Yuki Ohkita, Yuya Ohishi, Takahiko Furuya, Ryutarou Ohbuchi, Non-rigid 3D Model Retrieval Using Set of Local
Statistical Features, IEEE ICME 2012 Workshop on Hot Topics in 3D Multimedia (Hot3D), 2012.

Non-rigid 3D Model Retrieval Using Set of Local Statistical Features
Yuki Ohkita, Yuya Ohishi,

Takahiko Furuya

Ryutarou Ohbuchi

University of Yamanashi
Kofu, Yamanashi, Japan{g09mk004, g11mk008}AT
yamanashi.ac.jp

Nisca Cooporation
Minami-koma, Yamanashi, Japan
snc49925AT_gmail.com

University of Yamanashi
Kofu, Yamanashi, Japan
ohbuchiAT_yamanashi.ac.jp

Abstract—Various algorithms for shape-based retrieval of nonrigid 3D models, with invariance to articulation and/or global
deformation, have been developed. A majority of thesealgorithms assumes that 3D models have mathematically welldefined representations, e.g., closed, manifold mesh. Thesealgorithms are thus not applicable to other types of shapemodels, for example, those defined as polygon soup. This paper
proposes a 3D model retrieval algorithm that accepts diverse
3D shape representations and is is able to compare non-rigid
3D models. The algorithm employs a set of hundreds tothousands of 3D, statistical, local features to describe a 3Dmodel. These features are integrated into a feature vector per
3D model by using bag-of-features approach for efficiency incomparing 3D models and for invariance against articulationand global deformation. Experimental evaluation showed thatthe algorithm performed well for non-rigid 3D model retrieval.

Index Terms— 3D model retrieval; 3D geometricalmodeling; bag-of-words; shape descriptor, articulated model.
I. INTRODUCTION
Three-dimensional (3D) shape model has become amainstream multi-media data type. It used for mechanicaland architectural design, entertainment, medical diagnosis,
and for archaeology. Accordingly, effective and efficientmanagement methods for 3D models, especially via content
based retrieval by their shape, has become quite important.
In this paper, we propose a shape-based 3D model
retrieval algorithm that accepts diverse set of 3D shape
representations, and is invariant to similarity transformation
and articulation and/or global deformation. The algorithm
employs a 3D feature, not an appearance-based feature, sothat the algorithm is able to capture internal as well as
external structure of a 3D model. The algorithm accepts a
3D model represented as a set of oriented point sets. Almost
any surface-based 3D model representations can beconverted into oriented point set by sampling its surfaces by(oriented) points. The algorithm computes hundreds tothousands of Local Statistical Features, or LSFs, from a 3Dmodel. Each LSF is inherently invariant to similarity
transformation. By integrating large number of LSFs into afeature vector per 3D model, the algorithm assumes
invariance against articulation and/or global deformation.
Experimental evaluation has shown that, for an algorithm
that accepts polygons soup models, the algorithm has a very

good, if not the best, retrieval accuracy for non-rigid (i.e.,
articulated) 3D models.
II. RELATED WORK
Requirements for shape-based retrieval of 3D shapemodels vary depending on its application. Oftentimes,
invariance to similarity transformation, that is, acombination of 3DOF translation, 3DOF rotation, and1DOF (uniform) scaling, is required. Most 3D model
retrieval algorithms aim at invariance against similarity
transformation.
Invariance to shape representation may also be important,
as a 3D shape may be represented by using one of many
mutually incompatible 3D shape representations. Recent
mechanical CAD models are probably “solid” models,
which defines 3D volumes embedded in 3D space. A large
portion of 3D models, e.g., those used for movies and games,
however, uses polygon soup and/or (a set of) open manifold
mesh representations. A 3D model may also be defined byusing a set of (unconnected) oriented points placed at theassumed surfaces of a model. While some previous 3Dmodel retrieval algorithms (e.g., [8][1][3][5][4]) could
accept diverse set of shape representation, others are limited
to manifold or watertight mesh (e.g., [10][11]).
Another possible requirement is invariance, against
articulation and/or global deformation. A 3D model in
different pose is often required to be treated as the same orsimilar, as in the case of snake, pliers, or human (e.g.,
Figure 1) in McGill Shape Benchmark [7]. Earlier 3D model
retrieval algorithms (e.g., [8][1][5][3]) use global feature, sothey don’t have invariance to articulation; articulated figures
are recognized as different. More recent such methods as aset of local features [4] or diffusion-based distancecomputed on manifold mesh surface [10][11] to describe a
3D model for invariance to articulation. While [4] accepts
various shape representations, it is unable to consider
internal structure of 3D models as it compares 3D model

Figure 1. Human figures in McGill Shape Benchmark [7].

Yuki Ohkita, Yuya Ohishi, Takahiko Furuya, Ryutarou Ohbuchi, Non-rigid 3D Model Retrieval Using Set of Local
Statistical Features, IEEE ICME 2012 Workshop on Hot Topics in 3D Multimedia (Hot3D), 2012.
based on exterior view of 3D models. Diffusion distancebased algorithms (e.g., [10][11]) on the other hand, arelimited to manifold meshes.
Our proposed algorithm employs a set of 3D local
features to describe a 3D model. It is very tolerant of input
shape representation; the feature can be computed frompolygon soup or oriented point set models. By integrating
the set of local features into a feature vector per 3D model,
invariance to articulation is realized, and the cost ofcomparison between a pair of 3D models is reduced.
II. PROPOSED ALGORITHM
Proposed Local Statistical Feature (LSF) algorithm follows
the steps below to compare 3D shape models. (SeeFigure 2.) The algorithm natively compares 3D models inoriented point set representation. If the 3D models are insurface based representation, e.g., manifold surface or
polygon soup, step 1 is required to convert them intooriented point set models.
1.
2.
3.
4.

If a 3D model to be compare is in surface based
representation it is converted into oriented point set
representation consisting of m points.
Compute a set of LSFs from the set of m oriented
points of the 3D model.
Integrated n LSFs into a feature vector per 3D model.
Distances among feature vectors of a query model anddatabase models are computed. Top matches are
returned as retrieval result.

n feature
centersm oriented
pointsr…
…

…

Compute n LSFs
n feature vectors(LSF histograms)
Integrate features
A feature vector
per 3D model.
(BF- or LC-LSF)P  1  u1 t1  r1 1  u2  t 2  u1  u 2  t 3  .

In the equation, t1 , t 2 , and t3 are vertices of the triangle,
and u1 and u2 are pseudo-random number sequences(PRNS). We use the same equation, but instead of PRNS,
we use low-discrepancy sequence or quasi-random number
sequence (QRNS). The QRNS produces points that are more
uniformly distributed than PRNS. Our implementation uses
Sobol's QRNS [12].
Given a total number of samples per 3D model, a numberof points proportional to area of a triangle is generated forthe triangle. Assume that the total area of the 3D model is S.
Then, a point should be placed area s  S / m. For the ith
triangular facet fi having an area area( f i ) , its number ofpoints qi is computed as follows;(2)qi  floor  area  f i  s  qi 1  .
qi  area  f i  s  qi 1  qi

(3)

Each point i is associated with the position Pi as well as
the surface normal vector of the surface on which the pointis placed at.
B. Computing Local Statistical Features
For each 3D model, n  m feature points are selectedfrom the m oriented points, and n LSFs per 3D model arecomputed at these points. The algorithm selects specified
number n of feature points randomly from the m sample
points. Each LSF is computed using a set of sample points
enclosed in a sphere of radius r centered at the feature point.
The LSF radius r for a 3D model is set relative to the radiusof smallest sphere enclosing the 3D model. For example,
with r=0.5, the LSF sphere of influence is just large enoughto enclose the 3D model.
Assume that the feature center isand its normal vector
with an associated normal
is . Assume also that pointis a point within the sphere of influence of radius
vector. Usingand, a 4-tuple
r of the feature center(, , , ) consisting of a distance  and three values , , related to angles is computed. Here, point is the LSF
is the point otherfeature point (LSF sphere center), andthan the feature point but contained in the sphere of
influence. The four values  , , and are computed asfollows (See Figure 3);  arctan  wn1 , un2  .

Figure 2. A large number of local features are extracted, and they areintegrated into a feature vector per 3D model.

  v n 2
  u   p 2  p1 p 2  p1  .

A. Converting surfaces into oriented points
As described above, if a 3D model is represented as a setof surfaces, the model needs to be converted into oriented
point set model by sampling the surfaces (with normal
vectors) with m uniformly spaced points. First, if the model
contained non-triangular polygons, they are triangulated
prior to point generation.
Osada et al [8] uses the following equation to samples thesurface at position P ;(1)  p2  p1 .
whereu  n1 ,v   p 2  p1   u

 p 2  p1   u ,

(4)
(5)
(6)
(7)
and

w  u  v.
If there are k points within the sphere, a set of (k-1) tuples
are computed for a feature center, and (k-1) each of values , , and are collected into a 4-dimensional joint
histogram. If the histogram has 5 bins for each values of theYuki Ohkita, Yuya Ohishi, Takahiko Furuya, Ryutarou Ohbuchi, Non-rigid 3D Model Retrieval Using Set of Local
Statistical Features, IEEE ICME 2012 Workshop on Hot Topics in 3D Multimedia (Hot3D), 2012.
tuple, resulting LSF feature, if ‘flattened’ into 1D vector, is625 dimensional.
5
LSF is similar to Surflet Pair Relation Histograms(SPRH) [12] by Wahl, et al. LSF differs from SPRH in two
aspects; (1) SPRH is global, while LSF is localized within asphere of influence, and (2) LSF computes a statistics
between the center (“feature point”) and the other points
within the sphere of influence, while SPRH computes astatistics among all the pair of points.
Parameters m, n, and r have impact on retrieval
performance and computational cost. Being a statisticalfeature, one would like to have a large number m of sample
points and a large number n of feature points. But anincrease in m and/or n obviously increases computationalcosts. In the experiments described below, as a compromise,
we will use m=2,048 and n=512.
A small r ensures locality of features, which makes thealgorithm more invariant against global deformation andarticulation. A smaller r also means lower computationalcost. However, a small radius means a small number ofsample points in the sphere. Small number of sample points
results in an under-populated, less accurate histogram,
especially since the histogram has relatively large number ofbins (e.g., 625). A large r would produce a well-populated
histogram, but with a significant increase in cost ofcomputation. Also, reduced locality of the feature means
reduced invariance to global deformation and articulation.
v

Linear Combination Integration: LC-LSF

w

n2

v



w

u

p2

u= n1



to compare among sets, or bags of features [2]. The
approach first extracts a set of local features from an object(e.g., 3D model) to be compared. Each one of these features
is converted, by vector quantization, into a visual word in agiven visual codebook having vocabulary size k. After
vector quantization, the object is represented as a set ofvisual words. The set of visual word is integrated into ahistogram with k bins by counting population of each word.
This histogram is the feature describing the content. In theprocess of integration, location of features in the object isignored.
The visual codebook is learned by clustering of a large
set of local features extracted from the objects (e.g., images)
to be compared. In our implementation, we used k-meansclustering. Each visual word is a center of a cluster.
Depending on the diversity and complexity of objects to becompared, optimal number of words k in the codebook, thus
the dimension of feature vector per image, may vary. For
image recognition and retrieval, k varies from a few hundred
to hundreds of thousands.
The number of vocabulary k impacts both retrieval
performance and computational cost. For a large vocabulary
size k and a large number of samples to cluster, the k-meansclustering for codebook learning would take significant
amount of time. More importantly, cost of vector
quantization, incurred for each new query, would be verysignificant for a large k.

p1

Figure 3. Compute a 4-tuple () from a pair of oriented points.

C. Integrating LSFs into a feature per 3D model
At this point a 3D model is described by a set of hundreds
to thousands of LSF features. Comparing a pair of sets fortheir similarity (or distance) can be quite expensive. Ouralgorithm thus integrate the set (a large number of) offeatures per 3D model into a feature vector per 3D model byusing two approaches; well known Bag-of-Features (BF) or
Bag-of-Words approach, and a simpler, Linear Combination(LC) approach. In the following, a feature vectors per 3Dmodel produced by using BF integration and LC integration
is called BF-LSF and LC-LSF, respectively.

Given a set of local features, linear combination approachcombines the features by simply summing, component bycomponent, the histograms of local features into a single
histogram having same dimension as the local features.
Figure 4 illustrates this integration algorithm. This approach
may be considered as a regressed form of the Bag-ofFeatures approach.
LC-LSF does not require pre-computation of visualcodebook by clustering nor vector quantization. During
retrieval, LC-LSC does not require vector quantization. So
the cost of integration is significantly lower than BF-LSF.
625 bins
Bin-by-bin
summation

625 bins…
…

…
…
LSF features+

LC-LSF feature

Figure 4. Linear Combination (LC) approach for integrating local statisticalfeatures.

Bag-of-features integtation: BF-LSF

No integration: AC-LSF

In the field of object recognition for 2D images, Bag-ofFeatures (BF) approach is one of the most popular algorithm

We compared proposed BF-LSF and LC-LSF with anapproach that does not use integration of LSFs. That is,Yuki Ohkita, Yuya Ohishi, Takahiko Furuya, Ryutarou Ohbuchi, Non-rigid 3D Model Retrieval Using Set of Local
Statistical Features, IEEE ICME 2012 Workshop on Hot Topics in 3D Multimedia (Hot3D), 2012.

D. Computing Distance between 3D Models
Distances between integrated feature vectors x p and x q
produced by LC-LSF or BF-LSF are computed by using oneof the following two distance “metrics”, L1-norm DL1 and(symmetric version of) Kullback-Leibler Divergence (KLD)
DKLD.
m
(8)
D L1 x p , x q    x p ,k  x q ,kk 1

 x p,k 
D KLD  x p , x q     x p , k  x q , k  log  x q k 
k 1
 , 
m

(9)

In case of AC-LSF, a set of features, instead of a single
feature vector, is given per 3D model. The distance between
two models whose sets of features are Xp and Xq, respectively,
is computed by using the following equation;n(10)
D AC  X p , X q    m in d  x p , i , x q , j i 1

1 j  n

In the equation, d  x p , i , x q , j  is a distance among a pairof LSF features computed by using either L1-norm or KLD.
If the distance among a pair of LSF features d  x p , i , x q , j 
in (13) is computed by using L1-norm, the distance among
3D models using AC-LSF is called D AC , L1  X p , X q  . Ifd  x p , i , x q , j  is computed by using KLD, it is called
D AC , KLD  X p , X q .

IV. EXPERIMENTS AND RESULTS
We have conducted experiments to evaluate the proposed
algorithms LC-LSF and BF-LSF for shape-based 3D model
retrieval. They are compared against AC-LSF as well as
SPRH [13] (combined with the surface to point-set model
conversion described in section 2,1), Light Field Descriptor(LFD) [1], Spherical Harmonic Descriptor (SHD) [5], andBag-of-Features Dense SIFT (BF-DSIFT) [4].
Retrieval experiments are performed by using two
benchmark databases. McGill Shape Benchmark (MSB) [7]
is a set of highly articulated (non-rigid), watertight, highpolygon-count yet less geometrically varied/detailed models.
Princeton Shape Benchmark (PSB) [9] contains a set ofpolygon soup models having diverse shape variations.
Figure 5 shows examples of models from the databases.
(a) PSB

MSB consists of 255 models in 10 classes, and includes sucharticulated shape classes as “humans”, “octopuses”, “snakes”,“pliers”, and “spiders”. PSB contains two subsets of 907
models each, “training” set and “test” set. Both training set
and test set contains about 90 classes. Number of classes forthe two sets differs by a few, and while most of the classes
overlap, there are classes that are unique to one or the otherset. We used PSB test set partitioned into 92 classes forevaluation. Each model from the test set is presented as aquery to retrieve models in the test set.
As the numerical performance index, we use R-Precision,
which is a ratio, in percentile, of the models retrieved fromthe desired class Ck (i.e., the same class as the query) in the
top R retrievals, in which R is the size of the class Ck.
For the experiments below, we used the number ofsurface sample points m=2,048 and number of feature points
n=512. Visual codebook for BF-LSF is learned by k-meansclustering from 500k LSF features randomly drawn from all
the LSF features extracted from all the 3D models in a PSB
test set. Training samples are limited to 500k to contain timenecessary for k-means clustering.
A. LSF Radius of Influence and Retrieval Accuracy
In this experiment, impact of LSF radius of influence r
on retrieval accuracy is evaluated. Figure 6 plots the resultsof the experiment.
For MSB, performance peaks exists; the peak for BFLSF exists at r=0.3 and the peak for LC-LSF exists at
r=0.1~0.2. For PSB, on the other hand, no visible peakexists; both for BF-LSF and LC-LSF, the larger the LSF
radius, the higher the retrieval accuracy. These tendencies
can be explained as follows. The non-rigid, highly articulated
models of MSB database prefer bag-of-feature integration of
local (smaller radius) features. PSB database that consists
mostly of rigid 3D models, however, prefers global feature.
The plot also shows that, for the MSB, BF-LSF
significantly outperforms LC-LSF. However, for the PSB,
while BF-LSF consistently outperforms LC-LSF, thediscrepancy in performance is much smaller.
80
75
70
65
R-Precision [%]given n LSF features per 3D model, all pairs of LSF features
are compared among two sets of LSF features. We call thisapproach All pair Comparison LSF (AC-LSF). Thisapproach is computationally very expensive during a searchthrough a database; it requires O(n2) per pair of 3D models
for comparison. For this reason, only a few experiments will
be performed using AC-LSF.

60

BF-LSF (MSB)
LC-LSF (MSB)
BF-LSF (PSB)
LC-LSF (PSB)

55
50
45
40
35
30

(b) MSB

0.1

0.2

0.3

0.4

0.5

LSF radius r
Figure 5. Examples 3D models from Princeton Shape Benchmark (PSB)(a) [12] and McGill Shape Benchmark (MSB) (b) [7].

Figure 6. LSF radius of influence r and retrieval accuracy. (L1-norm for
distance among integrated features.)Yuki Ohkita, Yuya Ohishi, Takahiko Furuya, Ryutarou Ohbuchi, Non-rigid 3D Model Retrieval Using Set of Local
Statistical Features, IEEE ICME 2012 Workshop on Hot Topics in 3D Multimedia (Hot3D), 2012.
Note that the peak in retrieval accuracy may be affectedby other factors than r, most notably number of samples mand number of LSFs n per 3D model. There is interaction
between parameters m, n, and r. For an LSF with good
locality (i.e., small r) to have a well populated histogram, avery large m is needed. Also, a large n is necessary for BF
integration to work well. Thus, ideally, we'd like to haveexperimented with m>>2,048 and/or n>>512. However, due
mostly to computational cost concern, we settled for
m=2,048 and n=512 for the experiments in this paper.
B. Vocabulary size and retrieval performance
In this set of experiments, we explored the effect of
vocabulary size on retrieval accuracy. The result is plotted in
Figure 7 and Figure 8.
Recall that the k-means has a random component(initialization) so that the quality of vocabulary and thus
retrieval accuracy may vary depending on runs. Ideally, for agiven vocabulary size, an average of multiple runs should beplotted. In this experiment, however, retrieval accuracy of anindividual run is plotted, resulting in a plot with randomvariation in R-precision. Nonetheless, the plots for both MSB
and LSB suggest that a vocabulary size of more than a few
thousand words is necessary to achieve a plateau in retrieval
accuracy.

R-Precision [％]

80

0.500

78

0.333

76

0.250

Table 1. Execution timing for MSB benchmark.
MSB
AC-LSF
LC-LSF
BF-LSF

Extract
feature
21min
21min
21min

Integratefeature
n/a
10min
80min

Computedistance>48hrs6s
20s

R-precision
[%]
73.9
72.1
76.5

Table 2. Execution timing for PSB benchmark.
PSB
AC-LSF
LC-LSF
BF-LSF

Extract
feature
75min
75min
75min

Integratefeatureaborted
31min
95min

Computedistance
aborted
52s
186s

R-precision [%]
37.7
40.9

0.200

74
72

0.167

D. Performance comparison with other algorithms

0.143

We compared the retrieval accuracy of proposed LC-LSF
and BF-LSF algorithms with four other algorithms, SPRH
[13], SHD [5], LFD [1], and BF-DSIFT [4] algorithms. We
used our own original implementations of SPRH, LSF, and
BF-DSIFT. Executables for SHD and LFD are downloaded
from respective author’s web sites.
Figure 9 compares these methods for their retrieval
accuracy. The first three, SPRH, SHD, and LFD, are all
global features, while the latter three, BF-DSIFT, LC-LSF,
and BF-LSF employ sets of local features. As expected, thelatter three using sets of local features performed better for
articulated models of MSB. BF-DSIFT and BF-LSF tied forthe 1st place with R-Precision=76%. For rigid yet complex
and diverse models of PSB, view-based local feature BFDSIFT and view-based global feature LFD did better thanthe others including BF-LSF.

0.125

70
0

2000

4000

6000

8000 10000 12000

Vocabulary size
Figure 7. Vocabulary size and retrieval accuracy (MSB).
40

R-Precision [％]Feature extraction and integration are done once per 3Dmodel. To compute an LSF, the naive implementation
searched for points in a sphere of LSF influence by bruteforce linear search. Similarly, to vector quantize an LSF
feature, the naive implementation of BF-LSF searched forthe nearest visual word, i.e., cluster center, by using linear
search. Both of these could have been made much more
efficient, e.g., by using kd-tree and other algorithms for fastnearest neighbor search.
In an experiment to search through a database of 907
models, distance computation among a pair of 3D models is
performed 9072 =822,649 times. While AC-LSF performs
admirably in terms of accuracy, its computational cost is
prohibitive. We aborted the AC-LSF experiment for the PSB
database after a long time. Overall, while BF-LSF costs morethan LC-LSF for the feature integration, gain in retrieval
accuracy would justify the cost.

0.500

38

0.333

36

0.250
0.200

34

0.167

32

0.143

30
1000

2000

3000

4000

0.125

Vocabulary size
Figure 8. Vocabulary size and retrieval accuracy. (PSB)

C. Computational cost
Table 1 and Table 2 show computational costs of a naive
implementation of the algorithm. The timings for feature
extraction and integration are for all database models. The
timings are for processing all the database models and all thequeries of the benchmarks.

E. Comparison using SHREC 2011 Non-rigid 3D
We also compared the performance of BF-LSF withthose of participants in the SHape REtrieval Contest 2011
Non-rigid 3D Watertight Meshes track. Please refer to [6] fordetails of the results. Among 9 entrants of the track, BF-LSF
placed 3rd in Nearest Neighbor (NN) and 6th in the
Discounted Cumulative Gains (DCG) index. Note that anYuki Ohkita, Yuya Ohishi, Takahiko Furuya, Ryutarou Ohbuchi, Non-rigid 3D Model Retrieval Using Set of Local
Statistical Features, IEEE ICME 2012 Workshop on Hot Topics in 3D Multimedia (Hot3D), 2012.
entry by Sipiran, et al that employed HKS [11], performed
worst among the 9 entrants.
While not at the top in terms of retrieval accuracy among
the entrants, the proposed algorithm does have an advantage
other entrants did not have. As the name of the track suggests,
its benchmark database consists of watertight meshes
representing 3D solids. All but one of the algorithms in the
track requires 3D models to be represented as watertightmeshes. (Please note that this is not an issue for the track,since it is a track for “Non-rigid watertight meshes”.)
Consequently, those other algorithms can’t handle PSB that
contains polygon soup models. Our BF-LSF, on the other
hand, does not require the model to be watertight; it canhandle a diverse set of 3D model representations, e.g., thosefound in PSB.
52.5

SPRH

37.4
56.7

SHD

MSB
PSB

40.5
56.9

LFD

45.9
76.4

BF-DSIFT

55.8
72.2

LC-LSF

36.1
76.2

BF-LSF

39.1
0

20

40

60

80

R-Precision [%]Figure 9. Retrieval accuracies of various algorithms.

V. SUMMARY AND CONCLUSION
We proposed a 3D model retrieval algorithm thattolerates articulation and global deformation, in addition to
being able to handle a diverse set of shape representations
including polygon soup and oriented point set. A set of 3D
local geometrical features is integrated into a feature vector
per 3D model to achieve articulation invariance and efficientcomparison among 3D models.
Experimental evaluation using two 3D model retrieval
benchmarks showed that the algorithm is quite effective for
database consisting of highly articulated, yet simpler shapes,that is, McGill Shape Benchmark [7]. For rigid, highly
diverse set of shape models in Princeton Shape Benchmark[10], retrieval accuracy of the proposed algorithm is modest
compared to the other state of the art algorithms.
As a future work, we’d like to evaluate the method using
a benchmark that contains meaningful internal structure. In
such a benchmark, BF-LSF, which captures 3D geometricalfeature, might have an advantage over algorithms that are
based on external appearance of 3D models. We also need to
explore the impacts of various parameters, such as thenumber of sample points m and the radius r of influence of
LSF. We’d also like to experiment with distance metriclearning for improved retrieval accuracy.

ACKNOWLEDGMENT
This research has been funded in part by the Ministry of
Education, Culture, Sports, Sciences, and Technology ofJapan (No. 18300068).
REFERENCES[1] D.-Y. Chen, X.-P. Tian, Y.-T. Shen, M. Ouhyoung, On
Visual Similarity Based 3D Model Retrieval, Computer
Graphics Forum, 22(3), 223-232, (2003).
[2] G. Csurka, C.R. Dance, L. Fan, J. Willamowski, C.
Bray, Visual Categorization with Bags of Keypoints,
Proc. ECCV ’04 workshop on Statistical Learning in
Computer Vision, 59-74, (2004).
[3] P. Daras, A. Axenopoulos, A Compact Multi-view
Descriptor for 3D Object Retrieval, Proc. SeventhInternational Workshop on Content Based Multimedia
Indexing, 115-119, (2009).
[4] T. Furuya, R. Ohbuchi, Dense Sampling and Fast
Encoding for 3D Model Retrieval Using Bag-of-Visual
Features, Proc. ACM Intern’l Conf. on Image and Video
Retrieval 2009 (CIVR 2009), July 8-10, (2009).
[5] M. Kazhdan, T. Funkhouser, S. Rusinkiewicz, Rotation
Invariant Spherical Harmonics Representation of 3D
Shape Descriptors, Proc. Symp. of Geometry
Processing 2003, 167-175 (2003)[6] Z. Lian, et al., Shape Retrieval on Non-rigid 3D
WatertightMeshes,Proc.
Eurographics/ACM
SIGGRAPH Symp. on 3D Object Retrieval 2011, (2011).
[7] McGill 3D Shape Benchmark,<http://www.cim.mcgill.ca/~shape/benchMark/>[8] R. Osada, T. Funkhouser, B. Chazelle, and D. Dobkin,
Shape Distributions, ACM Trans. on Graphics, 21(4),pp. 807-832, October 2002.
[9] Princeton Shape Benchmark,<http://shape.cs.princeton.edu/benchmark/>[10] M. Reuter, F. E. Wolter, M. Shenton, M. Niethammer,
Laplace-Beltrami eigenvalues and topological featuresof eigenfunctions for statistical shape analysis,
Computer-Aided Design, 41(10), 739–755, (2009).
[11] J. Sun, M. Ovsjanikov, L. Guibas, A Concise andProvably Informative Multi-Scale Signature Based on
Heat Diffusion, Proc. Eurographics Symp. On
Geometry Processing 2009 (2009).
[12] W.H. Press et al. Numerical Recipes in C –The art of
Scientific Computing, Cambridge Universe Press,
Cambridge, UK, 1992, pp.309-315.
[13] E. Wahl, U. Hillenbrand, G. Hirzinger, Surflet-PairRelation Histograms: A Statistical 3D-Shape Representation for Rapid Classification, Proc. of the FourthIntern’l Conf. on 3-D Digital Imaging and Modeling(3DIM 2003), 474-481, Banff, Canada, (2003).