1. Iterative closest point algorithms
Tutorial 4
© Maks Ovsjanikov
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009 1

2. Outline
Closed-form solution of absolute orientation using unit quaternions.
Horn, B. K. P., Journal of the Optical Society of America, Vol
Problem Formulation.
Intuition.
Some Math.
Robust Global Registration
Gelfand N., Mitra N., Guibas L., Pottmann H., Proceedings of SGP, 2005
Formulation.
Problem and Motivation.
Multi-Scale point descriptors.
Global Matching – Branch and Bound.
Results.

3. Closed-form solution of absolute orientation using unit quaternions.
Horn, B. K. P., Journal of the Optical Society of America, Vol
Problem Formulation:
Given two sets points: in Find the rigid transform: and that minimizes:

4. Closed-form solution of absolute orientation using unit quaternions.
Horn, B. K. P., Journal of the Optical Society of America, Vol
Problem Formulation:
Given two sets points: in Find the rigid transform: and that minimizes:

5. Closed-form solution of absolute orientation using unit quaternions.
Horn, B. K. P., Journal of the Optical Society of America, Vol
Problem Formulation:
Given two sets points: in Find the rigid transform: and that minimizes:
In practice, have many points. Perform this step many times. Want a fast way to get an exact solution. Representation is important.

6. Given two sets points: in . Find the rigid transform:
and that minimizes:
Finding the translation part .
1. Let and : the centroids of and
and let and Then 2.
where:
minimized if

7. Given two sets points: in . Find the rigid transform:
and that minimizes:
Need to find the rotation .
1. Several ways to represent rotation: Orthonormal matrix, Euler angle, Gibbs vector,
quaternions, etc. Which one is most useful in this case?
2. Plan of action:
Want to minimize
Since rotation preserves lengths. Need to maximize:
Question: What unit vector maximizes if is symmetric?
Answer: Eigenvector corresponding to the maximum eigenvalue of .

8. Given two sets points: in . Find the rigid transform:
and that minimizes:
Need to find the rotation .
2. Plan of action:
Need to maximize:
3. If represent using orthonormal matrices, then get:
Would prefer: , where is a vector and matrices
Quaternions allow us to express rotation in such form.

9. Quaternions (two slides intro).
1. Vectors with 4 components, 3 of which are imaginary:
2. Multiplication is defined using the rules:
from which follows:
In matrix form:
3. Dot product:
Matrices associated with unit quaternions are orthonormal.

10. Quaternions (two slides intro).
4. Conjugation: Matrix corresponding to the conjugate is the transpose of the original.
5. Regular vectors can be represented with purely imaginary quaternions:
6. Rotation can be represented with composite product by a unit quaternion :
where
Conversely, every unit quaternion corresponds to a rotation.

11. Back to Alignment.
1. Want to maximize:
2. In quaternion form:
3. The optimal rotation quaternion is the eigenvector corresponding to the maximum eigenvalue of:

12. Summary
Closed-form solution of absolute orientation using unit quaternions.
Horn, B. K. P., Journal of the Optical Society of America, Vol
Efficient algorithm to find the optimal translation and rotation in the rigid alignment, provided that the correspondences are known.
Uniform scaling can be found similarly:
Quaternions provide a simple and efficient way to represent rotations.
Other methods: SVD, Orthogonal Matrices, Dual Quaternions
Estimating 3-D rigid body transformations: a comparison of four major algorithms. Eggert et al., Machine Vision and Applications, Vol

13. Outline
Closed-form solution of absolute orientation using unit quaternions.
Horn, B. K. P., Journal of the Optical Society of America, Vol
Problem Formulation.
Intuition.
Some Math.
Robust Global Registration
Gelfand N., Mitra N., Guibas L., Pottmann H., Proceedings of SGP, 2005
Formulation.
Problem and Motivation.
Multi-Scale point descriptors.
Global Matching – Branch and Bound.
Results.
Images by N. Gelfand

14. Problem: Robust Global Registration
Gelfand, Mitra, Guibas, Pottmann, Proceedings of SGP, 2005
Problem:
Given:
Two shapes P and Q which partially overlap.
Goal:
Using only rigid transforms, register Q against P by minimizing the squared distance between them.
No assumptions about initial positions!

15. Approaches
Rigid transform can be specified with small number of parameters
Try all possible transform bases
Find the one that aligns the most points
Guaranteed to find the correct transform
But can be costly

16. Approaches Use neighborhood geometric information
Descriptors should be
Invariant under transform
Local
Cheap
Improves correspondence search or used for feature extraction
[Mokh 01, Huber 03, Li 05]

17. We only need to align a few points correctly.
Registration Problem
Why it’s hard
Unknown areas of overlap
Have to solve the correspondence problem
Why it’s easy
Rigid transform is specified by small number of points
Prominent features are easy to identify
We only need to align a few points correctly.

18. Method Overview Use descriptors to identify features
Integral volume descriptor
Build correspondence search space
Few correspondences for each feature
Efficiently explore search space
Distance error metric
Pruning algorithms

19. Integral Descriptors Multiscale Inherent smoothing f(x) [Manay 04]
Br(p) p

20. Integral Volume Descriptor
Related to mean curvature.
Provably robust to noise.
0.20

21. Descriptor Computation
Approximate using a voxel grid
Convolution of occupancy grid with ball

22. Feature Identification
Pick as features points with rare descriptor values
Rare in the data [ rare in the model [ few correspondences
Works for any descriptor
Features

23. Multiscale Algorithm Features should be persistent over scale changeY N

24. Correspondence Space Search the whole model for correspondences
Range query for descriptor values
Cluster and pick representatives Q P

25. Evaluating Correspondences
Coordinate root mean squared distance
Requires best aligning transform
Looks at correspondences individually

26. Rigidity Constraint
Pair-wise distances between features and correspondences should be the same Q P

27. Rigidity Constraint
Pair-wise distances between features and correspondences should be the same Q P

28. Evaluating Correspondences
Distance root mean squared distance
Depends only on internal distances

29. Search Algorithm Few features, each with few potential correspondences
Minimize dRMS
Exhaustive search still too expensive P Q

30. Branch and Bound.
Find global minimum of a function.
Search Space:

31. Branch and Bound. Find global minimum of a function. Search Space:
Split (Branch).
Bound.
Prune:
If min2 > max1, can remove branch 2.
min2
max2
min1
max1

32. Search Algorithm Branch and bound
Initial bound using greedy assignment
Discard partial correspondences that fail thresholding test
Prune if partial correspondence exceeds bound
Spaced out features make incorrect correspondences fail quickly
Since we explore the entire search space, we are guaranteed to find optimal alignment
Up to cluster size Rc qi pi pj qj

33. Search Algorithm Branch and bound
Initial bound using greedy assignment
Discard partial correspondences that fail thresholding test
Prune if partial correspondence exceeds bound
Spaced out features make incorrect correspondences fail quickly
Since we explore the entire search space, we are guaranteed to find optimal alignment
Up to cluster size

34. Greedy Initialization
Good initial bound is essential
Build up correspondence set hierarchically …

35. Greedy Initialization
Good initial bound is essential
Build up correspondence set hierarchically … …

36. Greedy Initialization
Good initial bound is essential
Build up correspondence set hierarchically …

37. Partial Alignment
Allow null correspondences, while maximizing the number of matches points

38. Alignment Results
Input: 2 scans
Alignment
Refined by ICP

39. Alignment Results
Alignment
Refined by ICP
Input: 10 scans