International Journal of Computer Vision

, Volume 4, Issue 1, pp 59–78 | Cite as

Relative orientation

  • Berthold K. P. Horn


Before corresponding points in images taken with two cameras can be used to recover distances to objects in a scene, one has to determine the position and orientation of one camera relative to the other. This is the classic photogrammetric problem of relative orientation, central to the interpretation of binocular stereo information. Iterative methods for determining relative orientation were developed long ago; without them we would not have most of the topographic maps we do today. Relative orientation is also of importance in the recovery of motion and shape from an image sequence when successive frames are widely separated in time. Workers in motion vision are rediscovering some of the methods of photogrammetry.

Described here is a simple iterative scheme for recovering relative orientation that, unlike existing methods, does not require a good initial guess for the baseline and the rotation. The data required is a pair of bundles of corresponding rays from the two projection centers to points in the scene. It is well known that at least five pairs of rays are needed. Less appears to be known about the existence of multiple solutions and their interpretation. These issues are discussed here. The unambiguous determination of all of the parameters of relative orientation is not possible when the observed points lie on a critical surface. These surfaces and their degenerate forms are analyzed as well.


Image Processing Computer Vision Computer Image Iterative Method Relative Orientation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    L.U. Bender, “Derivation of parallax equation,” Photogrammetric Engineering 33 (10): 1175–1179, 1967.Google Scholar
  2. 2.
    A. Brandenberger, “Fehlertheorie der äusseren Orientierung von Steilaufnahmen,” Ph.D. Thesis, Eidgenössische Technische Hochschule, Zürich, Switzerland, 1947.Google Scholar
  3. 3.
    P. Brou, “Using the Gaussian image to find the orientation of an object,” Intern. J. Robotics Res. 3 (4): 89–125, 1983.Google Scholar
  4. 4.
    A.R. Bruss and B.K.P. Horn, “Passive navigation,” Comput. Vision, Graphics, Image Process. 21 (1): 3–20, 1983.Google Scholar
  5. 5.
    O.D. Faugeras and S. Maybank, “Motion from point matches: Multiplicity of solutions,” Proc. IEEE Workshop Motion Vision, March 20–22, 1989.Google Scholar
  6. 6.
    W. Forstner, “Rellability analysis of parameter estimation in linear models with applications to mensuration problems in computer vision,” Comput. Vision, Graphics, Image Process. 40 (3): 273–310, 1987.Google Scholar
  7. 7.
    R.B. Forrest, “AP-C plotter orientation,” Photogrammetric Engineering 32 (5): 1024–1027, 1966.Google Scholar
  8. 8.
    C.S. Fraser and D.C. Brown, “Industrial photogrammetry: New developments and recent applications,” Photogrammetric Record 12 (68): 197–217, October 1986.Google Scholar
  9. 9.
    M.A. Gennert, “A computational framework for understanding problems in stereo vision.” Ph.D. thesis, Department of Electrical Engineering and Computer Science, MIT, August 1987.Google Scholar
  10. 10.
    C.D. Ghilani, “Numerically assisted relative orientation of the Kern PG-2,” Photogramm. Enginner. Remote Sensing 49 (10): 1457–1459, 1983.Google Scholar
  11. 11.
    S.K. Ghosh, “Relative orientation improvement,” Photogrammetric Engineering 32 (3): 410–414, 1966.Google Scholar
  12. 12.
    S.K. Ghosh, Theory of Stereophotogrammetry, Ohio University Bookstores: Columbus, OH, 1972.Google Scholar
  13. 13.
    C. Gill, “Relative orientation of segmented, panoramic grid models on the AP-II,” Photogrammetric Engineering 30: 957–962, 1964.Google Scholar
  14. 14.
    B. Hallert, Photogrammetry. McGraw-Hill: New York, 1960.Google Scholar
  15. 15.
    D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination. Chelsea Publishing: New York, 1953, 1983.Google Scholar
  16. 16.
    L. Hinsken, “Algorithmen zur Beschaffung von Näherungswerten für die Orientierung von beliebig im Raum angeordneten Strahlenbündeln,” Dissertation, Deutsche Geodätische Kommission, Reihe C, Heft Nr. 333, München, Federal Republic of Germany, 1987.Google Scholar
  17. 17.
    L. Hinsken, “A singularity-free algorithm for spatial orientation of bundles,” Intern. Arch. Photogramm. Remote Sensing 27, B5, comm. v, pp. 262–272, 1988.Google Scholar
  18. 18.
    W. Hofmann, “Das Problem der ‘Gefährlichen Flächen’ in Theorie und Praxis,” Ph.D. Thesis, Technische Hochschule München. Published in 1953 by Deutsche Geodätische Kommission, München, Federal Republic of Germany, 1949.Google Scholar
  19. 19.
    B.K.P. Horn, Robot Vision. MIT Press: Cambridge, MA, and McGraw-Hill, New York, 1986.Google Scholar
  20. 20.
    B.K.P. Horn, “Closed-form solution of absolute orientation using unit quaternions,” J. Opt. Soc. Am. A 4 (4): 629–642, 1987.Google Scholar
  21. 21.
    B.K.P. Horn, “Motion fields are hardly ever ambiguous,” Intern. J. Comput. Vision 1 (3): 263–278, 1987.Google Scholar
  22. 22.
    B.K.P. Horn, “Relative orientation,” memo 994, Artificial Intelligence Laboratory, MIT, Cambridge, MA, 1987. Also, Proc. Image Understanding Workshop, 6–8 April, Morgan Kaufman Publishers: San Mateo, CA, pp. 826–837, 1988.Google Scholar
  23. 23.
    B.K.P. Horn, H.M. Hilden, and S. Negahdaripour, “Closed-form solution of absolute orientation using orthonormal matrices, J. Opt. Soc. Am. A 5 (7): 1127–1135, July 1988.Google Scholar
  24. 24.
    B.K.P. Horn and E.J. WeldonJr., “Direct methods for recovering motion,” Intern. J. Comput. Vision 2 (1): 51–76, June 1988.Google Scholar
  25. 25.
    H. Jochmann, “Number of orientation points,” Photogrammetric Engineering 31 (4): 670–679, 1965.Google Scholar
  26. 26.
    G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. McGraw-Hill: New York, 1968.Google Scholar
  27. 27.
    H.C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293: 133–135, September 1981.Google Scholar
  28. 28.
    H.C. Longuet-Higgins, “The reconstruction of a scene from two projections—Configurations that defeat the eight-point algorithm,” IEEE, Proc. 1st Conf. Artif. Intell. Applications, Denver, CO, 1984.Google Scholar
  29. 29.
    H.C. Longuet-Higgins, “Multiple interpretations of a pair of images of a surface,” Proc. Roy. Soc. London, 418: 1–15, 1988.Google Scholar
  30. 30.
    H.C. Longuet-Higgins and K. Prazdny, “The interpretation of a moving retinal image,” Proc. Roy. Soc. London B 208: 385–397, 1980.Google Scholar
  31. 31.
    E.M. Mikhail and F. Ackerman, Observations and Least Squares. Harper & Row: New York, 1976.Google Scholar
  32. 32.
    F. Moffit and E.M. Mikhail, Photogrammetry, 3rd ed. Harper & Row: New York, 1980.Google Scholar
  33. 33.
    S. Negahdaripour, “Multiple interpretation of the shape and motion of objects from two perspective images.” Unpublished manuscript of the Waikiki Surfing and Computer Vision Society of the Department of Electrical Engineering at the University of Hawaii at Manoa, Honolulu, HI, 1989.Google Scholar
  34. 34.
    A.N. Netravali, T.S. Huang, A.S. Krisnakumar, and R.J. Holt, “Algebraic methods in 3-D motion estimation from two-view point correspondences.” Unpublished internal report, A.T.&T. Bell Laboratories: Murray Hill, NJ, 1980.Google Scholar
  35. 35.
    A. Okamoto, “Orientation and construction of models—Part I: The orientation problem in close-range photogrammetry,” Photogramm. Engineer. Remote Sensing 47 (10): 1437–1454, 1981.Google Scholar
  36. 36.
    H.L. Oswal, “Comparison of elements of relative orientation,” Photogrammetric Engineering 33 (3): 335–339, 1967.Google Scholar
  37. 37.
    A. Pope, “An advantageous, alternative parameterization of rotations for analytical photogrammetry,” ESSA Technical Report CaGS-39, Coast and Geodetic Survey, U.S. Department of Commerce, Rockville, MA. Also Symposium on Computational Photogrammetry. American Society of Photogrammetry: Alexandria, Virginia, January 7–9, 1970.Google Scholar
  38. 38.
    K. Rinner, “Studien über eine allgemeine, vorraussetzungslose Lösung des Folgebildanschlusses,” Österreichische Zeitschrift für Vermessung, Sonderheft 23, 1963.Google Scholar
  39. 39.
    S. Sailor, “Demonstration board for stereoscopic plotter orientation,” Photogrammetric Engineering 31 (1): 176–179, January, 1965.Google Scholar
  40. 40.
    E. Salamin, “Application of quaternions to computation with potations.” Unpulbished Internal Report, Stanford University, Stanford, CA, 1979.Google Scholar
  41. 41.
    G.H. Schut, “An analysis of methods and results in analytical aerial triangulation,” Photogrammetria 14: 16–32, 1957–1958.Google Scholar
  42. 42.
    G.H. Schut, “Construction of orthogonal matrices and their application in analytical photogrammetry,” Photogrammetria 15 (4): 149–162, 1958–1959.Google Scholar
  43. 43.
    K. Schwidefsky, An Outline of Photogrammetry. Translated by John Fosberry, 2nd ed. Pitman & Sons: London, 1973.Google Scholar
  44. 44.
    K. Schwidefsky, and F. Ackermann, Photogrammetrie, Teubner: Stuttgart, Federal Republic of Germany, 1976.Google Scholar
  45. 45.
    C.C. Slama, C. Theurer, and S.W. Hendrikson (eds.), Manual of Photogrammetry. American Society of Photogrammetry: Falls Church, VA, 1980.Google Scholar
  46. 46.
    J.H. Stuelpnagle, “On the parameterization of the three-dimensional rotation group,” SIAM Review 6 (4): 422–430, October 1964.Google Scholar
  47. 47.
    R.H. Taylor, “Planning and execution of straight line manipulator trajectories.” In Robot Motion: Planning and Control, M.J. Brady, J.M. Hollerbach, T.L. Johnson, R. Lozano-Pérez, and M.T. Mason (eds.), MIT Press: Cambridge, MA, 1982.Google Scholar
  48. 48.
    E.H. Thompson, “A Method for the construction of orthonormal matries,” Photogrammetric Record 3 (13): 55–59, April 1959.Google Scholar
  49. 49.
    E.H. Thompson, “A rational algebraic formulation of the problem of relative orientation,” Photogrammetric Record 3 (14): 152–159, 1959.Google Scholar
  50. 50.
    E.H. Thompson, “A note on relative orientation,” Photogrammetric Record 4 (24): 483–488, 1964.Google Scholar
  51. 51.
    E.H. Thompson, “The projective theory of relative orientation,” Photogrammetria 23: 67–75, 1968.Google Scholar
  52. 52.
    R.Y. Tsai and T.S. Huang, “Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces,” IEEE Trans. PAMI 6 (1): 13–27, 1984.Google Scholar
  53. 53.
    S. Ullman, The Interpretation of Visual Motion. MIT Press: Cambridge, MA, 1979.Google Scholar
  54. 54.
    A.J.Van Der Weele, “The relative orientation of photographs of mountainous terrain,” Photogrammetria 16 (2): 161–169, 1959–1960.Google Scholar
  55. 55.
    P.R. Wolf, Elements of Photogrammetry, 2nd ed. McGraw-Hill: New York, 1983.Google Scholar
  56. 56.
    M. Zeller, Textbook of Photogrammetry. H.K. Lewis & Company: London, 1952.Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Berthold K. P. Horn
    • 1
  1. 1.Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridge

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