1. 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

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

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

4. 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]

5. 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

6. 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

7. 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

8. 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.

9. Feature Extraction Example

10. 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

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

12. 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.

13. 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

14. 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.

15. Experiments: 10 classes, 40 parts

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

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

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

19. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

37. 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.

38. 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

39. 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

40. 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

41. 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.

42. 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

43. Need: