Skip to main content
Log in

Review and analysis of solutions of the three point perspective pose estimation problem

  • Systems And Replication
  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript


In this paper, the major direct solutions to the three point perspective pose estimation problems are reviewed from a unified perspective beginning with the first solution which was published in 1841 by a German mathematician, continuing through the solutions published in the German and then American photogrammetry literature, and most recently in the current computer vision literature. The numerical stability of these three point perspective solutions are also discussed. We show that even in case where the solution is not near the geometric unstable region, considerable care must be exercised in the calculation. Depending on the order of the substitutions utilized, the relative error can change over a thousand to one. This difference is due entirely to the way the calculations are performed and not due to any geometric structural instability of any problem instance. We present an analysis method which produces a numerically stable calculation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • E. Church, “Revised Geometry of the Aerial Photograph,”Syracuse University Press, Bulletin 15, Syracuse, NY, 1945.

  • E. Church, “Theory of Photogrammetry,”Syracuse University Press, Bulletin 19, Syracuse, NY, 1948.

  • M. Dhome, M. Richetin, J. T. Lapreste and G. Rives, “The Inverse Perspective Problem from a Single View for Polyhedra Location,”IEEE Conference on Computer Vision and Pattern Recognition, 1988, pp. 61–68.

  • S. Finsterwalder and W. Scheufele, “Das Rückwärtsein-schneiden im Raum”Sebastian Finsterwalder zum 75.Geburtstage, Verlag Herbert Wichmann, Berlin, Germany, 1937, pp. 86–100.

    Google Scholar 

  • M. A. Fischler and R. C. Bolles, “Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography,”Graphics and Image Processing, Vol. 24, No. 6, 1981, pp. 381–395.

    Google Scholar 

  • S. Ganapathy, “Decomposition of Transformation Matrices for Robot Vision, ”Proceeding of International Conf. on Robotics and automation, Rome, Italy, pp. 130–119, Nov., 1984.

  • E. W. Grafarend, P. Lohse, and B. Schaffrin, “Dreidimensionaler Rückw ärtsschnitt Teil I: Die projektiven Gleichungen”Zeitschrift für Vermessungswesen, Geodätisches Institut, Universität Stuttgart, 1989, pp. 1–37.

  • J. A. Grunert, “Das Pothenotische Problem in erweiterter Gestalt nebst Über seine Anwendungen in der Geodäsie,Grunerts Archiv für Mathematik und Physik, Band 1, 1841, pp. 238–248.

    Google Scholar 

  • R. M. Haralick, H. Joo, C. N. Lee, X. Zhuang, V.G. Vaidya, and M. B. Kim, “Pose Estimation from Corresponding Point Data,”IEEE Trans. on Systems, Man, and Cybernetics, Vol. 19, No. 6, Nov./Dec. 1989.

  • B. K. P. Horn, “Closed-Form Solution of absolute Orientation Using Orthonormal Matrices,”J. Opt. Soc. Am. A, Vol. 5, No. 7, 1988, pp. 1127–1135.

    Google Scholar 

  • R. Horaud, “New Methods for Matching 3-D Objects with Single Perspective Views,”IEEE Transactions on the Pattern Analysis and Machine Intelligence, Volume PAMI-9, No. 3, May 1987.

  • S. Linnainmaa, D. Harwood, and L. S. Davis, “Pose Estimation of a Three-Dimensional Object Using Triangle Pairs”IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 10, No. 5, 1988, pp. 634–647.

    Article  Google Scholar 

  • P. Lohse, “Dreidimensionaler Rückwärtsschnitt Ein Algorithmus zur Streckenberechnung ohne Hauptachsentransformation,”Geodätisches Institut, Universität Stuttgart, 1989.

  • D. G. Lowe, “Three-Dimensional Object Recognition from Single Two-Dimensional Images,”Artificial Intelligence, Vol. 31, 1987, pp. 355–395.

    Article  Google Scholar 

  • E. L. Merritt, “Explicity Three-Point Resection in Space”Photogrammetric Engineering, Vol. XV, No. 4, 1949, pp. 649–655.

    Google Scholar 

  • E. L. Merritt, “General Explicit Equations for a Single Photograph”Analytical Photogrammetry, Pitman Publishing Corporation, New York, USA, pp. 43–79.

  • F. J. Müller, “Direkte (exakte) Losung des einfachen Rückwartseinschneidens im Raume”Allgemeine Vermessungs-Nachrichten, 1925.

  • P. H. Schonemann, and R.M. Caroll, “Fitting One Matrix to Another Under Choice of a Central Dilation and a Rigid Body Motion.”Psychometrika, 35(2):245–255, June 1970.

    Google Scholar 

  • P. H. Schonemann, “A Generalized Solution of the Orthogonal Procrustes Problem,”Psychometrika, 31(1):1–10, March 1966.

    Google Scholar 

  • G. H. Schut, “On Exact Linear Equations for the Computation of the Rotational Elements of Absolute Orientation,”Photogrammetria 16 (1), 1960, pp. 34–37.

    Article  Google Scholar 

  • C. C. Slama, ed., “Manual of Photogrammetry”American Society of Photogrammetry, Falls Church, Virginia, 1980.

  • A.D.N. Smith, “The Explicit Solution of Single Picture Resection Problem with a Least Squares Adjustment to Redundant Control,”Photogrammetric Record, Vol. V, No. 26, October 1965, pp. 113–122.

    Google Scholar 

  • E.H. Thompson, “Space Resection: Failure Cases,”Photogrammetric Record, Vol. V, No. 27, April 1966, pp. 201–204.

    Google Scholar 

  • R.Y. Tsai, “A Versatile Camera Calibration Technique for High-Accuracy 3D Machine Vision Metrology Using Off-the-Shelf TV Cameras and Lenses,”IEEE Journal of Robotics and Automation, Vol. RA-3, No. 4, 1987, pp. 323–344.

    Google Scholar 

  • J. Vlach and K. Singhal, “Computer Methods for Circuit Analysis and Design.” 1983.

  • J.H. Wilkinson, “Rounding Errors in Algebraic Process.” H.M. Stationery Office, London; 1963.

    Google Scholar 

  • P.R. Wolf,Elements of Photogrammetry, McGraw Hill, New York, USA, 1974.

    Google Scholar 

  • W.J. Wolfe, D. Mathis, C.W. Sklair, and M. Magee, “The Perspective View of Three Points,”IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, No. 1, 1991, pp. 66–73.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

About this article

Cite this article

Haralick, B.M., Lee, CN., Ottenberg, K. et al. Review and analysis of solutions of the three point perspective pose estimation problem. Int J Comput Vision 13, 331–356 (1994).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: