IEEE INTERNATIONAL CONFERENCE ON SHAPE MODELING AND APPLICATIONS (SMI) 2010 1

Visual Similarity based 3D Shape Retrieval UsingBag-of-Features

Zhouhui Lian1,2, Afzal Godil1, Xianfang Sun2

1National Institute of Standards and Technology, Gaithersburg, USA2School of Automation Science and Electrical Engineering, Beihang University, Beijing, China

Abstract—This paper presents a novel 3D shape retrievalmethod, which uses Bag-of-Features and an efficient multi-view shape matching scheme. In our approach, a properlynormalized object is first described by a set of depth-bufferviews captured on the surrounding vertices of a givenunit geodesic sphere. We then represent each view as aword histogram generated by the vector quantization ofthe view’s salient local features. The dissimilarity betweentwo 3D models is measured by the minimum distance oftheir all (24) possible matching pairs.

This paper also investigates several critical issues in-cluding the influence of the number of views, codebook,training data, and distance function. Experiments on fourcommonly-used benchmarks demonstrate that: 1) Ourapproach obtains superior performance in searching forrigid models. 2) The local feature and global feature basedmethods are somehow complementary. Moreover, a linearcombination of them significantly outperforms the state-of-the-art in terms of retrieval accuracy.

Keywords-3D Shape Retrieval; Bag-of-Features; SIFT;Local Descriptor

1. INTRODUCTION

How to efficiently and effectively retrieve 3D modelsbased on their shapes has become an active subjectin several research communities such as computer vi-sion, computer graphics, mechanical CAD, and patternrecognition. With the development of various kinds of3D shape retrieval benchmarks (e.g. PSB [1], ESB [2],McGill [3], NSB [4] etc.) and the successful organiza-tion of the 3D SHape REtrieval Contest (SHREC) [5],more and more researchers have been attracted to workin this area and a large number of algorithms have beenproposed.

One of the most important research directions in 3Dshape retrieval is Feature Extraction. Ideally, a goodshape descriptor has the following desirable properties,1) High discrimination; 2) Efficient shape matching; 3)Compact representation; 4) Efficient feature extraction;5) Invariance to similarity transformations (in somecases, descriptors should also be articulation invari-ant [6]); 6) Invariance to shape representation; 7) In-variance to shape degeneracies and noises. Generallyspeaking, existing 3D shape descriptors can be looselyclassified into four categories [7]: statistics-based, graph-based, transform-based, and view-based. Recent inves-tigations [8], [7], [9] illustrate that view-based methodswith pose normalization preprocessing get better perfor-mance in retrieving rigid models than other approaches

and more importantly they satisfy almost all character-istics mentioned above. Therefore, despite the fact thatthey have several intrinsic drawbacks (e.g. discarding in-visible information of an object), view-based approachesare still without doubt the most popular and practicalmethods in the field of 3D shape retrieval.

Up to now, probably because of the computationalcomplexity of shape matching for local features, mostof the view-based methods utilize only global shapedescriptors to represent 2D views, which hinders the fur-ther improvement of retrieval performance. In fact, localfeatures have been widely used in many computer vi-sion applications [10]. The utilizations of local featuresusually result in better performance than the traditionalmethods using only global descriptors. Intuitively, it isreasonable to infer that similar techniques can be appliedinto 3D shape retrieval, especially for the view-basedmethods.

Inspired by the work presented in [11] where Ohbuchiet al. reported two shape-based 3D object retrievalmethods using salient visual local features, the first onerepresents the whole 3D model by a word histogramwhile the second one directly expresses and matchestwo normalized models via the salient local features. Inthis paper, we propose a new visual similarity based 3Dshape retrieval approach which describes each view asa word histogram built using the vector quantization ofsalient local descriptors and employs an efficient multi-view shape matching scheme to compute the distancebetween two 3D objects. An overview of the methodis as follows: First, a 3D model is properly aligned tothe canonical coordinate frame so that the normalizedpose could be suitable for drawing standard three-viewimages and then depth-buffer views are captured on thesurrounding vertices of a unit geodesic sphere. After-wards, for each view we extract salient local features(e.g. SIFT [12]) which are subsequently quantized intoa word histogram using the Bag-of-Features approach.Finally, according to the properties of the geodesicsphere previously used, an efficient shape matching canbe carried out to measure the dissimilarity between twoobjects by computing the minimum distance of their all(24) possible matching pairs.

To some extent, the proposed method is quite differentfrom the BF-SIFT algorithm presented in [11]. Basically,our approach is a visual similarity based method, fol-lowing the idea that “if two 3D models are similar, they

also look similar from all viewing angles”; while BF-SIFT [11] is a “global Bag-of-Words based method”,that represents a 3D model as one histogram via thevector quantization of its local features using the Bag-of-Words approach. Moreover, several new techniqueshave been developed to make our method be well suitedfor practical applications of 3D rigid shape retrieval,and results also demonstrate that our method markedlyoutperforms BF-SIFT [11] in terms of retrieval accuracy.

The major contributions of this paper are twofold.1) A novel visual similarity based 3D shape retrieval

framework is proposed, where the Bag-of-Featuresmethod is utilized to describe each view as a wordhistogram and the objects are compared by anefficient multi-view shape matching scheme.

2) Exhaustive experiments are carried out carefully toinvestigate the influence of the number of views,codebook, training data, and distance function.Perhaps surprisingly, our results show that, incontrast to the traditional Bag-of-Features, thetime-consuming clustering is not necessary for thecodebook construction of our method.

The rest of the paper is organized as follows. Section2 discusses previous work. Section 3 presents an explicitdescription of our method. Experimental results are thendemonstrated and analyzed in Section 4. Finally, weprovide the conclusion of this paper in Section 5.

2. RELATED WORK

Based on the shape descriptors used, existing 3Dshape retrieval methods can also be classified into twocategories, global feature based methods and local fea-ture based methods. In this section, we only discuss themost relevant work with respect to our approach. Formore information about the development of 3D shaperetrieval, we refer the reader to a recent survey [13].

2.1 Global feature based 3D shape retrieval

Most of the existing 3D shape retrieval methods be-long to this category. So far, a large number of 3D globalshape descriptors have been proposed such as D1 [14],D2 [15], Spherical harmonic descriptor (SHD) [16], 3DWavelet descriptor [17], Skeleton descriptor [18], Reebgraph descriptor [19], Light field descriptor (LFD) [20],DESIRE [21], and so on. Since our 3D shape descriptoris designed to be able to measure the visual similaritybetween two objects, we pay more attentions to view-based methods, which have been considered as the mostdiscriminative approaches in the literature [1], [5].

Among these view-based methods, Light Field De-scriptor [20] (LFD) method is perhaps the most famousone, where a 3D model is represented by 100 silhouettes(10 views per group) rendered from uniformly dis-tributed viewpoints on the half side of a unit sphere andthe silhouette is encoded by a feature vector consistingof 35 Zernike moments and 10 Fourier coefficients. Theymeasured the dissimilarity between two objects by the

minimum distance of 6000 view group matching pairs,considering all possible situations. LFD method avoidspose normalization via an exhaustive searching whichinevitably aggravates computational cost. To address thisproblem, Lian et al. [9] developed a multi-view shapematching scheme for properly normalized generic mod-els. The experiments showed that, with the same imagedescriptors, retrieval performance including discrimina-tion, spatial requirement and searching speed could beconsiderably improved compared to the original LFDmethod. As a matter of fact, pose normalization has beenwidely applied in many view-based 3D shape retrievalmethods [22], [23], [8], [24], [25]. The major differencebetween them is the feature vectors they used to describeviews. For instance, 2D Fourier coefficients [22], theelevation descriptor [23], and the depth-line descrip-tor [8] have been employed to represent depth-bufferviews. Similarly, silhouette views have been expressedby the 2D shape distribution descriptor [24] and 1DFourier coefficients [25]. The methods discussed aboveall utilized the images captured from viewpoints locatedon the sphere. Recently, Papadakis et al. [26] proposeda 3D shape descriptor using a set of panoramic views,which were obtained by projecting a 3D model to thelateral surface of a cylinder. The panoramic views weredescribed by their 2D Discrete Fourier coefficients aswell as 2D Discrete Wavelet coefficients.

2.2 Local feature based 3D shape retrieval

2D local features have proven to be very successfulin many applications (e.g. image retrieval [27], objectclassification [28], and video data mining [29] etc.)and a vast number of 3D local features (e.g. 3D spinimage [30], harmonic shape context [31], and 2.5DSIFT [32] etc.) have also been developed, however, justa few works have been reported to apply local featuresin 3D shape retrieval. This is mainly because of the highcomputational cost of shape matching for huge amountsof local descriptors extracted from 3D objects. Localfeature based 3D shape retrieval is an interesting andpromising research direction, since it has intrinsic capa-bility of solving articulated shape retrieval and partialshape retrieval problems. Funkhouser and Shilane [33]selected distinctive multi-scale local features, whichare calculated via Spherical Harmonic transformation,and applied Priority-Driven search to efficiently achievepartial matching. Gal et al. [34] proposed a curvature-based local feature that describes the geometry of localregions on the surface and then constructed a salientgeometric descriptor by clustering together a set oflocal descriptors which are interesting enough accordingto a given saliency function. Geometric Hashing wasutilized to accelerate the partial matching of salient localfeatures. Tal and Zuckerberge [35] decomposed eachobject into meaningful components, and then, basedon the decomposition, they represented the 3D modelas an attributed graph that is invariant to non-rigidtransformations.

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Bag-of-Features, which is a popular technique tospeed up the matching of image local features, hasrecently been introduced into local feature-based 3Dshape retrieval. Liu et al. [36] presented a compact 3Dshape descriptor named “Shape Topics” and evaluatedits application to 3D partial shape retrieval in theirpaper [36], where a 3D object was represented as aword histogram constructed by vector quantizing thelocal features of the object. Spin images, calculated onpoints randomly sampled on the surface, were chosenas the local descriptor. Li et al. [37] introduced a weakspatial constraint to the method proposed in [36] bypartitioning a 3D model into different regions basedon the clustering of local features’ spatial positions,but the improvement was limited. Toldo et al. [38]applied a more sophisticated mesh segmentation methodto decompose a 3D object into several subparts. Eachsegmented region was represented by one descriptorand then a word histogram was generated by assigningall subpart descriptors of the object into visual words.Ohbuchi et al [11] reported a view-based method usingsalient local features (SIFT [12]). They represented a3D object by a word histogram derived from the vectorquantization of salient local descriptors extracted onthe depth-buffer views captured uniformly around theobject. Their experiments demonstrated that the methodresulted in excellent retrieval performance for both ar-ticulated and rigid objects.

3. METHOD DESCRIPTION

In this section, we first present an overview of ourmethod and then elaborate on the details of each step inthe corresponding subsections.

3.1 Overview

Since the method is mainly based on the Bag-of-Features approach and a multi-view shape matchingscheme (named Clock Matching for the sake of conve-nience and intuition), we call it “CM-BOF” algorithmin this paper. The CM-BOF algorithm, depicted inFigure 1, is implemented subsequently in four steps:

1) Pose Normalization: Normalize 3D objects withrespect to the canonical coordinate frame to ensurethat their mass centers coincide with the origin,they are bounded by the unit sphere, and they arewell aligned to three coordinate axes.

2) Local Feature Extraction: Capture depth-bufferviews on the vertices of a given unit geodesicsphere whose mass center is also located in theorigin and then extract salient local features fromthese range images.

3) Word Histogram Construction: For each view,quantize its local features into a word histogramusing a pre-specified codebook. Normally, thecodebook is obtained off-line by clustering thetraining data that are randomly sampled from thefeature set of all models in the target database.

However, the codebook of our method can also bedirectly built using randomly sampled Nw localfeature vectors. This has been verified by theexperiments described later.

4) Multi-view Shape Matching: Carry out an effi-cient shape matching scheme to measure the dis-similarity between two 3D models by calculatingthe minimum distance of their 24 matching pairs.

3.2 Pose Normalization

The key idea of our method is to measure the visualsimilarity between 3D objects with arbitrary poses, itis preferable if the models can be normalized in themanner that corresponds well with human perception.Therefore, we normalize the objects by a recently pro-posed approach [9] which combines the PCA (principalcomponent analysis) based and the rectilinearity basedpose alignment algorithms to obtain better normalizationresults. As we know, PCA is the most prominent toolfor accomplishing rotation invariance by solving threeprincipal axes of a 3D object. While the basic ideaof the rectilinearity-based method (only suitable for thepolygonal mesh) is to specify a standard pose throughthe calculation of the model’s rectilinearity value. Threesteps of the composite method are described as follows.

1) Translation and scaling. For a given 3D mesh,translate the center of its mass to the origin andthen scale the maximum polar distance of thepoints on its surface to one.

2) Rotation by two methods. Apply the PCA-basedand the rectilinearity-based method, respectively,to rotate the original model to the canonical coor-dinate frame and then store these two normalizedmeshes in memory;

3) Selection. Calculate the number of valid pixels ofthree silhouettes, projected on the planes Y OZ,ZOX , XOY , for the two normalized meshesgenerated in the previous step. And then selectthe model, which yields the smaller value, as thefinal normalization result.

Two normalization examples are displayed in Fig-ure 2. Note that, the method only performs incompletepose normalization for rotation transformation. Morespecifically, only the positions of three axes are fixedfor the model normalized by the composite method, thatis, the direction of each axis is still undecided and thex-axis, y-axis, z-axis of the canonical coordinate systemcan be located in all three axes. That also means 24different orientations are still plausible for the alignedmodels, or rather, 24 matching operations should becarried out when comparing two normalized objects. Formore details of the pose alignment algorithm, we referthe reader to the paper [9] where convincing experimen-tal results have been obtained to illustrate the advantageof this approach, in the context of pose normalizationand 3D shape retrieval, against other methods.

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Word - Histogram - Construction

Local - Feature - Extration Pose - Normalization

Multi-view - Shape - Matching (Clock - Matching)

Fig. 1. An illustration of our method. First, a model is properly normalized. Second, depth-buffer views are captured from the vertices ona given geodesic sphere and then, for each view, we calculate salient SIFT descriptors [12]. Third, a word histogram is obtained by vectorquantizing the view’s local features against the codebook, so that the object can be expressed by a set of histograms. Finally, an efficient shapematching is carried out to obtain the best match from all 24 matching pairs between two objects.

PCA Rectilinearity PCA Rectilinearity

Selection Selection

Final Final

Fig. 2. Two alignment examples of the pose normalization methodwe use. The final result is chosen, using a selection criterion, fromthe normalization results of two methods.

Fig. 3. Geodesic spheres generated from a regular octahedron.

3.3 Local Feature Extraction

After pose normalization, 3D meshes have been wellaligned to the canonical coordinate frame. Then their

depth-buffer views are captured on the vertices of agiven unit geodesic sphere whose mass center is alsolocated in the origin. The geodesic spheres used hereare obtained by subdividing the unit regular octahedronin the way shown in Figure 3. These kinds of polygonalmeshes are suitable for our multi-view based shaperetrieval mechanism, mainly because of the followingthree reasons. First, the vertices are distributed evenlyin all directions. Second, these geodesic spheres enabledifferent level resolutions in a natural manner. Thecoarsest (level-0) one is obtained using a unit regularoctahedron with 6 vertices and 8 faces. Higher levelscan be generated by recursive subdivisions. Third, sinceall these spheres are derived from an octahedron, giventhe positions of six vertices for the original octahedron,other vertices can be specified automatically. Moreover,all vertices are symmetrically distributed with respectto the coordinate frame axes. That means, when com-paring two models, only 24 matching pairs need to beconsidered for the feature vector in an arbitrary level.

After view rendering, a 3D object can be approx-imately represented by a set of depth-buffers fromwhich we extract salient SIFT descriptors, as presentedin [12]. The SIFT descriptor is calculated, using theVLFeat matlab source code developed by Vedaldi andFulkerson [39], in the following two steps. First, ob-tain the scale, orientation, and position information ofthe salient points detected by the Different-of-Gaussian

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Fig. 4. A demonstration on the robustness of the salient localdescriptor against small viewpoint changes.

(DoG) approach. Second, compute SIFT descriptors forthe local interesting regions which are determined bythe scale and position of the salient points. Here, theSIFT descriptor, which is actually a 3D histogram ofgradient location and orientation, is computed using itsdefault parameters, where the location is divided intoa 4 × 4 grid and the gradient angle is quantized intoeight orientations. This results in a feature vector with128 elements. The feature is designed to be robust,to some extent, against similarity transformation, affinedistortion, noise and illumination changes of images.Figure 4 shows some examples of SIFT descriptors ex-tracted from the range images which are scaled, rotated,and affine transformed. It can be seen that the SIFTdescriptor is stable to various changes of 3D viewpoints,which is a desirable property for our visual similaritybased 3D shape retrieval method to compensate itsreliance on the stability of pose normalization.

3.4 Word Histogram Construction

Directly comparing 3D models by their local visualfeatures is time consuming, especially for the 3D shaperetrieval methods using a large number of views. Toaddress this problem, we quantize the SIFT descriptorsextracted from a depth-buffer image into one wordhistogram so that the view can be represented in a highlycompact and distinctive way.

Before vector quantization, a codebook (also namedas vocabulary) with Nw visual words should be cre-ated. Usually, the codebook is generated via off-lineclustering. More specifically, huge numbers of featurevectors are first randomly sampled from the feature setof the target database to form a training set. Then, thetraining set is clustered into Nw clusters using the K-means algorithm. At last, centers of the clusters areselected as the feature vectors of visual words in thecodebook. Since the spatial requirement and calculating

Feature Vector

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Fig. 5. The data structure of our 3D shape descriptor that is composedof several word histograms.

time of the K-means clustering are significant, manyother algorithms [40] (e.g. kd-tree, ERC-tree, and Lo-cality sensitive hashing) have been applied to alleviatethe computational cost. However, as we can see from theexperiments described in Section 4.2, clustering is notnecessary for the codebook construction of our method.In other words, randomly sampled local feature vectorscan be directly used to create the vocabulary and thesetwo codebook construction approaches result in almostthe same discrimination for 3D shape retrieval.

By searching for the nearest neighbor in the code-book, a local descriptor is assigned to a visual word.Then each view can be represented using a wordhistogram whose ith bin records the number of ith

visual words in the depth-buffer image. In order toobtain satisfactory discrimination capability, usually thehistogram should have thousands of bins. Let the numberof views be 66 and the number of visual words in thecodebook be 1500, without optimization, the 3D shapedescriptor would be of dimension 99000. In fact, withthe observation that, for our method, the average numberof salient points in a view (with size 256× 256) is onlyabout 30, we can represent the histogram in a betterway that not only makes the shape descriptor highlycompact but also significantly improves the efficiencyof dissimilarity calculation.

Figure 5 demonstrates an example of the data struc-ture for our 3D shape descriptor, where only the in-formation (i.e. bin No. and bin value) of some bins,whose values are not equal to zero, appears in the featurevector. Experimental results show that, considering themethod with 66 views and a 1500-dimension codebook,on average the new data structure requires about 30times less spatial storage and performs approximately21 times quicker for feature comparison.

3.5 Multi-view Shape Matching

The last step of our 3D shape retrieval method is thedissimilarity calculation (also called shape matching)between two shape descriptors. The multi-view shapematching (Clock Matching) scheme we use is originallyproposed in [9], here we provide more details about thisapproach and apply it into our shape matching task withnew distance measures.

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The basic idea of Clock Matching is that, after weget the major axes of an object, instead of completelysolving the problem of fixing the exact positions anddirections of these three axes to the canonical coordinateframe, all possible poses are taken into account duringthe shape matching stage. The principle of the methodis simple and reasonable, moreover, our previous ex-periments [9] have already illustrated considerable im-provements against other approaches. As we mentionedabove, 24 different poses still exist for a normalizedmodel. Figure 6 shows all possible poses of a chair afterpose alignment processing in the canonical coordinateframe. For the sake of convenience x+, x-, y+, y-,z+, and z- axis are denoted as 0, 1, 2, 3, 4, and 5,respectively. When comparing two models, one of themwill be placed in the original orientation denoted asa permutation p0 = {p0(k)|k = 0, 1, 2, 3, 4, 5} whilethe other one may appear in 24 different poses de-noted as permutations pi = {pi(k)|k = 0, 1, 2, 3, 4, 5},0 ≤ i ≤ 23. From these 24 permutations (see theunderneath of each small image in Figure 6), all possiblematching pairs ((p0, pi), 0 ≤ i ≤ 23) between twomodels can be obtained. For instance, we can capture sixsilhouettes or depth buffers from the vertices of a unitregular octahedron and then extract 2D shape descriptorsfor these images to construct a view-based 3D featurevector. The vertices in the corresponding axes are alsodenoted as 0, 1, 2, 3, 4, and 5, respectively. Then wecompare all 24 matching pairs for two models and theminimum distance is chosen as their dissimilarity.

Generally speaking, Clock Matching performs in twosteps:

1) Initialization. Recursively subdividing the originalunit octahedron nd times, we get a geodesic spherewith the required resolution and the coordinatesof its vertices should be recorded consecutivelyaccording to the time they emerge. During theprocess of subdivision, two tables (named edgetable and vertex table, respectively) which indicatethe relationship between old and new vertices arealso obtained. An example of the edge table andthe vertex table, utilized to store the informationduring the stage of subdividing the octahedron, aredemonstrated in Figure 7. Note that we only needto process this step once.

2) Comparison.As mentioned above, when compar-ing two models represented by level-0 descriptors,we calculate the minimum distance among 24matching pairs ((p0, pi), 0 ≤ i ≤ 23) whichcan be derived using the permutations shown inFigure 6. If higher-level shape descriptors areapplied, we should use the edge table, vertex table,and pi, 0 ≤ i ≤ 23 to build new permutationsp′i = {p′i(k)|0 ≤ k < Nv}, 0 ≤ i ≤ 23 describing

all possible matching pairs (p′0, p

′i), 0 ≤ i ≤ 23

for two models represented by Nv views. Finally,the dissimilarity between the query model q and

{0,1,2,3,4,5} {0,1,4,5,3,2} {0,1,3,2,5,4} {0,1,5,4,2,3}

{1,0,3,2,4,5} {1,0,4,5,2,3} {1,0,2,3,5,4} {1,0,5,4,3,2}

{4,5,2,3,1,0} {4,5,1,0,3,2} {4,5,3,2,0,1} {4,5,0,1,2,3}

{5,4,2,3,0,1} {5,4,0,1,3,2} {5,4,3,2,1,0} {5,4,1,0,2,3}

{2,3,1,0,4,5} {2,3,4,5,0,1} {2,3,0,1,5,4} {2,3,5,4,1,0}

{3,2,0,1,4,5} {3,2,4,5,1,0} {3,2,1,0,5,4} {3,2,5,4,0,1}

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Fig. 6. All (24) possible poses of a chair after it has been incompletelyaligned to the canonical coordinate frame. The corresponding permu-tations are listed underneath. The permutation pi, i = 0, 1, . . . , 23denotes the positions of three major axes of the object in the contextof pose i.

the source model s is defined as,

Disq,s = min0≤i≤23

Nv−1∑

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D(FVq(p

′0(k)), FVs(p

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),

where FVm = {FVm(k)|0 ≤ k < Nv} denotesthe shape descriptor of 3D object m, FVm(k) isthe signature of view k, and D(·, ·) is the distancefunction. In Section 4.4, four metrics, denotedas DMaxHis, DMinHis, DAvgHis, and DL1, aredefined and compared. By default, we utilizeDMaxHis to measure the dissimilarity betweentwo views.

4. EXPERIMENTS

In this section, we first present and discuss experi-mental results to study the influence of the number ofviews, codebook, training data, and distance function onretrieval performance for our CM-BOF algorithm. Then,3D shape retrieval results are analyzed for the visualsimilarity based methods (CM-BOF and GSMD [9])using local features and global features, respectively. Fi-nally, we compare the retrieval accuracy of our methodswith other approaches.

We run the experiments on four publicly available 3Dshape benchmarks briefly described as follows:

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Fig. 7. The edge table and the vertex table generated whensubdividing the original octahedron into the geodesic sphere with 18vertices. The indexes of the vertices of the original edges are stored inthe edge table. New vertices’ indexes can be obtained by using pairs ofold vertices to search in the vertex table. A more intuitive illustrationof the relations between the original vertices (green circles) and thenew vertices (red pentagons) is given in the top-right of this figure.

• PSB: The test set of the Princeton Shape Bench-mark [1] contains 907 generic models which areclassified into 92 categories. The maximum numberof objects in a class is 50, while the minimumnumber is 4.

• NSB: The NIST (National Institute of Standardsand Technology) Shape Benchmark [4] is com-posed of 800 generic models which are classifiedinto 40 categories. Each class contains 20 objects.

• ESB: The Engineering Shape Benchmark [2] con-tains 867 CAD models which are classified into45 categories. It has maximum 58 and minimum 4objects in a class.

• McGill: The McGill Shape Benchmark [3] consistsof 255 articulated objects which are classified into10 categories. The maximum number of objects ina class is 31, while the minimum number is 20.

Since our methods are specifically designed for rigidmodels, most of our experiments (except the last one)are conducted only on the PSB and NSB databases.

We implement the feature extraction in Matlab R2007,while the shape matching code is written in C++ ofMicrosoft Visual Studio 2005. All programs are rununder windows XP on a personal computer with a2.66GHz Intel Core2 Quad CPU, 4.0GB DDR2 memory,and a 128MB NVIDIA Quadro Fx550 graphics card.

Note that: Unless otherwise specified, the defaultparameters of our CM-BOF method are selected as fol-lows: the resolution of depth-buffer images is 256×256,

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Fig. 8. Influence of the number of views. (a) and (b) show thePrecision-recall plots for the methods, with different numbers of views,run on the NSB and PSB databases, respectively.

the number of views Nv = 66, the size (i.e. the numberof visual words) of the codebook Nw = 1500, thesize (i.e. the number of local feature vectors) of thetraining set Nt ≈ 120000, the codebook is generatedby clustering the training set, which is derived from thetarget database, using the Integer k-means method whosesource code is available on the website [39].

4.1 Influence of the number of views

In the first experiment, we investigate the influenceof the number of views on retrieval performance.

Figure 8 shows the precision-recall curves calculatedfor our CM-BOF methods using geodesic spheres with6, 18, 66, and 258 vertices. It can be observed thatretrieval performance can be improved by increasing thenumber of views, especially when the number of viewsjumps from 6 to 66. But the improvements slow downas the number of views keeps growing, while the com-putational cost still increases sharply. This is because anupper limit exists for the retrieval performance of view-based methods, but more views involved always meansthat more time needs to be spent on calculating localdescriptors and more memories are required to storethe feature vectors. Consequently, in order to make thebalance between quality and cost, the number of viewsis experimentally chosen as 66 in the following sections.

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Fig. 9. Influence of the codebook size. (a) and (b) show the DCGversus codebook size curves for two methods (i.e. CM-BOF and BF-SIFT [11]) run on the NSB and PSB databases, respectively.

4.2 Influence of the codebook

In the second experiment, we study the influence ofthe codebook size and creation methods by comparingretrieval performance among the shape descriptors corre-sponding to codebooks with different sizes and differentconstruction methods.

Codebook size. Probably, we can say that the mostimportant parameter of the CM-BOF algorithm is thenumber of visual words (denoted as Nw) in the code-book. This is because the codebook size not only de-termines the spatial requirement but also significantlyaffects the retrieval performance of the method. Fig-ure 9 demonstrates results which report the DiscountedCumulative Gain (DCG, well-known as the most stableretrieval measure [41]) values for CM-BOF methodswith steadily increased codebook size. We observe that,as the codebook size enlarges, DCG values go up sharplyat the beginning and become stable approximately whenNw > 1000. Similar conclusions are obtained fromFigure 9 for the BF-SIFT method presented in the paper[11], where only one word histogram is used to describea 3D object. Here, both our CM-BOF and the BF-SIFTmethods utilize 66 depth-buffer views. According to theexperimental result, we set the number of visual wordsin the codebook as 1500 in this paper.

Construction methods. Next, two codebook buildingmethods are compared. The first one selects the centersof feature clusters to form the codebook, after the

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Fig. 10. Influence of the training data size. (a) and (b) show theDCG versus training data size curves for two methods (i.e. CM-BOF,BF-SIFT [11]) run on the NSB and PSB databases, respectively.

clustering of the train data set which is composed ofa large number of local descriptors randomly sampledfrom the database to be retrieved. The second methoddirectly uses the randomly sampled feature vectors asthe visual words in the codebook. Typically, the Bag-of-Features method is implemented with clustering. Theprevious work [42] has also demonstrated that thefirst codebook construction method results in betterperformance in image classification against the secondmethod. However, as we can see from Table 1, whichpresents the means and standard deviations of the DCGvalues over 10 runs of our CM-BOF algorithms withand without clustering. A conclusion can be made thatthe random sampling approach works as well as theclustering approach for our CM-BOF 3D shape retrievalalgorithm when the codebook size has been properlychosen. We infer that this is mainly due to the carefully-designed shape matching scheme and the fewer invalidinformation existing in the views captured from the 3Dobjects compared to the ordinary images used in other2D applications.

TABLE 1INFLUENCE OF THE CODEBOOK CONSTRUCTION METHOD. THETABLE GIVES THE MEANS AND STANDARD DEVIATIONS OF THE

DCG VALUES OVER 10 RUNS OF OUR CM-BOF ALGORITHMS ONTWO BENCHMARKS, FOR CODEBOOKS GENERATED USINGCLUSTERING (K-MEANS), AND FOR RANDOMLY SAMPLED

CODEBOOKS (RANDOM)

K-means RandomNSB 83.1± 0.1% 83.1± 0.2%PSB 71.7± 0.2% 71.7± 0.2%

4.3 Influence of the training dataIn the third experiment, we analyze the influence of

the training data size and training source on retrievalperformance.

Training data size. Figure 10 shows the curvesdepicting the relation between DCG values and the

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(b) - PSB (a) - NSB

Fig. 11. Influence of the training data source on the retrieval results.(a) and (b) compare the DCG values of our CM-BOF methods,corresponding to four different training data sources, run on the NSBand PSB databases, respectively.

number of feature vectors in the training set. It can beseen that the size of the training data has very littleimpact on the retrieval performance. No matter howlarge or how small the training set is, the correspondingretrieval performance remains stable, even when thesize of the training data is just a little bit larger thanthe codebook (e.g. Nw = 1500 and Nt = 1578).The experimental results provide an additional supportto the aforementioned conclusion that clustering is notnecessary for our method.

Training data source. It is worthy of investigatingwhether it is necessary to create the training set bysampling the feature vectors in the database to besearched. In other words, we want to study the influenceof the source database from which the local descriptorsare randomly sampled to form the training set. Here,retrieval performance is evaluated on the NSB and PSBdatabases for the CM-BOF methods corresponding tofour training sets generated from the PSB, NSB, ESB,and McGill databases, respectively. Figure 11 shows theresults. Apparently, as we expected, the NSB trainingdata gives the best result on the NSB database andthe PSB training data gives the best result on the PSBdatabase. Moveover, we also observe that better resultscould be obtained if more similar training data source,compared to the target database, is utilized.

4.4 Influence of the distance function

In the forth experiment, we compare the performanceof our methods, which use different metrics to calculatethe distance between two views.

Here we test four distance functions denoted asDMaxHis, DMinHis, DAvgHis, and DL1, respectively.Assume that view k is described by the word histogramHk = {Hk(j)|j = 0, 1, . . . , Nw − 1}, given twohistograms H1,H2, the distance between them can becalculated by the following four metrics, the first threeof which are modified from the histogram intersectiondistance presented in [43].

(b) - PSB (a) - NSB

Fig. 12. Influence of the dissimilarity measure on retrieval results.(a) and (b) compare the DCG values of our CM-BOF methods,corresponding to four different distance functions, evaluated on twobenchmarks. The metrics DMaxHis, DMinHis, DAvgHis, andDL1 are denoted as Max, Min, Avg, and L1, respectively.

1). Maximum dissimilarity histogram intersection dis-tance.

DMaxHis = 1−∑Nw−1

j=0 min(H1(j),H2(j))

max(∑Nw−1

j=0 H1(j),∑Nw−1

j=0 H2(j)).

2). Minimum dissimilarity histogram intersection dis-tance.

DMinHis = 1−∑Nw−1

j=0 min(H1(j),H2(j))

min(∑Nw−1

j=0 H1(j),∑Nw−1

j=0 H2(j)).

3). Average dissimilarity histogram intersection dis-tance.

DAvgHis = 1−∑Nw−1

j=0 min(H1(j),H2(j))

(∑Nw−1

j=0 H1(j) +∑Nw−1

j=0 H2(j))/2.

4). Normalized L1 distance.

DL1 =Nw−1∑

j=0

∣∣∣∣∣H1(j)∑Nw−1

j=0 H1(j)− H2(j)∑Nw−1

j=0 H2(j)

∣∣∣∣∣ .

As it can be seen from Figure 12, DMaxHis outper-forms other three metrics.

4.5 Comparison of local and global methods

In this section, results of our fifth experiment arepresented to discuss the advantages and disadvantages oftwo methods (CM-BOF and GSMD [9]), which utilizelocal and global features, respectively, to describe views.

These two methods both capture 66 views and applythe same shape matching scheme, the only differenceis that the local-based method (CM-BOF) uses a wordhistogram of local features to describe a view while theglobal-based method (GSMD [9]) represents the view bya global feature vector with 47 elements including 35Zernike moments, 10 Fourier coefficients, eccentricityand compactness. The comparison is performed usingthe precision-recall curve on each class of the NSBdatabase. Inspecting the comparison results shown in

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bed car cabinet biplane

sword missile ant glasses

sofa monitor hand - gun guitar

Fig. 13. Precision-recall curves for specific object types on the NSB database. In this Figure, Local denotes the local feature based method(CM-BOF) while the global feature based method (GSMD) is denoted as Global.

Figure 13, we could classify them into the followingthree categories and suggest several possible reasons.Four examples of the first category are displayed inrow 1, where the local-based method significantly out-performs the global-based method. 62.5% objects inthe NSB database belong to this category. We specu-late that, this is because, for these models in a sameclass, they have different global appearances but looksimilar when focusing on local regions, or their localdescriptors provide more details than the global features.For instance, cabinet, telephone, and biplane, etc. canbe better retrieved using the local-based method. Fourexamples of the second category are demonstrated inrow 2, where the global-based method obtains muchbetter results than the local-based one. Only 12.5%models belong to this category. Possible explanationsare twofold, on the one hand, local salient features areextracted from unimportant but locally different subpartsof these models (e.g. sword’s handle); on the otherhand, overall appearances of these models (e.g. missile,glasses, and ant) in the same class are similar but nottheir local regions. The last row shows the third category,where the local-based and global-based methods getalmost the same performance. 25.0% objects, such assofa, monitor, hand gun, guitar and so on, belong to thiscategory. To sum up, the local-based method (CM-BOF)is generally superior to the global-base method (GSMD)(the result comparisons for entire databases are shownin Figure 14), however, the global-base method mayoutperform the local-based method when searching forcertain kinds of models. Furthermore, these two methodsrepresent a depth-buffer view in quite different manners.

Therefore, to some extent, they are complementary andit is possible to create a more discriminative descriptorby using the combination of local feature and globalfeature to represent the depth-buffer views.

4.6 Comparison with the state-of-the-art

In our last experiment, we compare the performanceof our algorithms and other state-of-the-art methods.

Figure 14 demonstrates the precision-recall curveson aforementioned four benchmarks for seven methodslisted as follows,• D2: A method describing a 3D object as a his-

togram of distances between pairs of points onthe surface [15]. The number of histogram bins ischosen as 64.

• SHD: A method describing a 3D object as a featurevector consisting of spherical harmonic coefficientsextracted from three spherical functions giving themaximal distance from center of mass as a functionof spherical angle. See [16] [9] for details.

• LFD: A well-known visual similarity based methodproposed in [20]. See Section 2 for details. Thelength of the feature vector is 4700.

• GSMD: See Section 4.5 for details. The numberof viewpoints is selected as 66.

• BF-SIFT: A method representing a 3D object asa word histogram by vector quantizing the visualsalient SIFT descriptors [11]. Here, the depth-bufferviews are capture on the vertices of a unit geodesicsphere. The number of views is selected as 66 andthe length of the feature vector is 1500.

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Fig. 14. Precision-recall curves of seven methods run on the four benchmarks. (a), (b), (c), and (d) shows the results evaluated on the NSB,PSB, ESB, and McGill databases, respectively.

• CM-BOF: See Section 3 for details. The defaultsettings are chosen here and thus the average lengthof the feature vector is 3320.

• Hybrid (CM-BOF+GSMD): A composite methodbased on a linear combination of CM-BOF andGSMD. More specifically, in this method, a viewis expressed by a feature vector consisting of twodifferent kinds of shape descriptors, which areused in CM-BOF and GSMD, with pre-specifiedweights. We experimentally select the weights asWlocal = 7.0 and Wglobal = 1.0 for local andglobal features, respectively, by maximizing theretrieval accuracy on the PSB train set with baseclassification. The shape matching scheme andother parameters are exactly the same as the CM-BOF algorithm described above.

Several observations can be made from Figure 14. Forthe PSB and NSB databases which consist of genericmodels, the Hybrid method clearly outperforms othersix methods, among which CM-BOF and BF-SIFT takethe second place and the third place, respectively. Forthe ESB database which contains only CAD objects,although the Hybrid method still performs the best, BF-SIFT obtains slightly better result than CM-BOF. Forthe McGill database which is composed of articulatedmodels, three methods associated with local featuresmarkedly outperforms the methods only using globalfeatures. Despite applying pose normalization, CM-BOF

is still superior to all other methods, probably becauseof the salient local feature’s robustness against the smallchanges of viewpoints. The Hybrid method merely takesthe third place mainly due to the poor performance ofthe global descriptor it combines.

We also compare our Hybrid (CM-BOF+GSMD)method and CM-BOF method with the state-of-the-art approaches including PANORAMA [26], MDLA-DPD [8], GSMD+SHD+R [9], DESIRE [21], andLFD [20] on the PSB test set with base classification.As shown in Table 2, our Hybrid method significantlyoutperforms other methods compared here, while theCM-BOF algorithm, whose feature vector is only ofdimension 3320 on average, also obtains superior orcomparable 3D shape retrieval performance. Moreover,for our CM-BOF method, comparing a pair of 3Dobjects takes less than 1.0 millisecond and, with theGPU-based implementation [44], the feature extractionof an object can be finished within 5.0 seconds.

5. CONCLUSION

In this paper, we presented a novel visual similaritybased 3D shape retrieval method using Bag-of-Features.The key idea is to describe a view as a word histogram,which is obtained by the vector quantization of theview’s salient local features, and apply a multi-viewshape matching to calculate the dissimilarity between

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TABLE 2COMPARING RETRIEVAL RESULTS OF OUR METHODS (FIRST TWO

ROWS) WITH OTHER STATE-OF-THE-ART APPROACHES ON THE PSBTEST SET WITH BASE CLASSIFICATION.

1-NN 1-Tier 2-Tier DCGCM-BOF+GSMD 75.4% 50.9% 64.0% 74.6%

CM-BOF 73.1% 47.0% 59.8% 72.0%PANORAMA 75.3% 47.9% 60.3% -

GSMD+SHD+R 73.1% 47.2% 60.2% 72.1%MDLA-DPD 68.8% 43.6% 54.2% 67.8%

DESIRE 66.5% 40.3% 51.2% 66.3%LFD 65.7% 38.0% 48.7% 64.3%

two objects. A set of experiments were carried out to in-vestigate several critical issues of our CM-BOF method,including the impact of the number of views, codebook,training data, and distance function on the performanceof 3D shape retrieval. It can be seen that clusteringis not necessary for the method, and our local featurebased method is somehow complementary with respectto the global feature based method (GSMD [9]). Theexperimental results also demonstrated that our methods(the composite method and the CM-BOF algorithm) aresuperior or comparable to the state-of-the-art.

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