3D model retrieval scheme based on fuzzy clustering for physical descriptors

Abstract

With the rapid development of 3D scanners, graphic accelerated hardware and modeling tools, the application of 3D model databases is growing in both numbers and size. There is a pressing need for effective content-based 3D model retrieval methods. In this paper, a novel 3D model retrieval scheme called Fuzzy Clustering for Physical Descriptor is proposed. First, a multi-scale indexing of 3D model database is generated for fast matching by using fuzzy clustering for a concise descriptor named as partial physical descriptor, and then a novel and effectiveness descriptor named as the global physical descriptor is embedded into feature matching with the generated indexing for obtaining more accurate retrieval results. Several retrieval performance measures demonstrate that the proposed approach is superior to other methods and robust to the missing polygons and noise.

Keywords

3D model retrieval global physical descriptor partial physical descriptor fuzzy clustering

Introduction

With the rapid development of 3D scanners, graphic accelerated hardware and modeling tools, the use of 3D model database throughout the Internet is growing, and this trend should continue. 3D models play an important role in several domains, such as protein classification, computer-aided design, virtual reality, anthropometric data collection and so on. Duplicate designs consume a large amount of enterprise resources during product development, thus automatic search for similar parts is an effective solution for the design reuse. The popularization of image retrieval and 3D data development had created an urgent demand for the effective 3D model retrieval methods and systems. Since searching for similar models in 3D database can sometimes speed up the design and research process, thus, the content-based 3D model retrieval method have become a hot topic in the last few years.¹ Such method can be used to discover geometric relationships between 3D data and find the required data from the local databases or from the Internet.

Several retrieval methods for similar 3D model searching in databases are surveyed,² and a comparative study of these methods has been reported.³ Various 3D shape searching techniques are classified and compared based on shape representations, and the further research directions are introduced.⁴ The main challenge to a content-based 3D model retrieval system is the extraction of the most representative features of 3D models.⁵

The commonly adopted methods are distribution-based,^6–13 transform-based,^14–17 and 2D views-based.^18–22 Distribution-based methods collect numerical values from certain attributes of the 3D model, such as the local or global features, which are efficient and easy to implement. The D2 descriptor is proposed by using the probability distribution histogram of the two randomly selected points from the object’s surface,⁶ in order to improve the retrieval performance of D2, the angle-distance (AD) and absolute angle-distance (AAD) descriptors have been proposed by Ohbuchi et al., the Beta/Distance (BD) and Alpha/Beta/Distance (ABD) have been proposed by Reisert and Burkhardt.⁸ The concentric ABD (CABD) and symmetrical ABD,⁹ fractal D2 distribution,¹⁰ and the combined shape distribution have been proposed by Zou et al.¹¹ Although most of the above methods lack the fine grain discrimination required for the retrieval task, the retrieval performance could be greatly improved by combining multiple methods or extracting some novel descriptors.^12,13 Discovering some untapped and distinctive characteristics to improve the distribution-based methods is meaningful, hence, there have been many distribution-based methods proposed in recent years.

Transform-based methods register the surface points onto a 3D voxel or spherical grid by using a mapped function, which is then processed by transform tools, the compacted descriptors can be obtained by keeping the first few transform coefficients in the descriptor vector. Furthermore, the pose invariance can be obtained by discarding the “phase” of the transform coefficients at the expense of some shape information. The Gaussian Euclidean Distance Transform (GEDT) descriptor and Spherical Harmonic Descriptor (SHD) are proposed by Kazhdan et al.¹⁵ GEDT is a volumetric representation of the Gaussian Euclidean Distance Transform of a 3D object, expressed by the norms of spherical harmonic frequencies, where SHD is a rotation invariant representation of the GEDT. Radicalized Spherical Extent Function (REXT)¹⁶ is a collection of spherical functions giving the maximal distance from the center of mass as a function of spherical radius and angle. A Concrete Radicalized Spherical Projection (CRSP) descriptor was proposed by Papadakis et al.,¹⁷ which uses the Continuous PCA (CPCA) along with the Normal PCA (NPCA) to alleviate the rotation invariance problem and describes a 3D object using a volumetric spherical function based representation expressed by spherical harmonics.

2D views-based methods consider the 3D shape as a collection of 2D projections taken from different viewpoints of the 3D model, and each projection is then described by the standard image descriptors, such as the Fourier descriptor or the Zernike moments. Chen et al.¹⁸ proposed the Light Field Descriptor (LFD), which is comprised of the Zernike moments and Fourier coefficients computed on a set of projections taken from the vertices of a dodecahedron. Vranic proposed a shape descriptor where features are extracted from the depth buffers produced by six projections of the object, one for each side of a cube which encloses the object. In the same work, the Silhouette-based (SIL) descriptor is proposed which uses the silhouettes produced by the three projections taken from the Cartesian planes. Vranic¹⁹ developed a hybrid descriptor called DESIRE, which consists of the Silhouette, Ray and Depth buffer based descriptors, which are combined linearly by the fixed weights. A Bayesian 3D search engine using adaptive views clustering was proposed by Ansary et al.,²⁰ which is providing an “optimal” selection of 2D views from a 3D model, and a probabilistic Bayesian estimation was done for the above views. A novel 3D shape descriptor named as PANORAMA was proposed,²¹ which uses a set of panoramic views of a 3D model describing the position and orientation of the model’s surface in 3D space. The symmetrical depth images are proposed for retrieval 3D models in the literature.²² The above mentioned methods could obtain a good retrieval performance but have large feature size and high matching cost.

In this research, a novel 3D model retrieval method is proposed by using Fuzzy Clustering for Physical Descriptors (FCPD). The physical descriptor is defined as the physical features extracted from the 3D surface, which is the combination of the global physical descriptor (GPD) and partial physical descriptor (PPD). First, after preprocessing for the 3D models, each model is partitioned into several parts by the planes paralleling the XOY, YOZ and XOZ planes, respectively. Each sliced partial part is represented by a physical feature named as PPD, which is a combination of the inertia moment, the elastic potential energy and the density. Second, a PPD feature dataset is determined, related to the 3D model dataset. Fuzzy clustering is then done for the feature dataset, in order to display the closest clusters in priority, when a query is given. Third, the GPD is used for the feature matching after the indexing scheme. The proposed method has high discriminative power and robust to the missing polygons and noise.

The rest of the paper is organized as follows: The PPD is introduced in section Partial physical descriptor. The GPD is defined in section Global physical descriptor. FCPD is introduced in Fuzzy clustering for physical descriptor. The experimental results are given in section Experimental results. Finally, the conclusions are drawn in section Conclusion.

Partial physical descriptor

The PPD is an easy computed and effective shape-distribution feature.²³ The concept of the PPD is that if the two 3D models correspond to each other, then their sliced parts should also correspond to each other. The physical descriptor is proposed to represent each part of the 3D models, which is to synthesize of the main physical characteristics of 3D models: the inertia moments, the elastic potential energy and the density. Since a whole model is complex and irregular, it is partitioned into many regular parts at first, and then the Inertia Moments Descriptor (IMD), the Potential Energy Descriptor (PED) and the Density Descriptor (DD) are computed, respectively; finally the PPD is obtained by combining the three descriptors.

Global physical descriptor

Most of the proposed features are extracted from the whole 3D surface and generate sample points based on Monte Carlo method is widely used for its good characteristics.^6–8 Thus, the sample generation method is also used in this research. As shown in Figure 1, a 3D ant model is shown in Figure 1(a), and the sample representation is shown in Figure 1(b). The GPD is defined as follows before feature extraction.

Figure 1.

An ant model (a) and its sample representation (b).

GPD Definition: Assume that P = (P₁, P₂ … P_n), which is a sampled point set extracted from 3D surface. The center of mass of the 3D model is O, the 3D sampled point can be considered as electrons around with O. Take each pair of them, and compute the follow features (a–d), a combined features distribution is defined as the global physical descriptor (GPD).

As shown in Figure 2, p₁ and p₂ are the two electrons around the center of mass O, where Q and q_i are the electric quantity of O and electrons p_i. The following features can be calculated.

The distance r between electrons p₁ and p₂_.

The absolute value of electric potential difference E_d between p₁ and p₂. Assume that E₁ and E₂ are the electric potential energy of p₁ and p₂, $E_{d} = \frac{| E_{1} - E_{2} |}{Q}$ , where $E_{i}, (i = 1, 2)$ is defined as: $E_{i} = \frac{{kQq}_{i}}{r_{i}}$ .

The absolute value of angle $α_{e}$ among p₁, p₂ and the center O, which is defined as: $α_{e} = \frac{| n_{2} \cdot n_{1} |}{‖ p_{1} - p_{2} ‖ ‖ p_{1} - O ‖}$ .

The absolute value of angle $α_{l}$ between p₁ and p₂. Since each random sampled points located at one 3D triangle mesh surface, the normal vectors $n_{p_{1}}$ and $n_{p_{2}}$ of 3D triangle meshes describe the location of electrons p₁ and p₂. Thus, $α_{l}$ is defined as $| n_{p 1} \cdot n_{p 2} |$ .

Figure 2.

The two random sampled electronics p₁ and p₂ around the center of mass O, where r is the distance between p₁ and p₂, E₁ and E₂ are the electric potential energy of them. n₁, n₂, and n₃ are the direction vector of ${Op}_{1}$ , $p_{1} p_{2}$ , and ${Op}_{2}$ , $n_{p_{1}}$ $n_{p_{2}}$ are the normal vector of the 3D surface where the electrons located.

Group integration is an effective method to obtain invariant features,⁸ which is a constructive approach. It is supposed that f(x) is a kernel function for the object; an invariant feature can be constructed by integrating the kernel over the considered group

GPD = f (r, e, α_{e}, α_{l}) = \int_{(p_{1,} p_{2}) \in S \times S} δ_{r} (‖ p_{1} - p_{2} ‖) δ_{e} (‖ e_{1} - e_{2} ‖) \times δ_{l} (| n_{p 1} \cdot n_{p 2} |) δ_{o} (\frac{| (p_{1} - p_{2}) \cdot (p_{1} - O) |}{‖ p_{1} - p_{2} ‖ ‖ p_{1} - O ‖}) d r d e d α_{e} d α_{l}

(1)

Since it is an improved shape distribution-based method, it is also a rotation invariant descriptor as others.^12–14 The dimension of r, e, $α_{l}$ , and $α_{e}$ is selected as 16, 4, 4, and 4, respectively, and the dimension of GPD is 1024.

Fuzzy clustering for physical descriptor

In this part, the detail process of the proposed 3D model retrieval scheme will be introduced. The flow chart of the proposed method is shown in Figure 3. Users can do a fast or accurate retrieval based on the proposed scheme according their needs.

Figure 3.

The flow chart of the proposed methods.

Fuzzy clustering for PPD

The FCM²⁴ starts off with clustering 3D feature database in advance before a query is given. Assume that the 3D model dataset contains n models and they are classified into N_t categories. Thus, FCM is used to cluster the features of the dataset into N_c classes. It is not necessary to cluster the features into the only one category which they belong to. In addition, two or more categories are clustered into one class and different weights are given to different classes during the similarity matching. The average cluster accuracy is used to evaluate the feature clustering performance. Assume that an original category i containing K_i models is clustered into N_c classes, find the class j including the most models of category i, The accuracy of i-th category is defined as follows

A_{i} = max (\frac{K_{j}}{K_{i}}), j = 1, \dots, N_{c}

(2)

The average accuracy is defined as

A = \frac{1}{N_{t}} \sum_{j = 1}^{N_{c}} A_{m}

(3)

A novel clustering and retrieval measurement named as Clustering and Retrieval (CR) is defined as follows.

CR = \frac{1}{N_{t}} (1 - \frac{N_{c}}{N_{t}}) 2 + (1 - A) 2

(4)

S . t {0.8 \leq A \leq 1 1 < N_{c} < \frac{N}{max (K_{i})}, i = 1, 2, \dots, N_{t}

(5)

where N is the total number of 3D models in the database. The constraints in equation (5) are educed by experience. Both a lower value of A and a larger value of N_c are harmful to the retrieval results. It’s a quadratic programming problem. By repeatedly training, the optimal CR will be selected satisfying the constraints.

A symmetrical distance is used for PPD distance computation to weak the no orientation influence by CPCA. Let a, b, be the two 3D models. h_a and h_b are the two PPD features of them. Since each 3D model has a separate coordinate system, so the negative part of one model may be more similar to the positive part than the negative part of another one. Hence, the similarity measure is adopted as the minimum distance of these two types: the similarity of h_a and h_b, the similarity $h'_{a}$ and h_b. Where $h'_{a}$ is the symmetrical histogram of h_a. For example, the symmetrical PPD histogram of the 3D model a is $h'_{PPDa}$ , which is defined as

h_{PPDa}' = [IMD'_{1} EPED'_{1} DD'_{1} IMD'_{2} EPED'_{2} DD'_{2} IMD'_{3} EPED'_{3} DD'_{3}]

(6)

where

h'_{SAEDa}

h_{PPDa} = [{IMD}_{1} {EPED}_{1} {DD}_{1} {IMD}_{2} {EPED}_{2} {DD}_{2} {IMD}_{3} {EPED}_{3} {DD}_{3}]

(7)

The distance measure of PPD descriptor is compared using the following formula

d (b - a) = min {\sum_{i = 1}^{n} \frac{2 | h_{PPDa} (i) - h_{PPDb} (i) |}{| h_{PPDa} (i) + h_{PPDb} (i) |}, \sum_{i = 1}^{n} \frac{2 | h_{PPDa}' (i) - h_{PPDb} (i) |}{| h_{PPDa}' (i) + h_{PPDb} (i) |}}

(8)

The complete procedure of fuzzy clustering for the feature dataset is shown as follows

Algorithm I Fuzzy Clustering ()

Begin

Step 1

Obtain the feature dataset matrix F[n, m], where m is the length of one feature histogram, and n is the total number of features, and then initialize the number of clusters N_c and membership matrix U.

Step 2

Calculate the fuzzy cluster center c_i using

c_{i} = \sum_{j = 1}^{n} \frac{μ_{ij}^{m}}{\sum_{k = 1}^{n} μ_{ik}^{m}} x_{j}, j = 1, 2, \dots, n

Step 3

Calculate the weighted means M_ij using

M_{ij} = \sum_{\binom{k = 1}{k \neq j}}^{n} \frac{d (c_{i} - x_{k}) - 1 u_{ik}}{\sum_{\binom{l = 1}{l \neq j}}^{n} d (c_{i} - x_{l}) - 1 u_{it}}

Step 4

Update the Lagrange multiplier $ξ_{j}$ using

ξ_{j} = (\sum_{i = 1}^{c} (d (M_{ij} - x_{j}) 2 m \sum_{i = 1}^{c} u_{ij}^{m - 1}) \frac{1}{1 - m}) 1 - m

Step 5

upgrade the membership grade u_ij using

u_{ij} = ξ_{j}^{1 / m - 1} (d (M_{ij} - x_{j}) 2 m \sum_{i = 1}^{c} u_{ij}^{m - 1}) \frac{1}{1 - m} .

Step 6

If $max_{ij} {| μ_{ij}^{(k + 1)} - μ_{ij}^{(k)} |} < ɛ$ , go to step 7, else go to Step 2.

Step 7

repeatedly training until got the optimal CR value.

End

Similarity matching based on GPD

When a query is given, if the query model is selected from the local 3D model dataset, find its PPD feature q_PPD in the feature dataset. Then q_PPD is compared with N_c new cluster centers by L₁ distance, the distance value is in accordance with the order from small to large, and a related indexing sequence will be generated, which represents the priority of the searched model in the dataset to the query model. In other words, it is a weighted vector, giving different weights to the clusters, in order to make sure that the most similar clusters were searched out in priority, where it is noted as Seq.

{i \in [1, N_{c}] | Seq (j) = i, j = 1, 2, \dots, n}

(9)

where n is the total number of 3D models.

