1. Scale-Space Representation for Matching of 3D Models
Dmitriy Bespalov
Department of Computer Science
College of Engineering
Drexel University
3141 Chestnut Street
Philadelphia, PA 19104

2. Overview of the Thesis Introduction Approach Experimental results
Overview of feature decomposition
Variations of feature decomposition
Matching
Experimental results
Conclusions, contributions and future work

3. Motivation & Related Work?
query
Database
query
result ?
query

4. Goals of this Work Feature extraction technique
Models in polyhedral representation
Using local information
Tolerance to noise

5. CAD vs Shape Representation
CAD Representation
Shape Representation
conversion is hard
Topologically and geometrically consistent
Implicit surfaces
Analytic surfaces
NURBS, etc
Approximate representation, error prone
Mesh
Point cloud
Can produce with laser scanners
conversion is easy

6. What is a Scale Space Representation?
Commonly used for Coarse-to-Fine representations of an object
Very popular in computer Vision
Constructed via spatial filters: Gaussian pyramids, Wavelets…
Basic Idea:
At each scale, topologically relevant components will decompose the object into so called salient parts
Recursive application of this paradigm will create the object’s scale space hierarchy

7. Algorithm Overview (I)
Obtain mesh representation M
2. Define measurement function: assign distance measure to every pair of points or triangles

8. Algorithm Overview (II)
3. Decompose M into relevant components using a singular value decomposition of distance matrix D Note: this creates a clustering based on the angle between a vector Oti and the basis vectors (ck, ck-1)

9. Algorithm Overview (III)
Recursive feature decomposition using two principle components creates binary feature trees.
Use leaf nodes as features.

10. Algorithmic Complexity
Bisection process:
SVD decomposition takes O(n3).
Polyhedral representation creates a 3D lattice; if only neighboring vertices are used in construction of the distance matrix, SVD decomposition is faster and takes O(n2).

11. Variations of Feature Decomposition (I)
Use various distance measures to tune nature of extracted features
Global Distance Function
Geodesic Distance
Function
Angular Shortest Path
Max-Angle on Angular
Shortest Path
Global Feature
Extraction
Local Distance Function
Local Feature
Extraction

12. Controlling Feature Decomposition
Which “feature” is better?
Need to control decomposition process to get these features
Otherwise, get these features

13. Controlling Feature Decomposition: When to Stop?
Depends on the distance measure used
At each step of decomposition: decide whether to stop
Assign “quality” measure to each bisection

14. Controlling Global Feature Decomposition
f measures the “quality” of a bisection
Assume decomposition of M1 into M2 and M3
Bisect M1 into M2 and M3 if f(M1) < 0.5 M2 M2 M3 M3

15. Controlling Local Feature Decomposition
“Quality” of bisection is angle based
Assume decomposition of M1 into M2 and M3
Bisect M1 into M2 and M3 if angular distance between components M2 and M3 is large
angle across the border of the cluster is max on the path between most of the pairs of faces in M2 and M3. M2 M3

16. Matching for Feature Extraction
features
extract
features
set of features
set of features
How to match
extracted features?

17. Matching for Global Feature Extraction
Decomposition trees are near-to-balanced
Compare decomposition trees (bottom up dynamic programming) using sub-tree edit distances
Calculate model similarity based on an overall similarity of matched components

18. Matching for Local Feature Extraction
Decomposition trees can not be used
Compare feature graphs (leaves of decomposition trees)
Sub-graph isomorphism is used to asses similarity
Hill-climbing algorithm with random restarts

19. Retrieval Experiments
k-nearest neighbor classification (kNN)
Recall and precision measures
Precision against recall graphs
Relevant models: number of models that fall in the same category as query model
Retrieved models: number of models returned by a query
Retrieved and Relevant models: number of models returned and that fell into the same category as query model

20. Retrieval Using Functional Classification
Techniques used:
Reeb graph comparison (Reeb)
Global Feature Extraction (Scale-Space)
Local Feature Extraction (Local Scale-Space)

21. Functional Classification
Springs
Screws
Gears
Nuts
Brackets
Housings
Linkage arms
Functional
Classification

22. Experimental Results for Local Feature Extraction
Feature decomposition on CAD data
Feature decomposition on scanned data
Retrieval experiments on scanned data

23. Experimental Results: CAD Data

24. Experimental Results: Scanned Data
From Exact Representation
360° Scan
Single Scan

25. Experimental Results: Scanned Data
From Exact Representation
360° Scan
Single Scan
An example of one-to-many correspondence

26. Experimental Results: Scanned Data
From Exact Representation
360° Scan
Single Scan
An example of one-to-one correspondence

27. Experimental Results: Scanned Data
From Exact Representation
360° Scan
Single Scan
An example of one-to-one correspondence

28. Experimental Results: Scanned Data
From Exact Representation
360° Scan
Single Scan
An example of many-to-many correspondence

29. Experimental Results: Scanned Data
From Exact Representation
360° Scan
Single Scan
An example of one-to-many correspondence

30. Experimental Results: Scanned Data
From Exact Representation
360° Scan
Single Scan
An example of one-to-one correspondence

31. Retrieval on Partial and Scanned Data
Partial Data From Exact Representation
360° Scan
Single Scan
query
CAD Database

32. Retrieval on Partial and Scanned Data
Full Scan
Partial Scan
Partial CAD

33. Retrieval on Partial and Scanned Data
Full Scan
Partial Scan
Partial CAD

34. Retrieval on Partial and Scanned Data
Full Scan
Partial Scan
Partial CAD

35. Summary of Experimental Results
Feature extraction is acceptable
Matching could be improved
Matching for Global Feature Extraction:
Comparison of feature pairs is weak, drawn from Reeb Graph technique
Matching for Local Feature Extraction:
No feature pairs comparison
No many-to-many matching
No handling for noise features

36. Conclusions & Contributions
Parameterizable feature extraction for CAD
Features depend only on distance measure
Applicable to partial and scanned data
Query CAD database with scanned data
Attempted to address matching problem

37. Future Work Introduce matching for feature graphs
Better comparison for feature pairs
Handle many-to-many matching
Identify noise features
Develop various distance measures
That resemble traditional CAD features
Approximate B-Rep from polyhedral models

38. Q&A
Sponsored by:

39. Side Note: Compare Features
This Technique
CAD/CAM

40. The Eckart-Young Theorem
The Eckart-Young Theorem: Given an n by m matrix X of rank r ≤ m ≤ n, and its singular value decomposition, ULV', where U is an n by m matrix, L is an m by m diagonal matrix of singular values, and V is an m by m matrix such that U'U = In and V'V = VV' = Im with the singular values arranged in decreasing sequence λ1 ≥ λ2 ≥ λ3 ≥ ... ≥ λm ≥ 0 then there exists an n by m matrix B of rank s, s ≤ r, which minimizes the sum of the squared error between the elements of X and the corresponding elements of B when B = UΛsV' where the diagonal elements of the m by m diagonal matrix Λs are λ1 ≥ λ2 ≥ λ3 ≥ ... ≥ λs > λs+1 = λs+2 = ... = λm = 0