A Probability-Based Unified 3D Shape Search.

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Y. Avrithis et al. (Eds.): SAMT 2006, LNCS 4306, pp. 124 – 137, 2006.
© Springer-Verlag Berlin Heidelberg 2006
A Probability-Based Unified 3D Shape Search
Suyu Hou and Karthik Ramani
Purdue Research and Education Center of Information Systems in Engineering (PRECISE)
School of Mechanical Engineering
Purdue University, West Lafayette, U.S.A.
suhou@purdue.edu, ramani@purdue.edu
Abstract. We present a probability-based unified search framework composed of
semi-supervised semantic clustering and then a constraint-based shape matching.
Given a query, we propose to use an ensemble of classifiers to estimate the
likelihood of the query belonging to each category by exploring the strengths from
individual classifiers. Three descriptors driven by Multilevel-Detail shape descrip-
tions have been used to generate the classifier independently. A weighted linear
combination rule, called MCE (Minimum Classification Error), is adapted to sup-
port high-quality downstream application of the unified search. Experiments are
conducted to evaluate the proposed framework using the Engineering Shape
Benchmark database. The results have shown that search effectiveness is signifi-
cantly improved by enforcing the probability-based semantic constraints to shape-
based similarity retrieval.
1 Introduction
In recent years, 3D shape search has gained importance as a possible means for shape-
related engineering knowledge reuse by complementing text-based information sys-
tems [1], [2]. Various shape-based descriptors have been developed to support reuse
[3]. However, lower level shape representations generated by these techniques do not
themselves fully reflect the associated engineering semantics such as function or
manufacturing process, thus causing a semantic gap between human understanding
and system interpretation of the shape. As a result, the application of knowledge reuse
using shape-based retrieval is not very effective.
Various studies [4], [5], [6] have been conducted to reduce the semantic gap in
Content-based Search (CBS) by using results obtained from classification. In the en-
gineering domain, it is observed that implicit concurrence relations of geometry cues
exist among engineering parts from some semantic categories. However, the situation
for classifying an engineering model is different from that of multimedia mod-
els/images. There is no unique criterion for classifying engineering models. Even one
engineering model sometimes can be classified into different classes by various stan-
dards [26]. For example, four parts in Fig.1 have different functions thus belonging to
different part families. However they can be classified into the same class by their
look. It is hard to use a binary decision when facing with classifying engineering
models. Therefore, traditional binary classifier is not generally appropriate for the en-
gineering shape-based classification problem. Hence, it is important to choose a type

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A Probability-Based Unified 3D Shape Search 125
of automatic classifier suitable for an engineering context. The classifier ought to be
able to predict the classification with high accuracy and with no exclusion to possible
choices. Thus, the search engine can prune the database for optimal searching.

Fig. 1. Similar engineering model from different classes
In this paper, a strategy to reduce the semantic gap in our 3D engineering model
search system is investigated. The major contribution of this paper is the use of a
probability-based classifier in a unified search framework using both semantic con-
cepts and visual content. This paper addresses the problem of nondeterministic classi-
fication of engineering models, which can facilitate semantic-constrained shape
matching. The interpretation of the probability output is independent of the types of
classifiers. We take advantage of this fact to improve the confidence in decision mak-
ing by classifier combination. The details are elaborated in the following sections.
Section 2 introduces the related work. Section 3 outlines the system architecture. Sec-
tion 4 presents the classifier combination rule which is then employed by the unified
search in Section 5. Section 6 discusses the experimental results and concludes in
Section 7.
2 Related Works
Recent progress in pattern recognition has made the automatic classification of 3D
engineering parts possible by mapping part geometry to engineering semantics. One
way of part classification is to recognize the shape pattern embedded in a class by
supervised learning. One of the advantages of this way is its simple mathematical in-
terpretation of the pattern from a large set of data extracted from shape. As a result,
each class has a general shape pattern encoded in a mathematical configuration with a
limited number of parameters. In [9], a weighted k-nearest neighbor KNN is used for
engineering part classification. The same research group further applies Support Vec-
tor Machines (SVM) to classify the same database and demonstrate that SVM has a
better performance than KNN for their classification problem [8]. In [10], active
learning is used to employ the information from human labeling to annotate the 3D
models automatically. In [11], Bayesian network is used for hierarchical classification.
However, the above applications output a binary decision for classification which is
not appropriate for the engineering context. In addition, their efforts stop at the classi-
fication stage.
Classifier combination has recently been a popular method in various applications
of content-based classification [12], [13], [14], [15].Recent studies in combining

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126 S. Hou and K. Ramani
multiple classifiers for the classification problem has shown proof that the strategy of
taking advantage of various resources outperforms traditional monolithic classifiers
[16],[18]. Among the various approaches, it is popular to use a linear combination of
classifiers that output measurements representing the likelihood of a data belonging to
a class [17]. Linear combination is simple to implement and has shown a superior per-
formance over other methods such as majority vote, product rule, and Borda count
from many experimental and theoretical studies [18], [19], [20], [21]. Even though the
linear combination rules ignore the correlations among different resources, they can
still reach plausible results at low computational cost [18]. Linear combination rules
can be divided into two categories [17]: Simple Average (SA), which is a linear com-
bination with equal weights; and Weighted Average (WA), which includes rules such
as MSE (Minimum Square Error) [22], [23] and MCE (Minimum Classification Er-
ror) [24] to estimate the optimal weights for the linear combination model. The main
idea is that given a classifier, its contribution to the combined prediction of the testing
data is dependent on its performance with the training data. A classifier with better
classification accuracy is considered to have better predication capability and will be
given more weight for the combination model. However, MSE is criticized for its
derivation from regression context rather than for classification considerations [24].
3 System Architecture
In this paper, we use ShapeLab [25] and the Engineering Shape Benchmark (ESB) da-
tabase [26] as the test bed. ShapeLab has already been well developed for the shape-
based search, which provides experimental reference for the proposed research. The
classification criteria of ESB are from the methodology developed by Swift and
Booker for the purposes of cost estimation and process planning [7], thus reflecting
the engineering semantics in this work. Fig. 2 presents the system work flow. First,
training data from ESB is used to finalize the individual classifiers and the linear
combination model as shown inside the left dotted window of Fig. 2. Each shape de-
scriptor corresponds to a specific classifier. Different classifiers which output the

Supervised
Learning
Training
Models
.
.
.
Supervised
Learning
Testing
Models
Probability
Estimation 1
Probability
Estimation n
Combination
Rule
Combined
Probability
Estimation
Classifier 1
Classifier n
Shape Discriptor
1 Extraction
Shape Discriptor
n Extraction
.
.
.
.
.
.
Semantic
Clusters
Shape
Search
Individual ClassifierClassifier Combination Semantic Clustering Shape Search

Fig. 2. System architecture on probability-based unified shape search

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A Probability-Based Unified 3D Shape Search 127
probability estimation of data being classified to a particular class are developed
separately using supervised learning. Meanwhile we exploit the classification output
from the training data to estimate the optimal weights of the combination model used
later for the real application. The process of how to determine the weights will be
discussed in detail in Section 4. After offline initialization, each testing model will go
through the shape descriptor extraction and the estimations by individual classifiers
before reaching the combination stage. The combined estimation serves two purposes.
First, data from the database are deposited into corresponding clusters based on
the estimations. We further apply the resulting clusters to the shape matching. Sec-
ond, the probability-based output determines the share of each cluster for the shape
search, or it can allow the user to disambiguate the classification decision by provid-
ing a prior list [27]. In either case, the system conducts the shape-based retrieval with
preference for some particular clusters of models, thus reducing the semantic gap be-
tween the system retrieval and user expectations. In this paper, we mainly focus on
the combination rule and its downstream application of the unified search without
user interaction.
4 Probability-Based Classifier Combination
The combination rule applied in this paper linearly integrates the individual classifiers
which output probability. Before the proposed combination rule is presented, it is
necessary to introduce the theoretical background on the probability-based linear
combination model.
4.1 Linear Combination Model
Let
possible classes
probability-based classifiers. Each classifier is obtained by some training data.
{(), 1,... }
ii
X x yiN
==
, with
lth classifier
1
( )(( ),...
lllC
D xdxd
=
hood of x being classified to class
j
ω by the lth classifier
∑
.
n
x∈ℜ be the input observation vector. The task is to assign x to one of the C
{...}
C
ω ωω Ω =
. Define
D
1,2,
12
{,,...,}
L
D DD
=
to be an ensemble of
,
n
ix ∈ℜ and
iy ∈Ω . Let
( ))
x
( )
( )
lj
l D x denote the output of the
where (
l
dxP
ω
=
l D for x : | )
x
j
is the likeli-
l D . The estimation satis-
fies
1
[0,1] and 1
C
ljlj
j
dd
=
∈=
Define
( )
DP x
linear combination model under Equation (1) for its simple implementation and prac-
tical use to achieve plausible results.
Decision
( ),...,
D x
Profile
( ))T
D x
(DP) [28] for input data
x
as:
1
(
L CL
×=
where T denotes transpose. In this paper, we adopt the
( ; )
x w
( )
T
com
Dw DP x
=
where
1
1
L
l
l
w
=
=
∑
and (0,1)
lw ∈
.
(1)

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128 S. Hou and K. Ramani
Let
,1,
d
( ; )
x w
(( ; ),...,
x w
( ; ))
x w
com
combination
=
ω from all classifiers.
comcom C
Ddd
=
as the combined estimation using the
==
linear model.
,( ; )
com j
( ),1,...
T
j
x w w DP x jC
, where
1
( )(( ),....
x
( ))T
x
jjLj
DP xdd
is the ensemble of the likelihood of data x being as-
=
is the set of weight factors as-
∑
signed to class
j
1
( ,...)T
L
www
sociated with the L classifiers . The constraint of
1
1
L
l
l
w
=
=
with (0,1)
lw ∈
makes the
combined estimation satisfy the condition that
,
1
( )
x
1
C
com j
j
d
=
=
∑
.Generally, the weight
configuration is estimated by some predefined rules using the training output. The
commonly referred Simple Average (SA) is the linear combination rule un-
der1/
L
=
. In the next section, the process of how to determine the optimal weights
of the linear combination model in Equation (1) is explained.
lw
4.2 Weight Estimation by Adapting MCE
Let
ideal
with
per, we adopt the k
tion. The cross-validation divides the data into K groups. It uses
training and uses the remaining one group to test the classifier obtained from the
training data. This procedure repeats K times for each combination of training and
testing data. The output of the testing result at each run is collected for the weight es-
timation
The combination rule in [24] incorporates the MCE criterion for weight estimation.
MCE is based on the discriminant function of f in Equation (2) to measure how
likely
k
x
ω
∈
is misclassified as another class under the combination rule. In the dis-
criminant function, ( ; )0
f x w ≤
implies a correct decision and ( ; )
misclassification. The system then obtains the optimal weights by minimizing the
overall objective function derived from the discriminant function.
1
( )
b x
( ( ),...,
i
b x
output
1 and ( )
=
( ))
iiC
b x
=
be a C dimensional class index vector representing the
for the
where ,, 1,... ,and
ik
xjkjC
ω
∈≠=
cross-validation to obtain the data for the weight estima-
ith
data
N
ix
( )0
kiji
fold
b xb x
−
=
1,...
i
=
. In this pa-
1
K − groups for
0
f x w >
indicates a
( ; )
f x w
( )max
j k
≠
( ) ,
f x
kjk
f xx
ω= −+∈
.
(2)
The MCE rule is targeted to minimize the overall misclassification error from the
training data. However, the discriminant function in Equation (2) maximizes the
measurement of the right class and minimizes the maximal measurement of the wrong
class in a manner of equal importance. The minimization of the misclassification
error does not necessarily lead to an increase in the measurement of the right class. In
reality, the measurement for the right class is desired to be as big as possible, because
it is tightly related to the error between the ideal output and real estimation. Besides,