1. In the Rigid Kingdom Alexander Bronstein, Michael Bronstein
...He had Cinderella sit down, and, putting the slipper to her foot, he found that it went on very easily, fitting her as if it had been made of wax.
C. Perrault, Cinderella
In the Rigid Kingdom
Alexander Bronstein, Michael Bronstein
© 2008 All rights reserved. Web: tosca.cs.technion.ac.il

2. Imagine a glamorous ball…

3. A fairy tale shape similarity problem

4. Extrinsic shape similarity
Given two shapes and , find the degree of their incongruence.
Compare and as subsets of the Euclidean space
Invariance to rigid motion: rotation, translation, (reflection):
is a rotation matrix,
is a translation vector

5. How to get rid of Euclidean isometries?
How to remove translation and rotation ambiguity?
Find some “canonical” placement of the shape in
Extrinsic centroid (a.k.a. center of mass, or center of gravity):
Set to resolve translation ambiguity.
Three degrees of freedom remaining…

6. How to get rid of the rotation ambiguity?
Find the direction in which the surface has maximum extent.
Maximize variance of projection of onto
is the covariance matrix
Second-order geometric moments of :
is the first principal direction

7. How to get rid of the rotation ambiguity?
Project on the plane orthogonal to
Repeat the process to find second and third principal directions

8. Canonical basis
span a canonical orthogonal basis for in

9. How to get rid of the rotation ambiguity?
Direction maximizing = largest eigenvector of
and correspond to the second and third eigenvectors of
admits unitary diagonalization
Setting aligns with the standard basis
axes
Principal component analysis (PCA), a.k.a. Karhunen-Loéve
transform (KLT), or Hotelling transform.
Bottom line: the transformation
brings the shape into a canonical configuration in

10. Second-order geometric moments
Eigenvalues of are second-order moments of
In the canonical basis, mixed moments vanish.
Ratio describe eccentricity of
Magnitudes of express shape scale.

11. Higher-order geometric moments
Second-order moments allow some discrimination.
Use higher-order moments gives more discrimination.
-th order moment
Computed in the canonical basis.
Invariant to rigid motion.
Signature of moments
A fingerprint of the extrinsic geometry of .

12. A signal decomposition intuition
Moments are decomposition coefficients in the monomial basis
is a Dirac delta function for and
elsewhere.
span

13. A signal decomposition intuition
uniquely identify a shape (up to a rigid motion).
can be reconstructed exactly from
is the bi-orthonormal basis, i.e.
The monomial basis is not orthogonal.
The bi-orthonormal basis is ugly, but we do not need to reconstruct

14. Truncated signatures of moments
Compute the truncated moment signature
Construct a moments distance function, e.g.
A distance function on the shape of spaces.
Quantifies the extrinsic dissimilarity of and

15. Moments distance is small for nearly congruent and .
is large for strongly non-congruent and .
If and are truly congruent,
However, does not imply that and are
congruent (unless ).
Which shapes are indistinguishable by ?
Ideally, congruent at a coarse resolution (“low frequency”) and
differing in fine details (“high frequency”).
Degree of coarseness is controlled by the moments order
Geometric moments do not satisfy this requirement.

16. Other moments Instead of the monomial basis, other bases can be chosen
Fourier basis
Spherical harmonics, Zernike polynomials, wavelets, etc, etc.

17. Moments of joy, moments of sorrow
Shape similarity is translated to similarity of moment signatures.
Comparison of moments signatures is fast (e.g. Euclidean distance).
Sorrow:
Do not allow for partial similarity!

18. Iterative closest point (ICP) algorithms
Given two shapes and , find the best rigid motion
bringing as close as possible to :
is some shape-to-shape distance.
Minimum = extrinsic dissimilarity of and
Minimizer = best rigid alignment between and
ICP is a family of algorithms differing in
The choice of the shape-to-shape distance.
The choice of the numerical minimization algorithm.

19. Shape-to-shape distance
The Hausdorff distance
is the distance between a point and
the shape
is the distance between a point and

20. Shape-to-shape distance
A non-symmetric version is preferred to allow for partial similarity
The (max-min) formulation is sensitive to outliers.
Use the variant
is a point-to-shape distance.
Different possibilities to define

21. Point-to-point distance
Treat as a cloud of points.
Find the closest point to on
Define the distance as

22. Point-to-plane distance
Treat as a plane, and define the point-to-plane distance
is the normal to the surface at point .
Can be approximated as

23. Second-order point-to-shape distance
Point-to-plane distance is a first-order
approximation of the true point-to-shape distance.
Construct a second-order approximation
are the principal curvature radii at
are the principal directions.
is the signed distance to the closest point.

24. Second-order point-to-shape distance
The second-order distance approximant may become negative for
some values of .
Use a non-negative quadratic approximant

25. Second-order point-to-shape distance
“Near-field” case – point-to-plane distance
“Far-field” case – point-to-point distance

26. Second-order point-to-shape distance
Second-order distance generalizes the point-to-point and the point-to-
plane distances.
Gives more accurate alignment between shapes.
Requires principal curvatures and directions (second-order quantities).

27. Iterative closest point algorithm
Initialize
Find the closest point correspondence
Minimize the misalignment between corresponding points
Update
Iterate until convergence…

28. Closest points How to find closest points efficiently?
Straightforward complexity:
number of points on , number of points on .
divides the space into Voronoi cells
Given a query point , determine to which cell it belongs.

29. Closest points

30. Approximate nearest neighbors
To reduce search complexity, approximate Voronoi cells.
Use binary space partition trees (e.g. kd-trees or octrees).
Approximate nearest neighbor search complexity:

31. Best alignment Given two sets and of corresponding points.
Find best alignment
A numerical minimization algorithm can be used.
For some point-to-shape distances, a closed-form solution exists.

32. MATLAB® intermezzo
Iterative closest point algorithm

33. Until convergence… ICP should find the solution of Instead, it solves
Correspondence fixed to instead of
Not guaranteed to produce a monotonically decreasing sequence of
values of
Not guaranteed to converge!

34. Enter numerical optimization
Treat
as a numerical minimization problem.
Express the distance terms as a quadratic function
is a 3×3 symmetric positive definite matrix,
is 3×1 vector, and is a scalar.

35. Local quadratic approximant
Point-to-point distance:
Point-to-plane distance:

36. Local quadratic approximant
Minimize
over
Dependence of and on might be complicated.
For small motion , hence

37. Minimization variables
is required to be unitary (orthonormal).
Enforcing orthonormality is cumbersome.
Minimization w.r.t. to the rotation angles
involves nonlinear functions.
Under small motion assumption,
Linearize rotation matrix

38. Let Newton be!
Linearized rotation yields a quadratic objective w.r.t
Use a Newton step to find the steepest descent direction.
Approximation is valid only for small steps.
Use Armijo rule to find a fractional step ensuring sufficient
decrease of objective function.
What is a fractional step?

39. Fractional step
Let be a small transformation, which applied times gives
is a rotation by
Hence

40. Iterative closest point algorithm revisited
Initialize
Find closest point correspondence
Construct local quadratic approximant of
Find Newton direction
Use Armijo rule to find such that
Update
Iterate until convergence…

41. Iterative closest point algorithm revisited
Coefficients of the quadratic approximant can be
computed on demand using efficient nearest neighbor search.
Alternative: approximate the values of in the space
using a space partition tree.

42. Summary and suggested reading
Rigid shape similarity
R. C. Veltkamp and M. Hagedoorn, Shape similarity measures, properties, and constructions, (2001).
Moment-based shape descriptors
R. J. Prokop and A. P. Reeves, A survey of moment-based techniques for
unoccluded object representation and recognition, Graphical Models and Image
Processing 54 (1992), no. 5, 438–460.
D. Zhang and G. Lu, Review of shape representation and description
techniques, Pattern Recognition 37 (2004), 1–19.
Signal decomposition and reconstruction insights
M. Elad, P. Milanfar, and G. H. Golub, Shape from moments-an estimation
theory perspective, IEEE Trans. Signal Processing 52 (2004), no. 7, 1814–1829.
Partial shape similarity
More on partial similarity in upcoming lectures…

43. Summary and suggested reading
Iterative closest point algorithms
S. Rusinkiewicz and M. Levoy, Efficient variants of the ICP algorithm, Proc.
3D Digital Imaging and Modeling (2001), 145–152.
Point-to-shape distances: point-to-point, point-to-plane, second-order.
Fast and approximate nearest neighbor search
S. Arya and D. M. Mount, Approximate nearest neighbor searching, Proc. 4th
ACM-SIAM Symposium on Discrete Algorithms (1993), 271–280.
Convergence of ICP
E. Ezra, M. Sharir, and A. Efrat, On the ICP algorithm, Proc. Symp.
Computational Geometry (2006), 95–104.
D. Arthur and S. Vassilvitskii, Worst-case and smoothed analysis of the ICP
algorithm, with an application to the k-means method, Proc. Symp. Foundations
of Computer Science (2006), 153–164.

44. Summary and suggested reading
ICP from the point of view of numerical optimization
N. J. Mitra, N. Gelfand, H. Pottmann, and L. Guibas, Registration of point
cloud data from a geometric optimization perspective, Proc. Symp. Geometry
Processing (SGP), 2004, pp. 23–32.
ICP initialization and global optimality
N. Gelfand, N. J. Mitra, L. Guibas, and H. Pottmann, Robust global registration,
Proc. Symp. Geometry Processing (SGP) (2005).
H. Li and R. Hartley, The 3D-3D registration problem revisited, Proc. ICCV,
2007.