Skip to main content

Abstract

The terms Procrustes Analysis and Procrustes Techniques are referred to a set of least squares mathematical models used to perform transformations among corresponding points belonging to a generic k-dimensional space, in order to satisfy their maximum agreement.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The predicate \({{\,\mathrm{sym}\,}}[]\) is true when the argument is a symmetric matrix.

  2. 2.

    The vec operator transforms a matrix into a vector by stacking its columns.

References

  • J.L. Awange, Partial procrustes solution of the three dimensional orientation problem from gps/lps observations in Quo vadis geodesia? ed. by Krumm, F., Schwarze, V.S., pp. 41–51 (1999)

    Google Scholar 

  • J.L. Awange, E.W. Grafarend, in Solving Algebraic Computational Problems in Geodesy and Geoinformatics: The Answer to Modern Challenges (Springer, 2005)

    Google Scholar 

  • A. Beinat, F. Crosilla, Generalized procrustes analysis for size and shape 3D object reconstruction, in Optical 3-D Measurement Techniques (Wichmann Verlag, 2001), pp. 345–353

    Google Scholar 

  • A. Beinat, F. Crosilla, Procrustes statistics to test for significant ols similarity transformation parameters, in V Hotine-Marussi Symposium on Mathematical Geodesy. (Springer, 2004), pp. 113–119

    Google Scholar 

  • A. Beinat, F. Crosilla, D. Visintini. Examples of georeferenced data transformations in GIS and digital photogrammetry by procrustes analysis techniques. Int. Arch. Photogramm. Remote. Sens., XXXII(6W8/2), 2–5 (2000)

    Google Scholar 

  • A. Beinat. Tecniche di analisi procustiana e trasformazioni di datum in topografia e fotogrammetria. Ph.D. thesis, Dipartimento di Ingegneria Idraulica Ambientale e del Rilevamento, Politecnico of Milan (2000)

    Google Scholar 

  • M. Bennani Dosse, J.M.F. ten Berge, Anisotropic orthogonal procrustes analysis. J. Cl.Ification, 27(1), 111–128 (2010)

    Article  Google Scholar 

  • F. Boas, The horizontal plane of the skull and the general problem of the comparison of variable forms. Science 21(544), 862–863 (1905)

    Article  Google Scholar 

  • F.L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology (Cambridge University Press, 1997)

    Google Scholar 

  • F.L. Bookstein, Size and shape spaces for landmark data in two dimensions. Stat. Sci. 1(2), 181–222 (1986)

    Article  Google Scholar 

  • I. Borg, P. Groenen, Modern multidimensional scaling: theory and applications. J. Educ. Meas. 40(3), 277–280 (2003)

    Article  Google Scholar 

  • S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications (Wiley, 2013)

    Google Scholar 

  • N. Cliff, Orthogonal rotation to congruence. Psychometrika 31(1), 33–42 (1966)

    Article  Google Scholar 

  • T.M. Cole, Historical note: early anthropological contributions to “geometric morphometrics”. Am. J. Phys. Anthropol.: Off. Publ. Am. Assoc. Phys. Anthropol. 101(2), 291–296 (1996)

    Article  Google Scholar 

  • J.J.F. Commandeur, Matching Configurations (DSWO Press, 1991)

    Google Scholar 

  • F. Crosilla, Procrustean transformation as a tool for the construction of a criterion matrix for control networks. Manuscripta Geodetica 8, (1983b)

    Google Scholar 

  • F. Crosilla, Procrustes analysis and geodetic sciences, in Quo vadis geodesia...?, vol. 1, pp. 69–78. Department of Geodesy and GeoInformatics, University of Stuttgart (1999)

    Google Scholar 

  • F. Crosilla, A. Beinat, A forward search method for robust generalised procrustes analysis, in Data analysis, classification and the forward search (Springer, 2006) pp. 199–208

    Google Scholar 

  • F. Crosilla, A criterion matrix for the second order design of control networks. Bulletin géodésique 57(1–4), 226–239 (1983a)

    Article  Google Scholar 

  • F. Crosilla, A. Beinat, Use of generalised procrustes analysis for the photogrammetric block adjustment by independent models. ISPRS J. Photogramm. & Remote. Sens. 56(3), 195–209 (2002)

    Article  Google Scholar 

  • I.L. Dryden, K.V. Mardia, Statistical Shape Analysis (Wiley, New York, 1998)

    Google Scholar 

  • A. Fusiello, F. Crosilla, Solving bundle block adjustment by generalized anisotropic procrustes analysis. ISPRS J. Photogramm. Remote. Sens. 102, 209–221 (2015). ISSN 0924-2716

    Article  Google Scholar 

  • V. Garro, F. Crosilla, A. Fusiello, Solving the pnp problem with anisotropic orthogonal procrustes analysis, in Second Joint 3DIM/3DPVT Conference: 3D Imaging (Modeling, Processing, Visualization and Transmission, 2012), pp. 262–269

    Google Scholar 

  • C.R. Goodall, Procrustes methods in the statistical analysis of shape. J. R. Stat. Society. Ser. B (Methodol.) 53(2), 285–339 (1991)

    Google Scholar 

  • J.C. Gower, Generalized procrustes analysis. Psychometrika 40(1), 33–51 (1975)

    Article  Google Scholar 

  • J.C. Gower, G.B. Dijksterhuis, Procrustes Problems (Oxford University Press, Oxford Statistical Science Series, 2004)

    Book  Google Scholar 

  • E.W. Grafarend, J.L. Awange, Determination of the vertical deflection by gps/lps measurements. Zeitschrift für Vermessungswesen 125(8), 279–288 (2000)

    Google Scholar 

  • E.W. Grafarend, J.L. Awange, Nonlinear analysis of the three-dimensional datum transformation [conformal group \(\mathbb{C}_7(3)\)]. J. Geod. 77(1–2), 66–76 (2003)

    Article  Google Scholar 

  • B.F. Green, J.C. Gower, A Problem with Congruence, in Annual meeting of the psychometric society (Monterey, California, 1979)

    Google Scholar 

  • B.F. Green, The orthogonal approximation of an oblique structure in factor analysis. Psychometrika 17(4), 429–440 (1952)

    Article  Google Scholar 

  • M. Gulliksson. The partial procrustes problem: a first look. Technical Report UMINF 95.11., (Department of Computing Science, Umeå University, Sweden, 1995)

    Google Scholar 

  • M.T. Heath, A.J. Laub, C.C. Paige, R.C. Ward, Computing the singular value decomposition of a product of two matrices. SIAM J. Sci. Stat. Comput. 7(4), 1147–1159 (1986)

    Article  Google Scholar 

  • P.W. Holland, R.E. Welsch, Robust regression using iteratively reweighted least-squares. Commun. Stat.-Theory Methods 6(9), 813–827 (1977)

    Article  Google Scholar 

  • J.R. Hurley, R.B. Cattell, The procrustes program: producing direct rotation to test a hypothesized factor structure. Behav. Sci. 7(2), 258–262 (1962)

    Article  Google Scholar 

  • D.G. Kendall, Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)

    Article  Google Scholar 

  • K. Kenobi, I.L. Dryden, Bayesian matching of unlabeled point sets using procrustes and configuration models. Bayesian Anal. 7(3), 547–566 (2012)

    Article  Google Scholar 

  • J.T. Kent, The complex bingham distribution and shape analysis. J. R. Stat. Society. Ser. B (Methodol.) 56(2), 285–299 (1994)

    Google Scholar 

  • M.A. Koschat, D.F. Swayne, A weighted procrustes criterion. Psychometrika 56(2), 229–239 (1991)

    Article  Google Scholar 

  • S.P. Langron, A.J. Collins, Perturbation theory for generalized procrustes analysis. J. R. Stat. Society. Ser. B (Methodol.), pp. 277–284 (1985)

    Google Scholar 

  • R. Larsen, Functional 2d procrustes shape analysis, in Scandinavian Conference on Image Analysis (Springer, 2005), pp. 205–213

    Google Scholar 

  • R.W. Lissitz, P.H. Schönemann, J.C. Lingoes, A solution to the weighted procrustes problem in which the transformation is in agreement with the loss function. Psychometrika 41(4), 547–550 (1976)

    Article  Google Scholar 

  • F.T. Luk, Oblique procrustes rotations in factor analysis. SIAM J. Sci. Stat. Comput. 5(4), 764–770 (1984)

    Article  Google Scholar 

  • K.V. Mardia, R. Edwards, M.L. Puri, Analysis of central place theory. Bull. Int. Stat. Inst. 47(2), 93–110 (1977)

    Google Scholar 

  • E. Maset, F. Crosilla, A. Fusiello, Errors-in-variables anisotropic extended orthogonal procrustes analysis. IEEE Geosci. Remote. Sens. Lett. 14(1), 57–61 (2017)

    Article  Google Scholar 

  • C.I. Mosier, Determining a simple structure when loadings for certain tests are known. Psychometrika 4(2), 149–162 (1939)

    Article  Google Scholar 

  • H. Park, A parallel algorithm for the unbalanced orthogonal procrustes problem. Parallel Comput. 17(8), 913–923 (1991)

    Article  Google Scholar 

  • E.M. Phelps, A critique of the principle of the horizontal plane of the skull. Am. J. Phys. Anthropol. 17(1), 71–98 (1932)

    Article  Google Scholar 

  • P. Schönemann, A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1), 1–10 (1966)

    Article  Google Scholar 

  • P. Schönemann, R. Carroll, Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika 35(2), 245–255 (1970)

    Article  Google Scholar 

  • R. Sibson, Studies in the robustness of multidimensional scaling: perturbational analysis of classical scaling. J. R. Stat. Society. Ser. B (Methodol.), 217–229 (1979)

    Google Scholar 

  • R. Sibson, Studies in the robustness of multidimensional scaling: procrustes statistics. J. R. Stat. Society. Ser. B (Methodol.), 234–238 (1978)

    Google Scholar 

  • C.G. Small, The Statistical Theory of Shape. Springer Science & Business Media (2012)

    Google Scholar 

  • J.M.F. ten Berge, Orthogonal procrustes rotation for two or more matrices. Psychometrika 42(2), 267–276 (1977)

    Article  Google Scholar 

  • J.M.F. ten Berge, The rigid orthogonal procrustes rotation problem. Psychometrika 71(1), 201–205 (2006)

    Article  Google Scholar 

  • J.M.F. ten Berge, H.A.L. Kiers, J.J.F. Commandeur, Orthogonal procrustes rotation for matrices with missing values. Br. J. Math. Stat. Psychol. 46(1), 119–134 (1993)

    Article  Google Scholar 

  • J.M.F. ten Berge, D.L. Knol, Orthogonal rotations to maximal agreement for two or more matrices of different column orders. Psychometrika 49(1), 49–55 (1984)

    Article  Google Scholar 

  • R. Toldo, A. Beinat, F. Crosilla, Global registration of multiple point clouds embedding the generalized procrustes analysis into an ICP framework, in International Symposium on 3D Data Processing, Visualization and Transmission, pp. 109–122 (2010)

    Google Scholar 

  • G. Wahba, A Least Squares Estimate of Satellite Attitude. SIAM Review 7(3), (1965)

    Article  Google Scholar 

  • C. Wang, S. Mahadevan, Manifold alignment using procrustes analysis, in Proceedings of the 25th International Conference on Machine Learning (ACM, 2008), pp. 1120–1127

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fabio Crosilla .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 CISM International Centre for Mechanical Sciences

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Crosilla, F., Beinat, A., Fusiello, A., Maset, E., Visintini, D. (2019). Orthogonal Procrustes Analysis. In: Advanced Procrustes Analysis Models in Photogrammetric Computer Vision. CISM International Centre for Mechanical Sciences, vol 590. Springer, Cham. https://doi.org/10.1007/978-3-030-11760-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-11760-3_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-11759-7

  • Online ISBN: 978-3-030-11760-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics