative and

Interactive Approach”, 2001.

[7] E. Paquet and M. Rioux, “Content-Based

Access of VRML Libraries”, Lecture Notes in

Computer Sciences, Vol.1464, pp. 20-32, 1998.

[8] S. Jeannin, L. Cieplinski, J. R. Ohm and M.

Kim, MPEG-7 Visual part of eXperimentation

Model Version 7.0, ISO/IECJTC1/SC29/WG11

/N3521, July 2000.

[9] F.C. Wu, W.C. Ma and M. Ouhyoung,

“Skeleton Extraction of 3D Objects with

Radial Basis Function”, Technical Report

NTUCSIE 02-01, Dept. of CSIE, National

Taiwan University, Taipei, Taiwan, Apr. 2002.

[10] http://www.3dcafe.com

[11] http://www.3dm-mc.com

Fig. 3 Some results of 3D object retrieval in our system.

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Fig. 4 Our retrieval system can be used in PocketPC with wireless LAN card.