Orthogonal Procrustes problem

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The orthogonal Procrustes problem [1] is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A and B and asked to find an orthogonal matrix R which most closely maps A to B. [2] Specifically,

R = \arg\min_\Omega \|A\Omega-B\|_F \quad\mathrm{subject\ to}\quad \Omega^T\Omega=I,

where \|\cdot\|_F denotes the Frobenius norm.

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[edit] Solution

This problem was originally solved by Peter Schonemann in a 1964 thesis. The individual solution was later published. [3]

This problem is equivalent to finding the nearest orthogonal matrix to a given matrix M = ATB. To find this orthogonal matrix R, one uses the singular value decomposition

M=U\Sigma V^*\,\!

to write

R=UV^*.\,\!

[edit] Generalized/constrained Procrustes problems

There are a number of related problems to the classical orthogonal Procrustes problem. One might generalize it by seeking the closest matrix in which the columns are orthogonal, but not necessarily orthonormal. [4] Alternately, one might constrain it by only allowing rotation matrices with determinant 1 (special orthogonal matrices) In this case, one can write (using the above decomposition M = UΣV * )

R=U\Sigma'V^*,\,\!

where \Sigma'\,\! is a proper-sized identity matrix, with the smallest singular value replaced by det(UV * ) so that the overall determinant is positive.[5]


[edit] See also

[edit] References

  1. ^ Gower, J.C & Dijksterhuis, G.B. (2004), Procrustes Problems, Oxford University Press 
  2. ^ Hurley, J.R. & Cattell, R.B. (1962), “Producing direct rotation to test a hypothesized factor structure”, Behavioral Science 7: 258-262 
  3. ^ Schonemann, P.H. (1966). "A generalized solution of the orthogonal Procrustes problem". Psychometrika 31: 1–10. doi:10.1007/BF02289451. 
  4. ^ Everson, R (1997), Orthogonal, but not Orthonormal, Procrustes Problems, <http://citeseer.ist.psu.edu/everson97orthogonal.html> 
  5. ^ Eggert, DW; Lorusso, A & Fisher, RB (1997), “Estimating 3-D rigid body transformations: a comparison of four major algorithms”, Machine Vision and Applications 9 (5): 272-290