The similarity matching method of the descriptors presented is based on distance measuring, the most widely adopted is the L_n distance. However, it is not a sufficient dissimilarity metric for histogram-based matching.⁸ Some more sophisticated metrics, such as the normalized distance²⁵ and the diffusion distance²⁶ have been proposed for similarity measures. The normalized distance is used as the distance measure of the GPD. The similarity of the query model to a model in the searched dataset is listed as follows based on FCPD

S_{qj} = \frac{k}{Seq (j)} (1 - \frac{2 | q_{GPD} - f_{GPD} (j) |}{| | q_{GPD} + f_{GPD} (j) | |}) \times 100 %

(10)

where k is a constant, which is usually set as 1 in the experiment.

If the query model is out of the searched dataset, the PPD feature of the query model, which is noted as q’ _PPD , is calculated by using the aforementioned methods. Then it is compared with the feature dataset one by one, find the closest feature q_PPD. The closet 3D model is used to represent the query model, whose GPD feature is also used to represent the GPD descriptor of the query model. Thus, the similarity of the query model to one of the model in searched database is computed based on equation (10). The detail process is described in Algorithm II.

Algorithm II Similarity matching ()

Begin

Step 1

Select a query model q, and if the query model belonged to the searched dataset, find the its PPD feature $q_{PPD}$ , and go to step 2. else go to step 4.

Step 2

Then $q_{PPD}$ is compared with N_c new cluster centers by $D_{q} = ‖ q_{PPD} - c_{j} ‖$ . The distance value D_q is sorted by $[D'_{q}, Seq] = sort (D_{q})$ , and an indexing sequence Seq will be generated.

Step 3

compute the similarity value of the query model to the searched model by using equation (10).

Step 4

Calculate the PPD feature of the query model $q'_{PPD}$ , compared it with the feature dataset, find the closest feature $q_{PPD}$ . The closet 3D model is used to represent the query model, whose GPD feature is also used to represent the GPD descriptor of the query model, which is noted as $q_{GPD}$ . Then go to Step 2.

End

Experimental results

The databases of the Princeton Shape Benchmark (PSB),²⁷ SHREC2009²⁸ and SHREC2014²⁹ are used in the experiments. The PSB provides a test data set includes 907 models with 92 classes. SHREC2009 consists of the query and target data set. The target set is composed of 720 3D models classified in 40 categories. The query set consists of 80 3D models, two from each category. SHREC2014 is a large-scale 3D shape benchmark which contains 8987 3D models divided into 171 distinct classes. It was complied to be a superset of existing benchmarks and presents a new challenge to retrieval methods and it comprises generic models as well as domain-specific model types. The proposed method was tested on a Core2 Quad, 2.7GHZ system, with 2G of RAM.

The precision-recall plot and the DCG vs NN diagram were used for performance evaluation.

Precision-recall plot: For each query model in class C and any number K of top matches, recall is the percentage of models in class C accurately retrieved within the top K matches. Precision represents the percentage of the top K matches which are members of class C. The precision-recall plot indicates the relationship between precision and recall in a ranked list of matches. Curves closer to the upper right corner represent superior retrieval performance.

Nearest neighbor (NN): The percentage of models that are the closest to the query one and which are in the same category.

Discounted Cumulative Gain (DCG): A user is more likely to consider elements near the front of the ranked list. DCG is a statistic measure that weights correct results near the front of the ranked list more than correct results later in the list.

Robustness evaluation of the proposed method

In order to evaluate the robustness of the proposed FCPD, the following transformations are applied for all the 3D models from the PSB dataset. Each transformed 3D model is then used to query the test dataset. The average recall and precision of all the test dataset are used for the evaluation.

The robustness is evaluated by the following transformation.

Noise: randomly adding the noise –x to x to directions of x-, y-, and z-axis of all the 3D model vertices, where x is selected as 5%, 10%, 15%, and 20% the length of the model’s bounding box, respectively).

Decimation: for each 3D model, randomly select 20% polygons to be deleted.

A typical example of the deformation effect is shown in Figures 4–4(a) is an original 3D human model and Figure 4(b) shows a typical example of deletion, while Figure 4(c)–(f) show examples of noise effect.

Figure 4.

Deformation of noise and deletion from a 3D model. (a) Original 3D model. (b). Model with 20% deletion. (c) Model with 5% noise. (d) Model with 10% noise. (e) Model with 15% noise. (f) Model with 20% noise.

Precision versus Recall plots were computed for each value of noise and for decimation (Figure 5). From these results, it can be observed that the performance of our approach is robust to 5% and 10% noise addition and to decimation. Even if 15% and 20% are high noise addition, Precision versus Recall plots show that the proposed method remains quite robust.

Figure 5.

The precision-recall plots on PSB dataset with deformations.

Comparing with distribution-based methods

The proposed method is compared with the classic distribution-based 3D shape descriptors by using the precision versus recall plot. The D2, AAD, BD and ABD are implemented by us, and the normalized distance is used as the distance measure of these descriptors.

The average precision-recall plots of the FCPD and the classic distribution-based descriptors on the PSB dataset is shown in Figure 6, while on the SHREC2009 dataset is shown in Figure 7. Each 3D model is used to query the dataset on the PSB, while each model in the query set is applied to query the target set on the SHREC2009. In general, the FCPD is greatly improved on the two datasets, especially on the SHREC 2009. It is proved that when the query model is not selected in the dataset, FCPD is also effective.

Figure 6.

The precision-recall plots on the PSB dataset.

Figure 7.

The precision-recall plots on the SHREC2009.

Comparing with well-known and recently proposed methods

Retrieval performance of the proposed FCPD and other well-known descriptors on the PSB dataset is compared by using the DCG vs NN plots shown in Figure 8.

Figure 8.

The DCG vs. NN for descriptors on the PSB dataset.

As shown in Figure 8, a DCG versus NN scatter plot¹³ is illustrated the performance landscape, where each descriptor is indicated by a symbol. Generally speaking, 2D views-based methods obtain the best results, while the distribution-based methods are placed at the lower end of the performance landscape but can be effective by combining different types of shape information. DBF performs better than others except PANORAMA and the cluster-based methods, which applies the density estimation scheme to capture multivariate local surface information. By combing multivariate global and local shape information and adopting a cluster-based search, The FCPD performs the best than others.

The proposed FCPD was also compared with the recently published methods in the large-scale 3D model benchmark SHREC2014, e.g. the OPHOG proposed by Tatsuma,²⁹ the BF-fGALAF and CDMR proposed by Furuya,³⁰ the SBR-VC proposed by Li³¹ and BOF-JESC proposed by Zou,³² the experimental results are shown in Figure 9. Obviously, the retrieval performance of the proposed FCPD is better than the aboved methods.

Figure 9.

The precision-recall plots on the SHREC2014.

Conclusion

The main contributions of this paper are listed as follows.

First, a novel concept by mining the physical features to represent a 3D model is proposed the PPD is easy to compute, store, and cluster; the GPD is a rotation invariant descriptor, the random sampled points on the 3D surface is treated as the electronics, and the related electronic information are extracted for the GPD computation. The finally proposed FCPD, which represented the Fuzzy Clustering for Physical Descriptor, performs good retrieval results and robust to the missing polygons and noise.

Second, a novel combination of the two descriptors is proposed by embedding fuzzy clustering into them. Combining different descriptors together is also a hot topic these years. The mainly used methods are fixed weights of each descriptor. This paper proposed an inner integration of two descriptors by using fuzzy clustering.

Future work is to discover spatial relationships among local parts of a model adopt more distinctive physical descriptors and mine associations among these descriptors, in order to support a more effective retrieval; Suitable feature clustering methods will also be studied to improve the efficiency of retrieval; Since a pre-cluster process is effective for 3D model retrieval, the effective and general used novel clustering method can be studied. It can be extended to address the task of partial object matching. More intelligent segmentation methods can be used to slice a 3D model, features of each sliced parts could be extracted at first, and then intelligent matching could be considered to used for these sliced parts.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of China under Grant Nos. 61503164, 61305149, 61304175 and 61304153.The Natural Science Foundation of Jiangsu Province under Grant No. BK20140241 and the Natural Science Foundation of Jiangsu Higher Education Institutes of china under Grant No. 13KJB520007. This work is also partly supported by the Nature Science Foundation of Tianjin (No. 15JCQNJC04200), the High School Science and Technology Development Fund Project of Tianjin (No. 20120828).

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