Elsevier

Pattern Recognition

Volume 45, Issue 4, April 2012, Pages 1721-1738
Pattern Recognition

Partial retrieval of CAD models based on the gradient flows in Lie group

https://doi.org/10.1016/j.patcog.2011.09.017Get rights and content

Abstract

Based on the gradient flows in Lie group, a partial retrieval approach for CAD models is presented in this paper. First, a representation of the face Attributed Relational Graph (ARG) for a CAD model is created from its B-rep model and thus partial retrieval is converted to a subgraph matching problem. Then, an optimization method is adopted to solve the matching problem, where the optimization variable is the vertex mapping and the objective function is the measurement of compatibility between the mapped vertices and between the mapped edges. Different from most previously proposed methods, a homogeneous transformation matrix is introduced to represent the vertex mapping in subgraph matching, whose translational sub-matrix gives the vertex selection in the larger graph and whose orthogonal sub-matrix presents the vertex permutation for the same-sized mapping from the selected vertices to the smaller graph's vertices. Finally, a gradient flow method is developed to search for optimal matching matrix in Special Euclidean group SE(n). Here, a penalty approach is used to handle the constraints on the elements of the matching matrix, which leads its orthogonal part to be a permutation matrix and its translational part to have different integer elements. Experimental results show that it is a promising method to support the partial retrieval of CAD models.

Highlights

► Partial retrieval of CAD models is converted to a subgraph matching problem. ► The match is represented by a homogeneous transformation matrix. ► The translation part of the matrix gives the vertex selection in the larger graph. ► Its rotation part expresses the permutation of the selected vertices in a match. ► A gradient flow method is used to search for the optimal matching matrix in SE(n).

Introduction

In recent years, the popularity of three-dimensional (3D) CAD systems in the product design of manufacturing industry brings about the emergence of a large number of 3D CAD models. As the number of CAD models is continuously growing and a lot of CAD models are already available from public and proprietary databases, it becomes a problem how to search a model that is of reference value for a new design. Therefore, since recently, the CAD model retrieval has received extensive attention in the academic community.

An essential operation in the CAD model retrieval is model shape matching. Since direct comparison of 3D shape is not convenient, some intermediate shape description data generated from original models are usually adopted for the comparison purpose. As a matter of fact, 3D shape description has been widely studied in computer graphics area and some of the results can be directly applied to CAD models. Most shape description methods developed in computer graphics community can be classified into histogram-based, transform-based, view-based, graph-based and the combinations of the above. The most famous histogram-based method is called shape distributions which are probability distribution curves of distances between two points randomly sampled on model surface or angles between their normal vectors [1]. A newly proposed histogram-based method is the 3D shape impact descriptor in which histograms record the numbers of gravitational field values on the surfaces surrounding the 3D object at various equal-distances [2]. Typical transform-based methods include spherical harmonic descriptors and 3D Zernike descriptors, which describe a 3D shape with the moments of the 3D object's volumetric function, respectively, with respect to spherical functions and 3D Zernike functions [3], [4]. Both of them are a kind of generations of Fourier transforms but the moments are invariant under rotation, translation and scaling. Compared with spherical harmonic descriptors, 3D Zernike descriptors take into accounts the coherences of shapes on different spherical surfaces in radial direction. A specific transform-based method is symmetry descriptors that transform the volumetric function into a function on the unit sphere measuring reflective symmetry of the 3D object with respect to planes through the mass center and perpendicular to the sphere normal vectors [5]. Furthermore, rotational symmetry descriptors were also proposed in the similar way [6]. View-based methods use 2D projections of 3D objects in different directions and various existing shape descriptors for 2D images can be utilized to represent the shapes of the projections [7], [8]. Graph-based shape descriptors are some types of abstractions of 3D object's solid structures or surface region relations. The former include skeleton graphs [9], [10] and Reeb graphs [11], [12] while the latter have surface segment graphs [13], feature graphs [14], [15] and face graphs [16]. Compared with the previous methods, graph-based descriptors support more precise shape matching but they need more complicated matching algorithms. Since most of the above shape descriptors can only capture a part of shape characteristics of a 3D object, many 3D model retrieval systems combine them together to fulfill a model search task [17]. For comparison of their strength and weakness, some benchmarks have been developed [18], [19], [20]; the most famous one is the Princeton Shape Benchmark (PSB) [18]. Tangelder and Veltkamp have given a good survey on most of the shape analysis and description methods mentioned above [21].

Most of the above methods focus on the global shape description and support model retrieval based on global shape matching. However, the model retrieval based on partial shape matching may be used more frequently in product design. Funkhouser et al. [22] showed how partial models retrieved from databases can be utilized in synthesis of new product design and the main benefit of the method is that the retrieved models usually offer highly detailed geometric design for a user design concept. Fisher et al. [23] revealed another kind of applications of partial model retrieval, in which the strength of relationships between sub-models in databases provides a clue for finding a proper candidate sub-model in a given location. Some other usages have also been mentioned in literature. For example, multiple local-region matches may help the alignment of two models for their global matching and the precision of whole model retrievals may be improved by performing local feature matching as well. Most partial model matching approaches are based on model segmentation [22] or local feature extraction [24] and the bag of segments or features with their shape descriptors generated in the same ways as a whole model are employed for the partial matching. Gal and Cohen-Or [24] proposed a method for extracting salient geometric features by analyzing the variations of surface curvatures. The salient geometric features are some local surface regions whose curvatures are distinctive compared with their surrounding areas and their union may not form the whole model surface like model segmentation. Schreck et al. [25] used octant-based partitioning to decompose a 3D model into multiple parts for global and partial 3D object retrieval. Ferreira et al. [26] proposed a collection-aware segmentation approach which considers segments generated in the segmentations of other models. Philipp-Foliguet et al. [27] decomposed artwork 3D models in a database with watershed cut. For evaluation of different segmentation algorithms, Chen et al. [28] developed a benchmark comprising a data set with 4300 manually generated segments from 380 surface meshes of 19 different object categories. After segmentations, all the segments of the query model should be compared with a subset of segments of a model in database. Although most partial retrieval methods ignore the matching of the relations between the segments due to efficiency problem, some researchers has considered the relation matching in some way. Funkhouser et al. [29] handled the segment pair mapping problem with priority-driven search and Bronstein et al. [30] expressed and matched the relations with a matrix called bag of expressions.

While most methods for multimedia models can be used, some researchers have paid attentions to approaches that specialize in CAD models [31]. El-Mehalawi et al. [16] index and retrieve 3D models in CAD format based on face adjacency graphs whose nodes and links, respectively, correspond to faces with a shape of plane, cylinder, sphere or spline surface and edges connecting the faces. Cardone et al. [32] have studied CAD model retrieval methods that focus on their similarity of machining process. Li et al. [33] described CAD models and their decomposed components with feature dependency directed acyclic graph (FDAG) for reusable model retrieval, which can capture some related engineering knowledge besides their shapes. For the comparison of CAD model retrieval algorithms, Jayanti et al. [19] have developed an engineering shape benchmark called ESB and Bespalov et al. [20] have created a classified CAD model database called the National Design Repository. The former offers CAD models in mesh format while the latter provides models in B-rep format, which is a precise shape representation. For partial CAD model retrieval, graph-based model descriptions are usually preferred because they are more suitable for the representations of shapes in various scopes. Saber et al. [34] proposed a graph approach to represent 2D shapes with feature points on boundary and their distances. Bespalov et al. [15] introduced a graph method to describe 3D models with extracted local features and their adjacency relationships. Biasotti et al. [12] used the extended Reeb graphs to capture the structural information of 3D models. The feature-point graphs are adopted by Saber et al. is easy to construct, but the graph vertex number may become too large for 3D CAD models. The feature-adjacency graphs used by Bespalov et al. may be small but the local feature extraction based on scale-space decomposition is complicated. The extended Reeb graphs usually have a reasonable size but they are weak in description of model's surface shapes and their cost for graph generation is usually rather high. In this paper, we utilize the Attributed Relational Graph (ARG) of model faces as CAD model shape description similar to that presented by El-Mehalawi et al. [16]. Compared with the graphs mentioned above, the face ARG description has an advantage that it is convenient to be created directly from CAD model's boundary representation (B-rep).

For graph-based model descriptions, graph matching and subgraph matching are the major means for realizing the whole and partial model retrieval, respectively. Saber et al. [34] used the distance matrix to represent the feature-point graph and replace subgraph matching with sub-matrix matching. Since the feature points on a 2D boundary can be arranged sequentially, the sub-matrix matching is relatively easier than that for other subgraph matchings. In the work presented in literature [12], [13], [14], [15], the graph matching for the model retrieval based on both the feature-adjacency graphs and the extended Reeb graphs implements the largest common subgraph algorithm with some heuristics. Although existent graph matching algorithms can be used for the model comparison, they may not be effective and efficient enough. First, graphs like ARGs for CAD models may be large when their nodes represent the faces of solid models and the matching operation number may be huge for the retrieval in a large database. Second, the current approaches for efficiency improvement like search heuristics usually suffer from a problem of poor approximation to the exact solution. As a matter of fact, although heuristics is useful in avoidance of some unnecessary matching operations, graph matching is still indispensable and model retrieval still calls for a better graph matching algorithm.

Graph matching, which is to check graph isomorphism or sub-graph isomorphism, has been extensively studied in the pattern recognition field and mainly there are two kinds of approaches that have been developed. One is the technique based on the tree search with backtracking. In order to reduce the size of search space, different approaches for pruning the search branches like refinement procedures [35], forward-checking and looking-ahead [36], and discrete relaxations [37], have been proposed. Although these methods guarantee to find the optimal solution, their time costs are usually exponential due to the NP-complete property of the problem. The second kind of approaches is based on continuous optimizations that solve the graph matching problem with an optimization procedure. Some typical optimization methods used for graph matching include probabilistic relaxations [38], [39], neural networks [40], [41], genetic algorithms [42], [43], etc. These algorithms aim at providing a solution within a reasonable time, but they may not always find the optimal solution. Therefore, graph matching issue still attracts attentions from researchers.

Zavlanos and Pappas [44] recently presented a new optimization-based method for solving the weighted graph matching problem. In their research, they developed a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices. Since permutation matrices that are usually used to represent a matching between graphs are orthogonal with nonnegative elements, they defined two gradient flows in the space of orthogonal matrices. The first one minimizes the cost of the weighted graph matching over the manifold of orthogonal matrices whereas the second minimizes the distance from an orthogonal matrix to the set of all permutation matrices. The combination of the two gradient flows converges to a permutation matrix, which provides a suboptimal solution to the weighted graph matching problem. This method is created on the basis of a rigorous theory, but their work merely focuses on the case of matching between graphs with the same number of nodes, which cannot be applied to matching graphs with different sizes.

In this paper, we extend the dynamical systems approach to the case of subgraph matching and apply it to the partial retrieval of CAD models (see Fig. 1). Here, the face ARG representations for CAD models are created from their B-rep models first, and then an optimization procedure on the manifold of matrices representing transformations of rotations as well as translations in Rn is adopted to solve the sub-ARG matching problem. In the optimization formulation, the optimization variable is the graph vertex mapping and the objective function is a measure of the compatibility between the mapped vertices and between the mapped edges. Different from previously proposed methods, a homogeneous transformation matrix is introduced to represent the vertex mapping in subgraph matching, whose translational sub-matrix gives the vertex selection in the larger graph and whose orthogonal sub-matrix presents the vertex permutation for the same-sized mapping from the selected vertices to the smaller graph's vertices. To solve the optimization problem, a gradient flow method is developed to search the optimal matching matrix in Special Euclidean group SE(n). Here, a penalty approach is used to handle the constraints on the elements of the matching matrix, which leads its orthogonal part to be a permutation matrix and its translational part to have integer elements. To deal with the problem that the gradient-based methods may only produce a local optimum solution, multiple runs of the above optimization procedure starting from different initial matches are carried out and the initial matches are properly selected with a vertex-mapping filter that takes into account the requirement of the attribute consistence between the initial matched vertices as well as their adjacent edges. The experimental results have showed that the proposed approach is feasible.

The rest of this paper is organized as follows. After the face ARG for model description is introduced in Section 2, the related terminology definitions and problem statements are presented in Section 3. In Section 4, the gradient functions for the objective and for the constraint penalties are derived in detail, with which a dynamical system whose solution leads to the solution of the subgraph matching problem is developed as well. Following this, a few computation issues on implementation of the Runge–Kutta method for solving the matrix differential equation of the dynamical system are addressed in Section 5. Section 6 presents some results of experiments that apply the proposed approach to the partial retrieval of CAD models and Section 7 gives some discussions on the performance of the approach. Finally, the paper ends up with some conclusions in Section 8.

Section snippets

Face ARG for CAD models

Most of CAD systems use some forms of B-rep as internal representation, which can be conveniently exported into a file with the STEP standard CAD format. The methods for generating and expressing an ARG from a STEP file have appeared in literature [45]. In this context, ARG is formulated as an ordered pair G(V,E), where V is a set of ARG's nodes that represent model faces with their properties and E is a set of ARG's links that represent model edges with their properties as well. There are

Terminology definitions

Definition 1

Permutation matrix: A n×n matrix R=(rij) in an orthogonal group is a permutation matrix if rij∈{0,1} for i,j∈{1,2,3,…,n} and RTR=I,RRT=I. Here I is a unit matrix.

Definition 2

Translation matrix: For an integer-valued n×1 matrix d=(di)∈Rn×1 that satisfies 1≤dim (mn) and didj when ij, we define a translation matrix D=(Dij)∈Rn×m as follows:Dij={1whendi=j0otherwise,i=1,2,,n;j=1,2,,m.

For convenience, d is called translation matrix as well. Obviously, a translation matrix has the following properties:Dχ=d,χ=

Gradient flows

Gradient methods have been widely used in the optimization area because they can greatly reduce computations by choosing a proper search direction in the option space. But gradient methods are only suitable for the optimization problems with continuous variables. Unfortunately, problem (1) has a discrete variable T due to the fact that R in T is a permutation matrix and d in T is an integer-valued matrix. Ordinarily, the discrete variables should be relaxed to be continuous in some way for this

Solving the matrix differential equation

The dynamical system (8) is a matrix differential equation, which can be written as a more general formṪ(R,d)=T(R,d)f(T(R,d)),f(T(R,d))=[ΩRTδ00].T(t)|t=0=T0=[R0d001],T(t)SE(n),where Ω and δ are given in Eqs. (9), (10). Here, we adopt the algorithm of Runge–Kutta 4 with a constant step size to solve the above equation.

Implementation and experiments

Here, some implementation issues are addressed first and then an example is presented to illustrate the computation process of the proposed algorithm. After this, some experiments and their results are described.

Discussions

In partial model retrieval, subgraph isomorphism checking often plays an important role. Among the currently available approaches for the subgraph matching, complete heuristics methods cannot guarantee an acceptable precision, heuristics-based backtracking algorithms have a problem that the cost of the worst cases is still exponential, and optimization-based approaches usually produce a sub-optimal solution. Generally, the heuristics related methods are suitable for graphs having non-symmetry

Conclusions

In this paper, a partial retrieval approach for CAD models has been introduced. Since the ARGs are used to represent retrieval models as well as library models, partial model retrieval is converted to a subgraph matching problem. Unfortunately, subgraph matching is an extremely intractable problem when exact solutions are required. However, for such intractable problems, many good heuristics and approximation approaches have been developed, which can yield reasonable solutions for many

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Nos. 50875092, 61173115, 50935004) and the National High-Tech Research and Development Program of China (No. 2007AA04Z136).

Songqiao Tao received his M.S. degree in mechanical engineering from Wuhan University of Technology, China in 2005. He is currently pursuing research for a Ph.D. degree in mechanical engineering at Huazhong University of science and technology, China. His current research interests include: 3D model retrieval, geometry modeling and engineering optimization.

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      Then, a geometric correspondence identification in geometric design is to identify a shape correspondence between two models. Lots of model retrieval approaches have been proposed in the past decades, such as (a) the approaches that get the candidate models by inputting a query model [11,17,21–23,56,57], and (b) the works that discover the candidate model by deforming a template model (such as the work presented by Ovsjanikov et al. [58]). Usually, model retrieval approaches can determine whether two models’ shapes are similar or not on globally or partially, but they are difficult to obtain more matching details especially when the two models have different resolutions [9,59].

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    Songqiao Tao received his M.S. degree in mechanical engineering from Wuhan University of Technology, China in 2005. He is currently pursuing research for a Ph.D. degree in mechanical engineering at Huazhong University of science and technology, China. His current research interests include: 3D model retrieval, geometry modeling and engineering optimization.

    Zhengdong Huang is currently a professor at the School of Mechanical Science and Engineering, Huazhong University of Science and Technology. He received Ph.D. degree in mechanical engineering from Huazhong University of Science and Technology, China in 1997. He visited Engineering Research Center for Reconfigurable Machining Systems, University of Michigan, as a postdoctoral fellow from 1998 to 2001. His current research interests include: product modeling, CAX integration, CAD model retrieval and reuse, engineering optimization etc.

    Bingquan Zuo is a Ph.D. candidate. He received his B.S. degree in Mechanical Engineering from Huazhong University of Science and Technology, China in 2008.He joined the 3D model retrieval in 2007, and finished his undergraduate thesis on this study. Since then, 3D model retrieval became his research interest, and finished the engine of network service for 3D models.

    Yangping Peng received his B.S. degree in mechanical engineering from Hunan University of Technology, China in 2008. He is currently pursuing research for a Ph.D. degree in mechanical engineering at Huazhong University of science and technology, China. His current research interests include: geometry modeling and CG.

    Weirui Kang received her M.S. degree in electromechanical integration from Xi'an Technological University, Xi'an, China, in 2006. She is currently pursuing her Ph.D. in Mechanical design and theory at School of Mechanical Science and Engineering of Huazhong University of Science and Technology. Her research interests include computer-aided design, computer aided process planning.

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