1. Reflective Symmetry Detection in 3 Dimensions
Michael Kazhdan

2. Overview Introduction Related Work Definitions and Computation Results
Future Work

3. Goal Present a shape-descriptor for model analysis Tasks: Registration
Matching
Properties:
Parameterized over canonical domain
Insensitive to noise
Global

4. Overview Introduction Related Work Definitions and Computation Results
Future Work

5. Related Work Alignment: Locally Parameterized Features:
Generalized Hough Transform Ballard (1981)
Geometric Hashing Lamdan (1989)
Iterative Closest Point Besl, McKay (1992)
Locally Parameterized Features:
Spin Images Johnson, Hebert (1999)
Harmonic Shape Images Zhang, Hebert (1999)

6. Related Work Canonically Parameterized Features:
Extended Gaussian Images Horn (1984)
Spherical Attribute Images Dellinguette, Hebert, Ikeuchi (1993)
Orientation Histograms Sun, Si (1999)
Moments Elad, Tal, Ar (2001)
Shape Distributions Osada, Funkhouser, Chazelle, Dobkin (2001)

7. Overview Introduction Related Work Definitions and Computation Results
Future Work

8. Reflective Symmetry Descriptor
A function associating a measure of reflective symmetry to every plane through the origin
Need to address:
How do we measure symmetry?
How do we compute the measure efficiently?

9. Overview Introduction Related Work Definitions and Computation Results
Future Work

10. Measure of Symmetry
Q: How close is a function f to be symmetric w.r.t to a reflection r?
A: What is the distance to the nearest function g that is symmetric w.r.t. to r? f r
g = ?

11. Measure of Symmetry
Because the space of functions is a Hilbert space…
Because reflection preserves the inner product…
The closest symmetric function to f is the average of f with its reflection + = 2

12. Measure of Symmetry
So that the measure of symmetry of f w.r.t. the reflection r is the (scaled) distance of f from its reflection: - 2 2

13. Overview Introduction Related Work Definitions and Computation Results
Future Work

14. Functions on a Circle
If f(t), t[0,2], is a function defined on a circle then the measure of symmetry of f with respect to reflection about the angle  is: t 
2-t

15. Functions on a Disk f
{fr1,fr2,fr3,…}

16. Functions on a Sphere Step 1: “North pole” symmetries by projection.
Step 2: All symmetries by walking a great circle. f
projected f

17. Voxel Grids
Decompose the grid into concentric spheres, and apply the results for symmetry descriptors of spheres to the voxel grid.

18. Overview Introduction Related Work Definitions and Computation Results
Future Work

19. Distinguishing Between Classes

20. Similarity Within Classes

21. Symmetry Within Classes

22. How Well is “Shape” Captured?
Evaluate how well models can be:
Registered
Aligned
by only using their symmetry descriptors

23. Registration Experiment (Ideal)
Given a collection of models that are classified into groups and aligned:
For each pair of models within a group:
Find the rotation minimizing the L2-distance of the symmetry descriptors
Evaluate how close the minimizing rotation is to the registering rotation

24. Evaluating the Rotation
If M is the ideal registering rotation and N is the minimizing rotation found, how close is M to N?
What is the angle of the rotation of MN-1?

25. Registration Experiment (Practice)
Searching over the space of all rotations is computationally prohibitive:
We know the axis about which the ideal aligning rotation occurs
Search for best rotation about this axis

26. Registration Results Symmetry Principal Axes Model Database:
% of Models
% of Models
Rotation Error
Rotation Error
Symmetry
Principal Axes
Model Database:
Subset of Osada database that fully voxelized to 128x128x128
87 models, 24 Groups

27. Problems With Covariance
Multi-Dimensional eigenspaces:

28. Registration Results

29. Matching Experiment (Ideal)
Given a collection of models classified into groups:
For each pair of models:
Find rotation minimizing the L2-distance of the symmetry descriptors
Use the minimum L2- distance as a measure of the match quality.

30. Matching Experiment (Heuristic)
Searching over the space of all rotations is computationally prohibitive:
Find the principal axis of symmetry of each of the models
Search for minimizing rotation that maps principal axes of symmetry to each other

31. Matching Results
First Tier
First Two Tiers
Nearest Neighbor
Time
Shape Distributions
47%
67%
62%
7.4 seconds
Symmetries
48%
61%
69%
~4,500 seconds
First Tier: If the query model belongs to a class with n models, how many of the top (n-1) matches are also in that class?
Nearest Neighbor: How often is the top match in the same class as the query model?
Model Database:
Subset of Osada database that fully voxelized to 128x128x128
87 models, 24 Groups

32. Matching Results

33. Properties Parameterized over canonical domain: Insensitive to noise:
Parameterized over the (projective) sphere
Insensitive to noise:
Integration scales down high frequency Fourier coefficients
Global
For functions f and g, and any reflection r:

34. Overview Introduction Related Work Definitions and Computation Results
Future Work

35. Future Work
Consider applications of the L properties of the symmetry descriptor
Determine what information about shape can be easily extracted from the descriptor
Explore the potential orthogonality of different matching methods
Apply other alignment methods to the symmetry descriptors