1. Point set alignment
Closed-form solution of absolute orientation using unit quaternions
Berthold K. P. Horn
Department of Electrical Engineering, University of Hawaii at Manoa
Presented by Ashley Fernandes

2. Abstract
Finding relationship between coordinate systems (absolute orientation)
Closed-form
Use of quaternions for rotation
Use of centroid for translation
Use of root-mean-square deviations for scale

3. Disadvantages of previous methods
Cannot handle more than three points
Do not use information from all three points
Iterative instead of least squares

4. Introduction - Transformation
Transformation between two
Cartesian coordinate systems
Translation
Rotation
Scaling

5. Introduction - Method Minimize error Closed-form solution
Use of quaternions
Symmetry of solution

6. Coordinate systems

7. Selective discarding constraints
X axis
Y axis
Z axis l
Maps points from left hand to right hand coordinate system
Rotation

8. Finding the translation
Measured coordinates in left and right hand systems
Form of translation
Scale factor
Translational offset
Rotated vector from left coordinate system
Residual error
To be minimized

9. Centroids of sets of measurements
New coordinates
Error term
where
Sum of squares of errors

10. Centroids of sets of measurements
Translation, when r’o = 0
Error term, when r’o = 0
Total error term to be minimized

11. Finding the scale Total error term, since
To minimize w.r.t. scale s, first term should be zero, or

12. Symmetry in scale Suppose we tried to find , or so we hope. But, or
Instead, we use
Total error becomes
To minimize w.r.t. scale s, first term should be zero, or
Scale

13. Why unit quaternions
Easier to enforce unit magnitude constraint on quaternion than orthogonal constraint on matrix
Closely allied to geometrically intuitive concept of rotation by an angle about an axis

14. Quaternions
Representation If
Multiplication

15. Quaternions Multiplication expressed as product of
orthogonal matrix4x4 and vector4

16. Quaternions Dot product Square of magnitude Conjugate
Product of quaternion and its conjugate
Inverse

17. Unit quaternions and rotationif
We use the composite product
which is purely imaginary.
Note that this is similar to
Also, note that

18. Relationship to other notations
If angle is Θ
and axis is unit vector

19. Composition of rotations
First rotation
Second rotation
Since
Combined rotation

20. Finding the best rotation
We must find the quaternion
that maximizes
Let
and

21. Finding the best rotation
What we have to maximize

22. Finding the best rotation
where
and
Introducing the matrix3x3
that contains all the information required to solve the least-squares problem for
rotation.
where
and so on.
Then,

23. Eigenvector maximizes matrix product
Unit quaternion that maximizes
is eigenvector corresponding to most positive eigenvalue of N.
Eigenvalues are solutions of quartic in
that we obtain from
After selecting the largest positive eigenvalue
we find the eigenvector
by solving

24. Nature of the closed-form solution
Find centroids rl and rr of the two sets of measurements
Subtract them from all measurements
For each pair of coordinates, compute x’lx’r, x’ly’r, … z’lz’r of the components of the two vectors.
These are added up to obtain Sxx, Sxy, …Szz.

25. Nature of the closed-form solution
Compute the 10 independent elements of the 4x4 symmetric matrix N
From these elements, calculate the coefficients of the quartic that must be solved to get the eigenvalues of N
Pick the most positive root and use it to solve the four linear homogeneous equations to get the eigenvector. The quaternion representing the rotation is a unit vector in the same direction.

26. Nature of the closed-form solution
Compute the scale from the symmetrical form formula, i.e. the ratio of the root-mean-square deviations of the measurements from their centroids.
Compute the translation as the difference between the centroid of the right measurements and the scaled and rotated centroid of the left measurement.

27. Thank you.
The end.