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

Procrustes offers Theseus a bed for the night

Theseus gives Procrustes a dose of his own medicine.

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

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?

X Y

Map Registration Problem Coordinates of features known 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?

Common Thread: 1. Have two cartesian coordinate systems in space, X and Y. 2.Have points whose coordinates are known in both coordinate systems.

Common Thread: 1. Have two cartesian coordinate systems in space, X and Y. 2.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

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:

The above expression can be written as:

or, multiplying it out, as

We must minimize

So t must be chosen to minimize,

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

Introduce centers of gravity Now minimize

Introduce centers of gravity Now minimize This has the form where

We have the identity: Minimum is obtained when

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

Original expression to be minimized was:

This now becomes: where

This expression expands to

Choose Q to maximize the expression

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

For any two column vectors u and v, we have

So,

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

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

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 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.

To find Q maximizing tr(AQ):

Obtain SVD

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

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

Summary of Solution Steps 1.Find centers of gravity and. 2.Form displacements 3.Form the matrix 4. Obtain SVD 5. Take 6. Take