From Greek Mythology to Modern Manufacturing: The Procrustes Problem

1. From Greek Mythology to Modern Manufacturing: The Procrustes Problem By Dr. Dan Curtis Department of Mathematics Central Washington University

3. Procrustes offers Theseus a bed for the night

4. Theseus gives Procrustes a dose of his own medicine.

5. q p x 2 y 2 y 1 x 1 Y X

6. The Alignment Problem • We know the X-coordinates of the features and of p. • We know the Y-coordinates of the features , but not the Y-coordinates of q. • When the part is assembled, these points will coincide in space, so the and give the coordinates of the same point in two different coordinate systems. • What will be the Y-coordinates of q?

7. X Y

8. Map Registration Problem • Coordinates of featuresknown in X-coordinate system. Also, X-coordinates of feature p are known. • Y-coordinates of same features, are known. • What would the Y-coordinates of feature p be?

9. Common Thread: • 1. Have two cartesian coordinate systems in space, X and Y. • Have points whose coordinates are known in both coordinate • systems.

10. Common Thread: • 1. Have two cartesian coordinate systems in space, X and Y. • Have points whose coordinates are known in both coordinate • systems. • Find the transformation which maps the X-coordinates of a point • to the Y-coordinates of the same point. rotation matrix translation vector

11. The Orthogonal Procrustes Problem Given: points and in space, i = 1, …, n Find: optimal rotation Q and translation vector t does the best possible job of mapping the points “Best possible” means choose Q and t to minimize the following expression:

13. The above expression can be written as:

14. The above expression can be written as: or, multiplying it out, as

15. We must minimize

16. We must minimize So t must be chosen to minimize ,

17. We must minimize So t must be chosen to minimize or, equivalently,

18. Introduce centers of gravity Now minimize

19. Introduce centers of gravity Now minimize This has the form where

20. We have the identity: Minimum is obtained when

21. We have the identity: Minimum is obtained when Thus, take or

22. Original expression to be minimized was:

23. Original expression to be minimized was: This now becomes: where

24. This expression expands to

25. This expression expands to Choose Q to maximize the expression

26. This expression expands to Choose Q to maximize the expression Define the matrix A by

27. For any two column vectors u and v, we have

28. For any two column vectors u and v, we have So,

29. For any two column vectors u and v, we have So, New problem: Given a matrix A, find a rotation matrix Q which maximizes tr(AQ).

30. The Singular Value Decomposition U and V are orthogonal matrices (singular values)

31. Theorem 1: If A is an matrix and is the sum of the singular values of A, then with equality if and only if A is symmetric and positive semi-definite.

32. Theorem 1: If A is an matrix and is the sum of the singular values of A, then with equality if and only if A is symmetric and positive semi-definite. Theorem 2: If A is an matrix, then there is an orthogonal matrix Q such that AQ is symmetric and positive semi-definite. If Y is any other orthogonal matrix, then with equality if and only if AY is symmetric and positive semi-definite.

33. To find Q maximizing tr(AQ):

34. To find Q maximizing tr(AQ): • Obtain SVD

35. To find Q maximizing tr(AQ): • Obtain SVD • Take

36. To find Q maximizing tr(AQ): • Obtain SVD • Take Then: which is symmetric and positive semi-definite.

37. Summary of Solution Steps • Find centers of gravity and . • Form displacements • Form the matrix • 4. Obtain SVD 5. Take 6. Take