Shape Similarity Assessment Approach for CAD models Based on Graph Edit Distance

Transcription

1 Eurographics Workshop on 3D Object Retrieval (2010), pp 1 8 I Pratikakis, M Spagnuolo, T Theoharis, and R Veltkamp (Editors) Shape Similarity Assessment Approach for CAD models Based on Graph Edit Distance Paper 1022 Abstract Shape similarity assessment for CAD models is gaining more and more attention from both industry world and academic community In this paper, we propose a new shape similarity assessment approach based on graph edit distance Attributed graphs, which are derived from CAD models presented in boundary representation (Brep), are constructed according to the faces and their adjacency relationships in the models In case of expensive computation, a novel weighted suboptimal computational procedure of graph edit distance, considering only local, rather than global, topological structure is performed Furthermore, we train the query model in a training data set to get the salient local structures, which reveals the functions of the model These "salient local structures" are then used to retrieve the most similar models in the retrieval data set Experiment results show that our method provides solid retrieval performance on a real-world CAD model database Categories and Subject Descriptors (according to ACM CCS): I37 [Computing Methodologies]: Computer Graphics Three-Dimensional Graphics and Realism); Keywords: CAD model retrieval, Boundary representation, Graph edit distance, Functional classification 1 Introduction Over the last decade, shape-based retrieval of 3D objects in database has been an increasingly important task, which has broad application prospect in many disciplines such as computer vision, virtual reality, architecture and molecular biology [IJL 00] [YLZ07] Especially for CAD applications, such as product design and manufacturing industry, finding similar parts efficiently makes great sense in various manners Designers can improve both quality and efficiency of model designs by referring to existing similar ones [HLK06]; model archivists need to find parts sharing compatible shapes, in order to facilitate the process of part family formation; while for trading department, it is a modern marketing strategy to provide a favorable cost estimation of part that design customers submit online [RCC 02] [CGK03] In spite of extensive use, however, shape similarity assessment for CAD models didn t draw as much study as for generic models Substantial difference between them derives from discordance of model representation For generic models, meshes are usually used to give an approximate description, while most CAD models are represented in a complete and compact method, namely B-rep Specifically, meshes consist of a collection of triangles with definite location, and B-rep mainly record the topological structure of geometric objects, which is more useful for CAD models Accordingly, in the domain of CAD model retrieval, topological structure, instead of overall shape, tend to be recorded and compared As a result of different objectives, the majority of shape matching algorithms for generic models perform incompetently over CAD model database, although constrainedly applicable Among all the shape signatures of CAD models, attributed graph has shown superior adequacy The main advantage of description in graph over feature vector is that graph allows for a complete and powerful representation of structural relations Many methods for evaluating the dissimilarity of graphs, or graph matching namely, have been proposed, but only a very limited proportion has been applied for CAD model retrieval One of the most powerful and flexible methods for graph matching is graph edit distance Originally proposed in the context of string matching, edit distance aims at measuring the dissimilarity by counting the number of distortions one has to perform, to transform one pattern into another Compared to other approaches to graph matching, graph edit distance has the capability of handling arbitrary graphs Moreover, it does very well in capturing and measuring small local distortion [RB09] [EF84] In this paper, we apply graph edit distance to solve the

