Scale-Space Representation of 3D Models and Topological Matching Dmitriy Bespalov† Ali Shokoufandeh† William C. Regli†‡ Wei Sun‡ Department of Computer Science† Department of Mechanical Engineering & Mechanics‡ College of Engineering Drexel University 3141 Chestnut Street Philadelphia, PA 19104

Outline Project Goals Related Work Research Approach Algorithm Overview Experiments Summary Future Work

Project Goals Develop algorithms to compare CAD models in polyhedral representation (easiest to obtain) Integrate these algorithms into the National Design Repository (www.designrepository.org)

Selected Related Work Comparing Solid Models Comparing Shape Models Graph-based data structures to capture feature relationships [Elinson et al., 1997; Cicirello and Regli, 1999, 2000, 2001, 2002] Automatic detection of part families [Ramesh, 2000] Topological similarity assessment of polyhedral models [Sun, 1995; McWherter et al 2001, 2002] Comparing Shape Models Multiresolutional Reeb Graphs [Hilaga et al., 2001] Shape distribution [Osada et al, 2001, 2002, 2003]; [Ip et al, 2002, 2003] 2D views of 3D objects [Cyr and Kimia, 2001]

Research Challenges Techniques for solid models: Depend on model’s representation Feature extraction is not unique (!!?!!) Techniques for shape models: Unable to perform feature (or topology) matching Sensitive to model’s connectivity and refinement of approximation

Research Approach Start with CAD model Obtain polyhedral representation Perform geometry-based decomposition Obtain a segmentation into features Construct hierarchical feature graph ??? Use hierarchical graph matching to compare graphs

Algorithm Overview Polyhedral representations of CAD models are calculated. The shortest path between two points is used as a measurement function. Models obtained: The measurement function between two points of the model is illustrated in red. Swivel part Simple bracket part

Algorithm Overview 2. Each object is decomposed into relevant components using computation of SVD of the weight matrix. Decomposition (clustering) is based on the angle between a vector Opi and each of the two most significant basis vectors (ck, ck-1). Distance weight matrices are constructed, eigenvectors are computed and clustering is performed.

Feature Extraction Example

Algorithm Overview 3. Binary feature trees are obtained by recursively applying feature decomposition routine with the branching factor 2. Sample binary tree for simple bracket Sample binary tree for swivel

Algorithm Overview 4. Binary trees that correspond to the models are compared from bottom up. Parents are matched only if their children matched.

Arrangement of matching nodes for both models. Algorithm Overview 5. Total similarity value is calculated based on the similarity between each matched pair of nodes. Simple bracket part Swivel part Arrangement of matching nodes for both models.

Complexity Bisection process: Graph matching: SVD decomposition takes O(n3) Polyhedral representation provides us with planar graph (2D manifold); if only neighboring vertices are used in construction of the Distance matrix, SVD decomposition is faster and takes O(n2) Graph matching: , where n1 and n2 are the number of features in both graphs

Experiments We have selected a total of 40 CAD models in polyhedral representation. Models were manually clustered in 10 groups. Similarities for all possible pairs of models were computed using both algorithms and similarity matrices were constructed to visualize the results.

Experiments: 10 classes, 40 parts

Experiments: 10 classes, 40 parts Each pixel in the matrix correspond to similarity value of a pair of models. Darker color represents higher similarity.

Experiments: 10 classes, 40 parts Each pixel in the matrix correspond to similarity value of a pair of models. Darker color represents higher similarity.

Current Issues Controlling Feature Decomposition Comparing with Existing Techniques New node similarity function for better tree matching

Controlling Feature Decomposition To make feature extraction process fully automated, we assign a measurement to each bisection. Let M be the original model’s point set and E be a set of all edges connecting points in M; M1 be some partition of M; M2 and M3 denote the partitions of M1 after bisection. D(u,v) denotes the distance between u and v on the model’s surface. We say that bisection of M1 into M2 and M3 is good if f(M1) < 0.5

Controlling Feature Decomposition Feature tree and a sample view for Spring model Feature tree and a sample view for Part 10 model

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented.

Controlling Feature Decomposition Illustrates decomposition process for Fork model. Pictures of the point sets for three bisections are presented. A small pin is present. There is no pin on opposite side.

Comparing with Existing Techniques Scale-Space technique with automated feature extraction Multiresolutional Reeb Graph based technique Original distribution based technique Enhanced distribution based technique Learning enhanced distribution based technique ITV feature extraction based technique

Comparing with Existing Techniques We used the highest similarity value in classification. Each model was compared to the rest of the models. The group that the model is assigned (by the technique) was determined based on the highest similarity value. False-Positives and False-Negatives were determined by comparison to the real grouping (manually done). This is a very robust classification. Best performance: G0: Reeb Graph and ITV G5: Scale-Space, Orig. and Learn. Dist. G1: Scale-Space, Orig. and Enh. Dist. G6: Enh. and Learn. Dist. G2: Reeb Graph G7: Scale-Space and Reeb Graph G3: Scale-Space, Reeb Graph, ITV G8: Orig. and Learn. Dist. G4: Orig. and Enh. Dist. G9: Orig. Dist and ITV

Comparing with Existing Techniques Scale-Space technique G5: FP: 1, FN: 1 G6: FP: 1, FN: 2 G7: FP: 2, FN: 1 G8: FP: 5, FN: 1 G9: FP: 3, FN: 0 G0: FP: 0, FN: 3 G1: FP: 0, FN: 2 G2: FP: 1, FN: 1 G3: FP: 1, FN: 1 G4: FP: 1, FN: 3 Total Errors: 15 G=Group FP=False-Positives FN=False-Negatives Scale-Space technique with automated feature extraction G5: FP: 0, FN: 1 G6: FP: 1, FN: 1 G7: FP: 0, FN: 1 G8: FP: 4, FN: 1 G9: FP: 3, FN: 0 G0: FP: 0, FN: 3 G1: FP: 0, FN: 1 G2: FP: 2, FN: 1 G3: FP: 1, FN: 1 G4: FP: 1, FN: 2 Total Errors: 12 Multiresolutional Reeb Graph based technique G5: FP: 3, FN: 1 G6: FP: 1, FN: 1 G7: FP: 0, FN: 1 G8: FP: 4, FN: 1 G9: FP: 4, FN: 0 G0: FP: 0, FN: 3 G1: FP: 0, FN: 1 G2: FP: 0, FN: 1 G3: FP: 0, FN: 2 G4: FP: 1, FN: 2 Total Error: 13 Original distribution based technique G5: FP: 0, FN: 1 G6: FP: 1, FN: 1 G7: FP: 3, FN: 1 G8: FP: 1, FN: 1 G9: FP: 2, FN: 0 G0: FP: 0, FN: 2 G1: FP: 0, FN: 2 G2: FP: 1, FN: 1 G3: FP: 3, FN: 1 G4: FP: 0, FN: 1 Total Errors: 11 Enhanced distribution based technique G5: FP: 0, FN: 2 G6: FP: 0, FN: 1 G7: FP: 1, FN: 1 G8: FP: 2, FN: 1 G9: FP: 4, FN: 0 G0: FP: 0, FN: 5 G1: FP: 0, FN: 1 G2: FP: 1, FN: 1 G3: FP: 6, FN: 1 G4: FP: 0, FN: 1 Total Errors: 14 Learning enhanced distribution based technique G5: FP: 0, FN: 1 G6: FP: 0, FN: 1 G7: FP: 2, FN: 1 G8: FP: 1, FN: 1 G9: FP: 3, FN: 0 G0: FP: 0, FN: 4 G1: FP: 0, FN: 2 G2: FP: 1, FN: 1 G3: FP: 5, FN: 1 G4: FP: 1, FN: 1 Total Errors: 13 ITV feature extraction based technique G5: FP: 1, FN: 2 G6: FP: 1, FN: 1 G7: FP: 2, FN: 1 G8: FP: 7, FN: 2 G9: FP: 2, FN: 0 G0: FP: 0, FN: 2 G1: FP: 0, FN: 2 G2: FP: 1, FN: 1 G3: FP: 1, FN: 1 G4: FP: 0, FN: 3 Total Errors: 15

Summary & Contributions We introduced an algorithm that can be efficiently used in indexing CAD models. We are able to perform feature decomposition on CAD models given only geometry information. The algorithm only works if a model has one connected component.

Future Work Integrate the technique into the National Design Repository From 40 to 100s to thousands of parts Develop information theoretic measures to compare performance of matching techniques

Need: