OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 4, Iss. 4 — Apr. 1, 1987
  • pp: 629–642

Closed-form solution of absolute orientation using unit quaternions

Berthold K. P. Horn  »View Author Affiliations

JOSA A, Vol. 4, Issue 4, pp. 629-642 (1987)

View Full Text Article

Enhanced HTML    Acrobat PDF (1437 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. It finds applications in stereophotogrammetry and in robotics. I present here a closed-form solution to the least-squares problem for three or more points. Currently various empirical, graphical, and numerical iterative methods are in use. Derivation of the solution is simplified by use of unit quaternions to represent rotation. I emphasize a symmetry property that a solution to this problem ought to possess. The best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points. The unit quaternion representing the best rotation is the eigenvector associated with the most positive eigenvalue of a symmetric 4 × 4 matrix. The elements of this matrix are combinations of sums of products of corresponding coordinates of the points.

© 1987 Optical Society of America

Original Manuscript: August 6, 1986
Manuscript Accepted: November 25, 1986
Published: April 1, 1987

Berthold K. P. Horn, "Closed-form solution of absolute orientation using unit quaternions," J. Opt. Soc. Am. A 4, 629-642 (1987)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. C. C. Slama, C. Theurer, S. W. Henrikson, eds., Manual of Photogrammetry (American Society of Photogrammetry, Falls Church, Va., 1980).
  2. B. K. P. Horn, Robot Vision (MIT/McGraw-Hill, New York, 1986).
  3. P. R. Wolf, Elements of Photogrammetry (McGraw Hill, New York, 1974).
  4. S. K. Gosh, Theory of Stereophotogrammetry (Ohio U. Bookstores, Columbus, Ohio, 1972).
  5. E. H. Thompson, “A method for the construction of orthogonal matrices,” Photogramm. Record 3, 55–59 (1958). [CrossRef]
  6. G. H. Schut, “On exact linear equations for the computation of the rotational elements of absolute orientation,” Photogrammetria 16, 34–37 (1960).
  7. H. L. Oswal, S. Balasubramanian, “An exact solution of absolute orientation,” Photogramm. Eng. 34, 1079–1083 (1968).
  8. G. H. Schut, “Construction of orthogonal matrices and their application in analytical photogrammetry,” Photogrammetria 15, 149–162 (1959).
  9. E. H. Thompson, “On exact linear solution of the problem of absolute orientation,” Photogrammetria 15, 163–179 (1959).
  10. E. Salamin, “Application of quaternions to computation with rotations,” Internal Rep. (Stanford University, Stanford, California, 1979).
  11. R. H. Taylor, “Planning and execution of straight line manipulator trajectories,” in Robot Motion: Planning and Control, M. Bradey, J. M. Mollerbach, T. L. Johnson, T. Lozano-Pérez, M. T. Mason, eds. (MIT, Cambridge, Mass., 1982).
  12. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw Hill, New York, 1968).
  13. J. H. Stuelpnagle, “On the parameterization of the three-dimensional rotation group,”SIAM Rev. 6, 422–430 (1964). [CrossRef]
  14. G. Birkhoff, S. MacLane, A Survey of Modern Algebra (Macmillan, New York, 1953).
  15. P. H. Winston, B. K. P. Horn, LISP (Addison-Wesley, Reading, Mass., 1984).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited