1. Correspondence & Symmetry
Tutorial 7
© Maks Ovsjanikov, Alex & Michael Bronstein
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009 1

2. } } Outline Shape Similarity: Self Similarity
Articulated Shape Matching by Robust Alignment of Embedded Representations Mateus D. et al., Workshop on 3DRR, 2007
Laplace-Beltrami Eigenfunctions for Deformation Invariant
Shape Representation Rustamov R., SGP, 2007
Self Similarity }
Symmetries of Non-Rigid Shapes Raviv D., B., B., K.,, NRTL, 2007
Global Intrinsic Symmetries of Shapes Ovsjanikov. et al., SGP, 2008

3. Images by Q.-X. Huang et al. 08
Articulated Shape Matching by Robust Alignment of Embedded Representations Mateus D. et al., Workshop on 3DRR, 2007
Problem:
Given 2 articulated shapes in different poses, find point correspondences :
Many degrees of freedom, cannot apply rigid alignment.
Images by Q.-X. Huang et al. 08

4. Articulated Shape Matching by Robust Alignment of Embedded Representations Mateus D. et al., Workshop on 3DRR, 2007
Approach:
Embed each shape into a feature space, defined by the Laplacian.
The embedding is isometry invariant:
for any isometric deformation .
The embedding is only defined up to a rigid transform in the feature space.
Find the optimal rigid transform in the feature space to find the correspondences.

5. Articulated Shape Matching by Robust Alignment of Embedded Representations Mateus D. et al., Workshop on 3DRR, 2007
Approach:
The shape is defined as a point cloud. Approximate the Laplacian:
Solve the generalized eigenvalue problem:
Find the most significant eigenvalues/vectors.
For each data point , let
Where is the i-th eigenvector of

6. Articulated Shape Matching by Robust Alignment of Embedded Representations Mateus D. et al., Workshop on 3DRR, 2007
Approach:
For each data point , let
Where is the i-th eigenvector of
Would like to have for corresponding points. However, each eigenvector is only defined up to a sign. Reflection:
If correspond to the same eigenvalue, then for any
is also an eigenvector. Rotation:
Points from the two point sets can be aligned using:
where is orthogonal.

7. Articulated Shape Matching by Robust Alignment of Embedded Representations Mateus D. et al., Workshop on 3DRR, 2007
Approach:
Given point correspondences it is easy to obtain the optimal orthogonal matrix: SVD approach from optimal rigid alignment.
Let , and compute its singular value decomposition:
The optimal solution is given by:
With this step, can perform ICP in the feature space to find the optimal correspondences.

8. Articulated Shape Matching by Robust Alignment of Embedded Representations Mateus D. et al., Workshop on 3DRR, 2007
Results:

9. Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation Rustamov R., SGP, 2007
Main Goal: Find a good, isometry-invariant shape descriptor.
Good: Efficient, Easily Computable, Insensitive to local topology changes (unlike MDS)

10. Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation Rustamov R., SGP, 2007
Main Idea: For every point define a Global Point Signature
Where is an eigenvector of the Laplace-Beltrami operator.
GPS is a mapping of the surface onto an infinite dimensional space. Each point gets a signature.

11. Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation Rustamov R., SGP, 2007
Properties of GPS: If
GPS is isometry invariant (since Laplace-Beltrami is)
Given all eigenfunctions and eigenvalues, can recover the shape up to isometry (not true if only eigenvalues are known).
Euclidean distances in the GPS embedding are meaningful:
K-means done on the embedding provides a segmentation.

12. Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation Rustamov R., SGP, 2007
Properties of GPS:
Euclidean distances in the GPS embedding are meaningful:

13. Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation Rustamov R., SGP, 2007
Comparing GPS:
Given a shape, determine its GPS embedding.
Construct a histogram of pairwise GPS distances (note that GPS is defined up to sign flips, distances are preserved)
For any 2 shapes, compute the norm difference between their histograms.
For refined comparisons use more than one histogram.

14. Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation Rustamov R., SGP, 2007
Results:

15. Conclusions
Laplacian embedding is useful because of its isometry-invariance. Can be used for comparing non-rigid shapes under isometric deformations.
Sign flipping and repeated eigenvalues can cause difficulties (no canonical way to chose them).
Limitations:
Embeddings are not necessarily stable or mesh independent.
Difficult to compute for large meshes (millions of points)
Both topological and geometric stability is not well understood.

16. } } Outline Shape Similarity: Self Similarity
Articulated Shape Matching by Robust Alignment of Embedded Representations Mateus D. et al., Workshop on 3DRR, 2007
Laplace-Beltrami Eigenfunctions for Deformation Invariant
Shape Representation Rustamov R., SGP, 2007
Self Similarity }
Symmetries of Non-Rigid Shapes Raviv et al., NRTL, 2007
Global Intrinsic Symmetries of Shapes Ovsjanikov. et al., SGP, 2008

17. Self-similarity (symmetry)
Shape is symmetric, if there exists
a rigid motion
such that
In , the set of isometries
is only rotations,
translations and reflections.
Am I symmetric?
Yes, I am symmetric.

18. Symmetry
I am symmetric.
What about us?

19. Symmetry Shape is symmetric, if there exists a rigid motion
such that
Alternatively:
Shape is symmetric if there exists an automorphism
such that
Said differently:
Shape is symmetric if has a non-trivial self-isometry.
Substitute extrinsic metric with intrinsic counterpart
Distinguish between extrinsic and intrinsic symmetry.

20. Symmetry: extrinsic vs. intrinsic
Extrinsic symmetry
Intrinsic symmetry

21. Symmetry: extrinsic vs. intrinsic
I am extrinsically symmetric.
We are all intrinsically symmetric.
We are extrinsically asymmetric.

22. Extrinsic Symmetry Detection (1 slide)
Two General Approaches:
Optimize over all transformations. Encode the strength of each symmetry.
Vote for the best symmetry transform. Clustering in transformation space.
A Planar-Reflective Symmetry Transform for 3D Shapes, Podolak et al., SIGGRAPH 2006
Partial and Approximate Symmetry Detection for 3D Geometry, Mitra et al, SIGGRAPH 2006

23. Symmetries of Non-Rigid Shapes Raviv D., B., B., K., NRTL, 2007
Idea:
Find the best non-trivial self embedding. Minimize distortion:
Use GMDS to find the optimal non-trivial self-embedding.
Difficulty: want to stay away from the trivial solution.
Need a good initial guess for .

24. Symmetries of Non-Rigid Shapes Raviv D., B., B., K., NRTL, 2007
Initial Guess:
Adapt the Global Rigid Matching idea to non-rigid setting:
For each point on the surface find a non-rigid descriptor.
Match points with similar descriptors.
Merge disjoint pairs of correspondences into sets of 4.
Compute the distortion of the partial solution.
Branch and bound global optimum
Incrementally add points to get a partial solution.
If the distortion is greater than the known solution, disregard it.
Depends on the quality of the initial greedy guess.

25. How many points within each level set
Symmetries of Non-Rigid Shapes Raviv D., B., B., K., NRTL, 2007
Non-rigid Descriptor:
At each point compute the histogram of geodesic distances.
Comparing Descriptors:
Non-trivial. Comparing is bad because of binning. Use instead:
where : distance between bins.
Geodesic level sets
How many points within each level set

26. Symmetries of Non-Rigid Shapes Raviv D., B., B., K., NRTL, 2007
Results:
Limitations:
1. Not easy to explore multiple symmetries.
2. Need a better descriptor.

27. Global Intrinsic Symmetries of Shapes Ovsjanikov et al., SGP, 2008
Problem:
Given a shape, detect and quantify intrinsic symmetries.
Detect multiple symmetries efficiently. Avoid doing a non-linear optimization. Find efficient encoding for symmetries.

28. Global Intrinsic Symmetries of Shapes Ovsjanikov et al., SGP, 2008
Main Observation:
Eigenvalues and eigenvectors of the Laplace-Beltrami operator are invariant with isometric deformations:
If is an eigenfunction of , and is an isometry:
If and T is a self-isometry (symmetry). Then:
Recall that non-repeating eigenfunctions are defined up to a sign, and repeating eigenfunctions span a linear subspace.

29. Main Theorem:
If O has an intrinsic symmetry T, then GPS(O) has a Euclidean symmetry. We prove:
For eigenfunctions, associated with non-repeating eigenvalues. Only 2 possibilities: or
Positive
Negative

30. Main Theorem:
If O has an intrinsic symmetry T, then GPS(O) has a Euclidean symmetry. We prove:
For eigenfunctions, associated with non-repeating eigenvalues. Only 2 possibilities: or
Positive
Negative

31. Unstable under non-isometric deformations.
Repeating Eigenfunctions
Unstable under non-isometric deformations.
Not necessary, or sufficient for graph automorphisms
Must exist, if an intrinsic symmetry: or if two symmetries: for some

32. Only include eigenvectors associated with non-repeating eigenvalues.
Restricted Signature Space:
Only include eigenvectors associated with non-repeating eigenvalues.
In the restricted space, intrinsic symmetries become combinations of reflective symmetries around principal axes:
Detecting intrinsic symmetries reduces to detecting simple reflection symmetries.

33. Euclidean symmetries when present.
Results:
Euclidean symmetries when present.
Two different symmetries for human shape.

34. As-Rigid-as-possible:
Articulated Motion
SCAPE:
As-Rigid-as-possible:

35. Change in GPS after geodesic shortcuts:
Topological Noise:
Change in GPS after geodesic shortcuts:
Correspondences

36. Incorporate Repeating eigenvalues
Features and Limitations:
Simple algorithm for detecting intrinsic symmetries
Intrinsic symmetries become Euclidean in GPS
Laplace-Beltrami operator encodes metric properties
Limitations:
Incorporate Repeating eigenvalues
Disambiguate between mixed symmetries
Local and partial intrinsic symmetries