Content-based 3D Model Classification using Geometric Graph Representation by ISERP ISERP

Chin-Chia Wu*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 5, Issue No. 2, 116 - 122

Content-based 3D Model Classification using Geometric Graph Representation Chin-Chia Wu*

Keywords-model classification; spin-image signature; geometric structure; graph matching

INTRODUCTION

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Due to highly developed scanning devices and technologies, acquiring three dimensional (3D) models becomes much easier. With the increasing usage of 3D models, a great growth of requirements for finding the appropriate models that users want have become more critical. Therefore, a large variety of research works have appeared that tackle the issue of 3D model classification. A basic approach called text-based classification uses human semantic in filenames, captions, or context to describe and classify 3D models since it is simple and intuitive. Unfortunately, this method fails in many cases, such as bad annotations, different language definitions, and related keywords too common to describe the difference in subtle geometry. This means the text-based method is too limited and ambiguous to be suitable for applications with a large number of 3D models. In contrast, content-based methods for 3D model classification use the model shape data to group models into several categories. In recent years, a growing body of research about content-based retrieval has been conducted in many fields, and the vast literature devoted to this topic has been reviewed on several occasions. Many research studies have been made to 3D model retrieval, and most of them are based on geometric shape similarity [1]. Hence, the efficiency and

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Dept. of Electrical Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C sflin@mail.nctu.edu.tw accuracy of content-based methods for model classification are considered important. Generally, content-based methods for model classification can be divided into two broad categories: appearance-based methods and appearance-and-structure-based methods. A general step of approaches in both two categories is that to extract local features for representing the appearances of models. However, the difference between the two types of methods originates from the way that generating the local features. The appearance-based methods only extract features without position information or topological information. A famous approach is called bag-of-words model, which is used originally to classify documents in natural language processing [2, 3]. A great deal of research has applied the bag of words model to achieve the recognition and the classification, for example, texture recognition [4-6] and image categorization [7]. Csurka [8] has proposed an approach of image categorization used the bag of words model with a NaĂŻve Bayes classifier. Recently, a growing number of research studies utilize the bag of words model in 3D model retrieval and classification. Toldo [9] has proposed a method for 3D object categorization using the bag of words model with sub-part descriptors. Li [10-12] has proposed a spatially enhanced bag of words model for 3D shape retrieval. The result has suggested that the spatial constrain improves the performance on 3D shape retrieval and classification.

Abstractâ&#x20AC;&#x201D;To classify 3D models is an important problem in computer vision. This work attempts to deal with the problem of classifying models into objects of knowing classes. In this paper, an approach based on the geometric graph representation is proposed for classifying 3D models in point clouds. The approach first uses a moving-least-squares (MLS) technique to calculate the geometric information. The point feature extraction is achieved by using spin image signatures. Then a surface segmentation is performed to cluster the point features. After that, a geometric graph model is introduced to generate the representation of model content for each model. To classify an incoming model with an appropriate class, an efficient algorithm for inexact graph matching is employed. In summary, the experimental results show that our approach outperforms the original spin image signature method, and has good performance in model classification.

Sheng-Fuu Lin

Dept. of Electrical Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C chia.ece93g@nctu.edu.tw

Another category is appearance-and-structure-based methods. These approaches extract features with geometric information or shape information. Many of these approaches use a structure to describe the shape information, like skeletons or graphs [13, 14]. In order to handle the partial matching or incomplete models, various methods for local graph matching are proposed [15-18]. The processing of 3D models in point clouds is a significant challenge. First, 3D models represented by triangle meshes are of benefit for calculating and describing the geometry of surfaces. On the contrary, point cloud data are unconstrained with only information of positions. It is difficult to describe the geometry of surfaces explicitly. Moreover, some methods of feature extraction, like spin image [19], are proposed for 3D mesh models. The innate limitations in methods make them cannot be adopted for point cloud data directly. Levin [20, 21] has proposed a surface representation based on moving-least-

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Chin-Chia Wu*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 5, Issue No. 2, 116 - 122

This paper attempts to deal with the problem of classifying models into objects of knowing classes. The main contribution of this work is that a geometric graph representation is proposed to classify 3D models. This geometric graph consists of three parts, including point feature extraction, surface segmentation, and the construction of the geometric graph. This part-and-structure based method expects to overcome the drawback of the part based methods.

P  { pi 

}, i  1,

,n ,

This paper is organized as follows. Section II discusses the graph representation model for 3D models, which includes preprocessing and feature extraction for point cloud data. Section III introduces the inexact graph matching method and the proposed method for model classification. The experimental results with discussions are presented in section IV. The conclusions are given in section V.

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II. GRAPH REPRESENTATION OF 3D MODEL This section mainly discusses the representations of 3D models from point cloud data. First, this study employs MLS surfaces to describe the surface from the unconstrained point set. After that, the spin image signature is introduced to describe point features. Then a surface segmentation is performed to cluster the point features. Lastly, the geometric graph is constructed by using the result of segmentation. The model classification algorithm will be discussed in the next section. A. Surface Preprocessing Levin [20, 21] has proposed MLS surfaces introduced in many applications, such as computer graphics and surface reconstruction. Furthermore, a rich variety of improvements and extensions have been presented and widely used in various scenarios. An important property of MLS surfaces is to handle the noisy input; however, most of them do not preserve the details of the surfaces. To keep the information of sharp surface, this study adopts robust implicit MLS (RIMLS) surface proposed by Öztireli [22] as the definition of point set surfaces, which is briefly presented as follows.

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(1)

with the normal ni at point pi , the iterative minimization of RIMLS definition at k -th iteration is shown as: f

n ( x) 

( x  pi )i ( x)w(rik 1 )wn (nik 1 )

 ( x)w(r i

k 1

)wn (nik 1 )

(2)

with the residuals: ri k 1  f k 1 ( x)  ( x  pi )T ni ,

(3)

where x is an input point near the surface, and i () is a spatial weight function approximated to a Gaussian:  x  pi i ( x)  1   hi 2 

  ,  

(4)

where hi is the kernel bandwidth related to the radii of the sampled points. The refitting weight term in Eq. (2) is defined as:

In this paper, an approach based on the geometric graph representation is proposed for classifying 3D models in point clouds. The approach first uses a RIMLS technique to calculate the geometric information. The point feature extraction is achieved by using spin image signatures. Then a surface segmentation is performed to cluster the point features. After that, a geometric graph model is introduced to generate the representation of model content for each model. To classify an incoming model with an appropriate class, an efficient algorithm for inexact graph matching is employed. In summary, the experimental results show that our approach outperforms the original spin image signature method, and has good performance in model classification.

The RIMLS algorithm is derived from the implicit MLS definition (IMLS) [25] and a robust local kernel regression approach. For a sampled point set defined as:

squares (MLS) method for point cloud data. Furthermore, many variants and extensions have been presented [22-25] and provided the ability to keep the details of surfaces. In short, MLS surfaces help solve the problems in both calculating the geometric information and surface reconstruction. Additionally, MLS surfaces support to extract precise features.

 r2 w( ri )  exp   i 2  ( h ) r i 

  , 

(5)

where  r is a scale term and can be set to a constant. The last weight function in Eq. (2) is the normal refitting weight defined as:  ( ni k ) 2 wn ( ni k )  exp     n2 

  , 

(6)

where  n is a width of the filter to control the sharpness of surfaces, and ni k is the distance between the current gradient and an input normal, which is given by:

ni k  f k ( x)  ni .

(7)

The gradient  f k can be calculated by:

f

 w  ( x)n   w  ( x)  n ( x)   w  ( x) i i

( x  pi )  f k ( x)

 (8)

i i

with wi  w(rik 1 )wn (nik 1 ) . To project an input point onto the RIMLS surface, the Eq. (2) is computed iteratively until it is converged or the termination criteria are reached. In short, the projected point is calculated by:

x  x  f k ( x)f k ( x) .

(9)

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Chin-Chia Wu*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 5, Issue No. 2, 116 - 122

B. Spin Image Signatures The spin image [19] is a well-known method to describe geometric data around an oriented point as a 2-D histogram. To represent a 3-D model, it needs as many spin images as the number of the oriented points on the model. Massive amount of data can cause high computational cost and large storage spaces. Johnson and Hebert used principal component analysis (PCA) to reduce the redundancy of spin images. However, PCA generates separate feature space for each 3-D model to extract significant features. A query model needs to calculate different features for each model based on its feature space. This brings additional computational efforts. Thus, spin image signatures [27] is considered as the point feature to overcome the above drawbacks. The method is introduced briefly as follows.

psi  ns j 

x , yPSi

1 N SI

x , yNS j

1 N SI

x , ySTk



SI ( x, y ), i  1,



SI ( x, y ), j  1,



SI ( x, y ), k  1,

, np,

, nn, , nt ,

where PSi is the i-th shell in the positive-  plane, NS j is the j-th shell in the negative-  plane, and STk is the k-th sector in the spin image. The number of positive-  shells, negative-  shells and sectors denote as np, nn, and nt, respectively. A spin image signature SIS is defined as: SIS  { ps1 ,

, psnp , ns1 ,

, nsnn , st1 ,

, stnt } .

(11)

The spin image signature has the same properties as the spin image as a pose-independent descriptor.

(a)

(b)

(c)

Figure 1. Spin image signature: (a) positive shells, (b) negative shells, and (c) sectors.

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Data clustering is one of the classic problems in data mining and pattern recognition. Most clustering algorithms are based on iterative square-error partitioning or on hierarchical techniques. The simplest square-error partitioning method, kmeans, is adopted in this work. However, two difficulties of the k-means algorithm, selecting initial cluster centers and determining the parameter k, limit practical applications and reduce the accuracy of clustering. The k-means++ [28] algorithm is introduced to overcome the initial seeding problem. It improves the poor clustering results found by the standard kmeans algorithm. Figure 2. shows the color-encoded result of the surface segmentation applied to a cow model.

(10)

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stk 

1 N SI

C. Surface Segmentation The surface segmentation is the significant part of the graph representation. The basic idea of surface segmentation is that points with similar spin image signatures should be clustered in a group. Moreover, the construction of geometric graph model will be derived from the surface segments. Since the appropriate segmentation cannot be determined in advance, this study uses an unsupervised learning method to perform the surface segmentation. Hence, given a great amount of lowlevel features form all models of point cloud data, the surface segmentation can be done by applying vector quantization or data clustering.

The spin image signature is a region-based descriptor of spin image content. It is defined as an n-dimensional vector including three parts: positive shell (PS), negative shell (NS), and sectors (ST), as shown in Figure 1. These three parts are computed form a spin image SI as follows:

The point feature extraction is applied to each point in point cloud data. First, the spin image is generated for an oriented point. Then the spin image signature is calculated from the spin image. Additionally, the parameters of spin image signatures are illustrated in the experiment in section IV.

Since MLS algorithms depend on normals significantly, the initial normals are calculated approximately by using neighbor points. The refine normals will be obtained from MLS surfaces. However, the orientation of the yielded normal is not determined. To deal with this problem, the minimum spanning tree (MST) algorithm [26] is always employed to maintain the consistent orientation of normals.

(a)

(b)

Figure 2. The color-encoded result of surfaec segmentation for the cow model: (a) front view and (b) top view.

Another problem is determining the parameter k with lack of information about the extracted features. It is hard to identify the distribution of the high dimensional features. Therefore, for the training data, k-means algorithm is applied several times with different parameters to find a proper number of clusters. Furthermore, the silhouette validation method [29] is employed to measure the goodness of the result of clustering. The silhouette for i-th sample is defined as: s(i ) 

b(i )  a(i ) , max a(i ), b(i )

(12)

where a(i) is the average dissimilarity of i-th sample with all other data within the same cluster, and b(i) is the minimum of average dissimilarity of i-th sample to all objects in other cluster. The range of silhouette values is form -1 to 1. While the value closed to 1 means that the sample is assigned to a very appropriate cluster. On the contrary, the value closed to -1

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Chin-Chia Wu*, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 5, Issue No. 2, 116 - 122

D. Geometric Graph Model Most point features and local features only describe the neighbors around the points, and it is an arduous task to represent an entire surface or a complete model. On the other hand, global features describe complete object but ignore the details of objects. The proposed geometric graph model combines these two characteristics to describe a 3D model. The vertices of a geometric graph associate with local features, and the edges of the graph associate with the spatial distances among the surface segments. The geometric graph is defined as a weighted graph G  (V , E, L, A) composed of vertices and edges. V is the set of vertices and E  V V is the set of edges of graph G . The attribute A denotes the weight associated with the edge. Each vertex represents a segment on the surface and is labeled the clustered spin image signature L , that is, cluster centroid. In addition, a vertex has spatial position according to the barycenter of the surface segment. Hence, the weight describes the Euclidean distance between two vertices.

To match geometric graphs, an inexact graph matching method is adopted. The main reason is that it is impossible to guarantee that geometric graphs have exactly the same number of vertices. Moreover, different results of surface segmentation can cause various geometric graphs. Besides, the complexity of inexact graph matching is NP-complete. Hence, an efficient algorithm for inexact graph matching [18] is employed to match geometric graphs and discussed in detail below. The primary concept of this algorithm is that to explore iteratively the best possible vertex mappings and to select the best mapping at each iteration phase. The advantage of this algorithm is that the iterative process can often find the optimal mapping within a few iterations, significantly reducing the time for matching. To perform such match, the algorithm selects the best possible mappings that minimize the matching error due to vertices only in the first iteration. The first iteration of the algorithm is summarized as follows. Given two graphs, the scene graph Gs  (Vs , Es ) and the model graph Gm  (Vm , Em ) , in order to detect the most possible mappings, a matrix Pnm  ( pij ) is introduced, where n and m are the numbers of vertices in the scene graph and the model graph, respectively. Each element pij in P denotes

An adjacent criterion is proposed to determine the connection of the graph. If two surface segments are adjacent, the vertices associated with these two segments are connected by an edge e  E . Furthermore, the edge has no orientation.

weights of corresponding edges. The final matching error consists of these parts.

means that the sample is incorrectly classified. Consequently, the number of clusters with highest average silhouette is used to is determined the proper number of the clusters.

III. MODEL CLASSIFICATION This section presents an approach of the model classification that uses an inexact graph matching method. First, an efficient algorithm for inexact graph matching is introduced. After that, a classification procedure is proposed to classify a coming model with an appropriate class.

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A. Inexact Graph Matching The type of graph matching problems can be grouped into two classes: exact graph matching and inexact graph matching. Given two graphs, the scene graph Gs  (Vs , Es ) and the model graph Gm  (Vm , Em ) , with the same number of vertices, the exact graph matching problem is to find an isomorphism, that is, a one-to-one mapping f : Vs  Vm . If the isomorphism exists, it is said that Gs is isomorphic to Gm , or Gs matches Gm . In contrast, inexact graph matching means that it is impossible to find an isomorphism between the two graphs to be matched. This case happens when the number of vertices is different in both the model and scene graphs. Additionally, the found one-to-one mapping cannot preserve the edge structures. Therefore, in these cases no isomorphism can be obtained between both graphs, the graph matching problem becomes to search the best matching between them, as known as the subgraph isomorphism problem. Given two graphs, the goal is to find the best matching with the smallest matching error between their vertices. The smallest matching error between the two graphs can be determined as the distance between them. To calculate the matching error, two components should be considered: the vertex pairs and the

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the distance between vertex i in scene graph and vertex j in model graph. The matrix P is initialized by setting the element:

pij  d (vi , v j ) ,

(13)

where d () is the distance function of the attributes of vertices. In this study, the 2-norm distance is used. Another n  m matrix B is introduced to detect the most promising mapping. The matrix B is first initialized by setting bij  0 , and then the algorithm sets specific elements of B to 1. For each row in matrix B , the elements corresponding to the minimum elements in the same row of matrix P are selected as the specific elements. After that, for each possible mapping extracted from B , the algorithm computes the error generated by vertices and the error generated by edges. The mapping that gives the smallest matching error will be recorded. In the second iteration, the algorithm will reset some elements in each row of matrix B to zero. These elements correspond to the second smallest elements in each row of matrix P . The algorithm will extract those isomorphisms from matrix B that contain at least one vertex-to-vertex matching added to matrix B at this iteration. Of these isomorphisms and the isomorphisms obtained in the first iteration, those with the smallest cost are retained. The algorithm then proceeds to the next iteration until that the number of iterations reaches the limit or all elements in matrix B are marked as 1. B. Classification Procedure The proposed 3D model classification contains a training stage and a classifying stage. Figure 3. illustrates the two stages

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of the proposed approach. The two stages will be explained in detail to show how this classification procedure works.

appears to be an effective and stable measure for implementing evaluations.

The training stage is to create the classification database. All models in the training set are processed as follows. First, each 3D model in point clouds is applied the RIMLS algorithm to retrieve the surface information. Then, use the approach proposed in section II to extract the geometric graph representation of the model. Finally, the graph is stored in the database with its class information. In short, after the training stage, all models are represented as geometric graphs and class information.

Other details of the experiment are illustrated as follows. First, all models in PSB database are triangle mesh models. A procedure simulating the function of a range finder is used to obtain the point clouds on surfaces. Each model is scanned with the resolution about 4,000 points. Some parameters in the proposed method are summarized as follows: The models are scaled to unity.



The dimension of spin images is 32  32, and the support angle is 90 .



The parameter (np, nn, nt ) of spin image signatures sets to (6,6,6) .

TABLE I.

THE OVERALL PERFORMANCES OF MODEL CLASSIFICATION FROM THE PSB DATABASE

The classifying stage is to provide a quick classification for a coming model. As same as training models, the geometric graph representation of a coming model is brought by applying the approach mentioned above. Subsequently, the inexact graph matching is used to find the best match across the model database. As the result, the class belonging to the best match is assigned to the incoming model.



Method

NN (%)

DCG (factor)

proposed

75.8

0.641

SIS

63.2

0.625

Table 1 summarizes the overall performances of the 3D model classification across the entire database. The second column shows that the proposed method achieves 75.8% of the NN measure, and the third column shows that the proposed method achieves 0.641 in DCG measure, which is higher than the score of SIS. Indeed, our method shows the better results than SIS in both evaluation measures.

Figure 3. The process of the proposed 3D model classification.

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IV. EXPERIMENTAL RESULTS AND DISCUSSIONS To demonstrate the performance of the proposed approach, we have performed several experiments for 3D model classification by using PSB database [30]. PSB is the wellknown benchmark for 3D model retrieval and classification. It contains 1,814 objects in general categories like animal, building, vehicle, and so on, and is divided into training and testing two sets. The training set that has 907 objects classified in 90 categories is used to create the ground-truth of the classification database. The other that includes a different 907 objects in 92 classes is used for computing the classification performances.

The performance of model classification is evaluated by computing quantitative statistics of match results. The spin image signature (SIS) [27] is employed as a competing method to verify the performance of our proposed method. According to the characteristics of different methods, measurement results will show retrieval effectiveness under certain conditions. Two commonly used evaluation measures, which are nearest neighbor (NN) and discounted cumulative gain (DCG), are used to measure the quality of model classification. All scores of these evaluation methods are in range [0, 1]. Higher scores represent better results. It is important that DCG takes the position of relevant models into account. Therefore, DCG

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To show more detailed classification comparisons, four commonly used datasets, which are animal, furniture, common tool, and vehicle in PSB, are chosen. Each dataset includes several classes. The classification performances are measured by DCG histogram. Moreover, the performance is compared with spin image signature method. Figure 4. shows the averaged DCG scores of each classification for all datasets in the experiment. The proposed method performs better retrieval precision than SIS methods in about 70% of classes. Nevertheless, a small part of results compares poorly with the results from SIS method. The classification performance will have different degrees of accuracy in different categories. The major causes of this phenomenon are as follows. The objects in the same class usually have a number of different poses and many variations in shape, which increases the difficulty in obtaining an accurate categories result. For example, the snake has variable shape in the class, and therefore does not have a uniform shape. However, the proposed method uses a geometric graph representation to overcome the ambiguity of the non-uniform shape. As Figure 4. (a) shows, the performances of the proposed method is better than SIS method. In summary, the proposed method has good performance compared with SIS method both in terms of model classification in the entire testing dataset or in a specific class.

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Figure 4. The histograms of DCG for model classification performance on four test datasets. (a) Animal, (b) furniture, (c) common tool, and (d) transportation dataset.

V. CONCLUSION In this paper, an approach based on the geometric graph representation is proposed for classifying 3D models in point clouds. The approach first uses a RIMLS technique to calculate the geometric information. The point feature extraction is achieved by using spin image signatures. Then a surface segmentation is performed to cluster the point features. After that, a geometric graph model is introduced to generate the representation of model content for each model. To classify an incoming model with an appropriate class, an efficient algorithm for inexact graph matching is employed. In summary, the experimental results show that our approach outperforms the original spin image signature method, and has good performance in model classification. Future research is obviously required, but this is an exciting first step. Additional research in this area of feature classification should prove quite beneficial. In addition, this work might be extended to retrieve 3D models and 3D model recognition.

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