2 2 Paper 1022 / Shape Similarity Assessment Approach for CAD models Based on Graph Edit Distance problem of shape similarity assessment for CAD models In case of expensive computation of exact graph edit distance, a suboptimal, but much faster procedure is taken Our retrieval method is composed of two phases In the training phase, the query model are trained in a training data set, and some salient local structures of the model are derived Then in the retrieval phase, these salient local structures are used to retrieve the most functionally similar models from the retrieval data set The remainder of this paper is organized as follows: In Section 2, we briefly review already existing shape similarity assessment approaches for CAD models, and formally provide the definition and notation of graph edit distance Our retrieval method based on graph edit distance is presented in Section 3 Section 4 presents the experiment results on a real-world model sets, and related discussions The paper concludes in Section 5 with a short summary and a few remarks on future works 2 Background and related works In this section, we first give a brief survey of existing retrieval methods for CAD models, then formally describe the definition of graph edit distance Several methods to calculate graph edit distance including both exact and approximate ones are also involved 21 Existing retrieval methods In recent years, considerable works on shape similarity assessment has been published The prevailing methods are constructing reasonable shape signatures from the models, and then comparing these signatures using suitable functions According to the representation of models and the type of corresponding shape signatures, we provide a classification of these techniques into statistic based, image based, topological based and manufacturing feature based techniques Some of them are originally proposed for generic model retrieval, but also applicable for CAD models Statistic based technique The most classical method in the category is shape distribution [OFCD02], where a group of geometric functions are applied on random points obtained on the surface Out of the few shape functions mentioned in the literature, D2, which calculates the distance between two random points, has been found most suitable because of its robustness, efficiency and invariance to rotation and translation From another perspective, Spherical Harmonic Function (SHF) [KFR03] is introduced, whose basic idea is to establish a spherical function to depict the energies at different frequencies Image based technique Techniques in this category usually transform similarity assessment of 3D shapes into the problem of a series of 2D images A typical image-based technique is Light field descriptor (LFD) [COTS03], in which a set of uniformly distributed cameras were lay up to capture the collection of 2D images The final 3D model retrieval results were combined from the matching results of all those acquired 2D images by integrating 2D Zernike moment and Fourier descriptors Topological based technique As reasons in term of representation, topological based technique are well suitable for CAD model retrieval An effective way to derive shape signatures is constructing an attributed graph according to the faces and their adjacency in the models A convincing example is Model Signature Graph (MSG), whose detailed definition and concept is presented in [MPRS01] Two signatures, Invariant Topological Vector (ITV) and Eigenvalue Spectrum, are also proposed in the article Just as the names imply, ITV denotes a combination of certain graph topological invariants, while Eigenvalue Spectrum deals with the problem based on spectrum graph theory Manufacturing feature based technique Although conveying topological information well, B-rep does not capture the part geometry in a relevant way to modeling In such a case, it s usually converted into representation of Feature Based Model (FBM) [RYHD01] The related methods won t be discussed deeper in the present paper, due to the complexity of feature recognition 22 Graph edit distance The basic definitions and notations of graph edit distance are given below, referring to their practical sense Assume that attributed graph has been constructed according to a CAD model Definition 1:(Graph) Provided that L denotes a set of labels for nodes and edges A graph g is a quaternion {V,E,µ,ν}, and where V denotes the finite set of nodes, E V V denotes the finite set of edges, µ : V L is the node labeling function, ν : E L is the edge labeling function The definition above leads to capability of handling arbitrary graphs with unconstrained labeling functions As for the specific domain of CAD model retrieval, the nodes in the corresponding graph can be labeled with attributes that further describe the properties of the faces in the model, with the edges also labeled with some attributes associated with curves For example, we can record for the face the geometric type of surface, the relative area, or set of surface normals, and attach the type, concavity and convexity to the B-rep edges In an alternative way, edges could only represent adjacency relationships of nodes, with the geometric information appended to their adjacent nodes The essential idea of graph edit distance is to define the distance between graphs by the minimum amount of distortion which needed to be made, in order to transform one graph to another A standard set of distortion operations is given by insertions, deletions and substitutions of both nodes

3 Paper 1022 / Shape Similarity Assessment Approach for CAD models Based on Graph Edit Distance 3 Figure 1: A possible edit path between graphs with nodes labeled with different colors and edges The operations are logically equivalent to insertions, deletions and substitutions of faces and their adjacency, ie local modifications Given the source graph g 1 and the target graph g 2, a sequence of edit operations transform g 1 into g 2 is defined as an edit path between both graphs every pair of graphs (g 1,g 2 ), we introduce the notation γ(g 1,g 2 ) to stand for set of all such edit paths Since different edit operations may provide modification to a different extent, a cost function c(e i ) is defined with respect to the underlying set of labels Definition 2:(Graph edit distance) Given the definition and notation declared above, graph edit distance d(g 1,g 2 ) is define by k d(g 1,g 2 ) = min (e 1,e 2,,e k ) γ(g 1,g 2) c(e i ) (1) i=1 Obviously, the graph edit distance of two similar graphs under specific cost functions will be very close, which indicates there exist an inexpensive edit paths, and vice versa Therefore, graph edit distance makes up an metric of graph similarity assessment As for the applications on CAD models, it provides an effective measurement of the significance of distortions The computation of the edit distance is usually carried out by means of a tree search algorithm, which explores the space of all possible mapping of the nodes and edges of one graph to those of another graph Optimal edit path can be found in various tree search strategy Unfortunately, the computational complexity of the edit distance is exponential in the number of the involved graphs This means that the running time and space complexity may be rather huge even for reasonably small graphs In order to address this issue, quite a few approximate, but faster methods, ranging from means of binary linear programming [JH06] to designing proper prune strategy [NRB06] have been proposed The latest research [RB09] transformed the suboptimal computation into the assignment problem, or weighted bipartite matching namely The approach is well instructive and makes up one base of our works 3 Retrieval based on graph edit distance In this section, we present a new similarity assessment method for CAD models, with a vector of preferences highly related to application not specified at this point Detailed technical approach to construct attribute graphs from B-rep files is firstly stated Then, we calculate the suboptimal graph edit distance using a modified version of algorithm proposed in [RB09] At the end of the section, strategies for weighting each element are discussed 31 Constructing attributed graphs As emphasis for comparing CAD models is placed on the topological structure, we develop techniques for archiving and comparing CAD models by exploiting intrinsic graph properties Each face is transformed into one attributed node, and the type histograms of both the surfaces and edges associated with the original face are recorded as a labeling function There are two main reasons we rule out a diverse collection of attributes On the one hand, we regard structural properties more important than local geometric attributes, such as convexity, circular variance or rectilinearity On the other hand, too many complex attributes will lead to expensive computation When it comes to edges, no geometric attributes are put into use along the same vein In fact, edges are only labeled with their adjacent faces implicitly, instead of attached to other properties We take into account both effectiveness and computational amount when selecting the attributes, so that they can give a substantial description of topological structure Consequently, attributed graphs special used in our method are formally equivalent to undirected graphs with labeled nodes and edges In manufacturing industry, some operations during either designing or producing process, such as chamfering angle or rounding off, often produce a large number of trivial faces However, most of them are not important for models functionality Accordingly, we can ignore these trivial faces during the construction of attributed graphs A simple but effective algorithm, by calculating the relative areas, is performed to remove the trivial faces Given the face s surface area s, and all surface area of its adjacent faces, s i,(i = 1,2,,n) The relative area s r is defined as: s r = s/ n i=1 s i If s r fails to exceed the threshold set beforehand, the current face will be viewed as trivial and ignored We find 01 a proper value of the threshold according to our experimental analysis An example of deleting trivial faces and the impact of deletions are showed in Figure 2

4 4 Paper 1022 / Shape Similarity Assessment Approach for CAD models Based on Graph Edit Distance Definition 3:(Cost Matrix) Given the same notations above, the cost matrix C is defined as (a) (b) (c) Figure 2: (a) CAD model with trivial faces colored in red; (b) Corresponding attributed graph before removing trivial faces; (c) Attributed graph after removing trivial faces C = c 1,1 c 1,2 c 1,m c 1,ε c 2,1 c 2,2 c 2,m c 2,ε c n,1 c n,2 c n,m c n,ε c ε, c ε, c ε,m (2) where c i, j denotes the cost of a substitution, c i,ε denotes the cost of a deletion, c ε, j stands for the cost of a insertion 32 Suboptimal computational procedure As is mentioned above, the latest research [RB09] calculates graph edit distance approximately based on the assignment problem, which refers to the problem of finding the optimal alignment of two sets, with the numerical costs for each pair element given Attributed graphs are decomposed into sets of subgraphs, which consist of one node and its adjacent structure The algorithm proposed in the article [RB09] use Munkre s algorithm to find the optimal alignment of both sets of subgraphs The similarity between two subgraphs, or the assignment cost equivalently, is defined and calculated first Then, the notation ε is introduced to represent an empty subgraph Consequently, insertions and deletions can be performed logically by assign ε to an actual subgraph or the inverse process It is not only theoretically analyzed but also empirically verified that the accumulated assignment costs of the subgraphs serve us an approximate value of graph edit distance In this paper, we make a substitution of the Munkre s algorithm by LAPJV [JV87], which is more efficient in this context LAPJV characterizes by making massive use of preprocessing procedures to determine the first primer-dual pair For detailed description of the algorithm, we refer readers to [JV87] The left upper of the cost matrix represents the cost of all possible element substitutions, with left lower insertions and right upper deletions The combination of the solution to the assignment problem and the definition of cost matrix compose a method for estimation of our edit operations cost We measure the similarity between each pair of nodes first by comparing the type of corresponding faces and type histograms of their associated edges Edges of the same geometric type on inner loops and outer loops are viewed as different types here We use L1 distance to calculate the difference between both edge type histograms, then normalize it to the interval from 0 to 1 After the similarity between each pair of node is evaluated, the edit cost of graph edges, which implies the change in the node s neighborhood, is calculated The computational procedure is also carried out based on the assignment problem and LAPJV algorithm Specifically, for each pair of subgraph, an individual cost matrix is established and an optimal assignment of adjacent nodes can be performed The similarity each pair of subgraphs is calculated by summing up the cost of nodes and edges Each entry in the general cost matrix can be specified, and LAPJV algorithm is applied again to work out the final result While our original idea coming from the approximate and fast computation of graph edit distance, the proposed algorithm has its underlying practical meaning In fact, models are locally decomposed into a number of local structures, each of which consists of one central node and its adjacent nodes Figure 3 illustrates the practical significance of our computational procedure

5 Paper 1022 / Shape Similarity Assessment Approach for CAD models Based on Graph Edit Distance 5 Training Phase Training i Retrieval Data Set Data Set Figure 3: Model deposition and alignment of local structures Local Structures Set Weight Local Structures Set 33 Weighting strategy For many applications, different faces in the model, or local structures extensionally, may not play an equally important role for retrieval For instance, when we are to find models similar in overall shape to the query, larger-area faces are likely to overweigh those with small sizes Some instructive attributes to measure the importance of faces for specific applications as far as our consideration include: 1) surface area proportion; 2) the number of adjacent faces; and 3)discriminability Unlike the first two attributes, the discriminability of a face is somehow difficult to understand and calculate A detailed explanation will be made in Section 4 We extend the definition of graph edit distance by assigning a weight for each local structure in the model, or the subgraph equivalently Formally, if a substitution e i, j is performed, weight of both ends of the alignment is introduced into the computation While if a insertion or deletion is made, i = ε or j = ε equivalently, w ε is also assigned a much smaller weight value k S(g 1,g 2 ) = min (e i1, j 1,,e ik, j k ) γ(g 1,g 2) w i w j Φ(c(e i, j )) (3) i=1 In the equation, S(g 1,g 2 ) denotes the similarity instead of distance, to ensure elements with larger weight can make major contributions Accordingly, we use Φ(c(e i, j )) to transform the cost to the similarity 34 Retrieval framework After an elaborate observation and survey, we discover that different faces and their neighborhood in the models are not equally important for the functional feature of the model In other words, the functionality does not completely correspond to the shape Semantically, only a small collection of local structures are essential to the model s functionality, which should be assigned a larger weight when retrieval for models with similar functionality In order to bridge the semantic gap, we lead in a training phase before retrieval, in which the discriminability of each local structure is evaluated with the help of priori category knowledge A structure chart of training phase and test phase is given in Figure 4 Note the fact that there tend to exist a small collection of similar local structures, or salient local structures namely, between two models in the same functional category Our Query Retrieval Phase Local Structures Weighted Local Structures Match Retrieval List Figure 4: Retrieval framework for part functional classification aim is to find out salient local structures by priori category knowledge and the similarity value between each pair of local structure Let s review the computational procedure of graph edit distance given in the last section We have made an approximate approach, where the original task is transformed into the assignment problem The alignment relationship between each other s local structures is obtained during the process If one local structure i are frequently found in the list of alignment relationships, and the assigning cost of i is quite small for almost each time, we can conclude that models in this functional category share a similar local structure to i In such a case, the local structure i is viewed as a salient one Formally, the weight for each face is assigned as follows: w i = ( C j=1 (ξ c i, j)) 1 C (4) The definition ensures that a frequently mapped node can be assigned a larger value We choose multiplication over addition to increase discrimination ability Meanwhile, we introduce a number ξ larger than the maximal possible distance The notation C denotes the size of the category a model belongs to In the actual retrieval phase, we use the formation in Equation 3 to get final results Obviously, the salient local structures are expected to play major role in the computational procedure 4 Experiment results The effectiveness and flexibility of the proposed method is empirically verified in the following section The performance of classification is compared against a range of existing algorithms We also adapt our method to other applications, component design reuse and modified cost estimation

6 6 Paper 1022 / Shape Similarity Assessment Approach for CAD models Based on Graph Edit Distance 41 Data set A sum of 200 models in IGES format are collected from industry We provide a hierarchical classification of these models, mainly according to the functional and semantic concepts, which are related but not equivalent to the overall shape Some geometric attributes are also taken into account secondarily Final classification includes 10 categories Since our solution to the problem of part functional classification includes training and retrieval phase, we partitioned the models into training and test sets evenly Figure 5 shows some samples from each category in order most effective on generic model sets, LFD and SHF, perform incompetently Both methods provide a solid performance when retrieving models with similar overall shapes, but are not suitable for CAD models, because of lack of topological information Additionally, there is a remarkable difference between ITV, together with eigenvalue based algorithm and our method Although especially designed to describe the similarity of topological structure, the signatures, ITV and eigenvalues contain only a small amount of information, and thus are too rough to be applied for retrieval From the bars in Figure 7, we can see that our method performs at least 15% higher than the second best algorithm for each sort of measurement Our method LFD SHF ITV Eigen D2 Figure 5: Samples from each category in order 42 Functional classification We have compared our method against a set of algorithms, which includes either algorithms which originally proposed for generic models, or typical algorithms specially for CAD models We choose a vector of 10, 08, 08, 03, 02 and 02 for each cost of operations Experiment results has verified that our method performs best under a real-world CAD model set Precision Our method LFD SHF ITV Eigen D Recall Figure 6: Precision-recall plot compared against classical algorithms As is shown by the precision-recall plot in Figure 6, our method (GED) is superior to all the algorithms mentioned In particular, the couple of algorithms which proves to be Nearest neighbor First tier Second tier DCG Figure 7: Measurement comparison It is proved that, however, when handling models with some local distortions, the effectiveness and efficiency can meet demands under most circumstances See the tier image in Figure 8, we have applied our method on the set composed of nuts, the first three categories in database, and obtained promising results Tier image visualizes nearest neighbor, first tier and second tier matches Specifically, pixel(i, j) is black if model i is model j, red if model i is among the first tier, and blue if model i is among the second tier It is clear that most black or red pixels distribute on the diagonal, indicating a favorable result, and our method give better results than LFD Our main purpose is to apply a powerful and flexible paradigm, graph edit distance, to the domain of CAD model retrieval The effectiveness of graph edit distance is not only logically reasonable, but also experimental proven As the nature of graph edit distance, this approach has the following major advantages Discriminability: Experiment results have indicated that, our method has apparent advantage over classical retrieval methods when used to find models in the same functional category It has also been verified a small proportion of local structures play an essential role to models functionalsubmitted to Eurographics Workshop on 3D Object Retrieval (2010)

7 Paper 1022 / Shape Similarity Assessment Approach for CAD models Based on Graph Edit Distance 7 Table 1: Database of our experiments Id Category Amount(test set only) V E max(v) max(e) 1 screw/slotted screw screw/inner hexagon screw screw/cruciform slot screw screw/self-tapping screw bolt/hexagon bolt bolt/snap-head bolt rivet/snap rivet washer flange piston we can made a independent modification to the function computing distance between faces Despite of remarkable results provided by our method, there exist one main limitation, which almost all retrieval method based on B-rep share With B-rep giving an detailed topological structure, it may not work for some special cases Therefore, the edit distance between gears with different number of cogs could be illogical large How to avoid this disadvantage could be a task worthwhile research in future works Additionally, further study may include methods of speeding up our algorithm Figure 8: Retrieval results Figure 9: Tier images on nuts set only, with our method on the left and LFD on the right ity, which is instructive to a certain degree to later research in the domain Suitable for multiple applications: We can adopt this technique to other CAD applications with convenience only by adjusting cost functions and weight strategies High flexibility: As little restriction has made within the framework, customized function can be easily added in For instance, when non-geometric attributes, including material, texture or tolerances need to be taken account, References [CGK03] CARDONE A, GUPTA S K, KARNIK M: A survey of shape similarity assessment algoriths for product design and manufacturing applications Journal of Computing and Information Science in Engineering 3 (2003), [COTS03] CHEN D, OUHYOUNG M, TIAN X P, SHEN Y T: On visual similarity based 3d model retrieval In Computer Graphics Forum (2003), no 5, pp [EF84] ESHERA M A, FU K: A graph distance measure for image analysis IEEE Transactions on Systems, Man and Cybernetics 14, 3 (1984), [HLK06] HONG T, LEE K, KIM S: Similarity comparsion of mechnical parts to reuse exsiting designs Computer-Aided Design 38 (2006), [IJL 00] IYER N, JAYANTI S, LOU K, KALYANARAMAN Y, RAMANI K: Three-dimensional shape searching: state-of-theart review and future trends Computer-Aided Design 37 (200), [JH06] JUSTICE D, HERO A: A binary linear programming formulation of the graph edit distance IEEE Transactions on Pattern Analysis and Machine Intelligence 28, 8 (2006), [JV87] JONKER R, VOLGENANT A: A shortest augmenting path algorithm for dense and sparse linear assignent problems Computing 38 (1987), [KFR03] KAZHDAN M, FUNKHOUSER T, RUSINKIEWICZ S: Rotation invariant spherical harmonic representation of 3d shape descriptors In In Symposium on Geometry Processing (2003), no 4 2

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