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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The predicate \({{\,\mathrm{sym}\,}}[]\) is true when the argument is a symmetric matrix.
- 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)
J.L. Awange, E.W. Grafarend, in Solving Algebraic Computational Problems in Geodesy and Geoinformatics: The Answer to Modern Challenges (Springer, 2005)
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
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
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)
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)
M. Bennani Dosse, J.M.F. ten Berge, Anisotropic orthogonal procrustes analysis. J. Cl.Ification, 27(1), 111–128 (2010)
F. Boas, The horizontal plane of the skull and the general problem of the comparison of variable forms. Science 21(544), 862–863 (1905)
F.L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology (Cambridge University Press, 1997)
F.L. Bookstein, Size and shape spaces for landmark data in two dimensions. Stat. Sci. 1(2), 181–222 (1986)
I. Borg, P. Groenen, Modern multidimensional scaling: theory and applications. J. Educ. Meas. 40(3), 277–280 (2003)
S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications (Wiley, 2013)
N. Cliff, Orthogonal rotation to congruence. Psychometrika 31(1), 33–42 (1966)
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)
J.J.F. Commandeur, Matching Configurations (DSWO Press, 1991)
F. Crosilla, Procrustean transformation as a tool for the construction of a criterion matrix for control networks. Manuscripta Geodetica 8, (1983b)
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)
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
F. Crosilla, A criterion matrix for the second order design of control networks. Bulletin géodésique 57(1–4), 226–239 (1983a)
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)
I.L. Dryden, K.V. Mardia, Statistical Shape Analysis (Wiley, New York, 1998)
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
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
C.R. Goodall, Procrustes methods in the statistical analysis of shape. J. R. Stat. Society. Ser. B (Methodol.) 53(2), 285–339 (1991)
J.C. Gower, Generalized procrustes analysis. Psychometrika 40(1), 33–51 (1975)
J.C. Gower, G.B. Dijksterhuis, Procrustes Problems (Oxford University Press, Oxford Statistical Science Series, 2004)
E.W. Grafarend, J.L. Awange, Determination of the vertical deflection by gps/lps measurements. Zeitschrift für Vermessungswesen 125(8), 279–288 (2000)
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)
B.F. Green, J.C. Gower, A Problem with Congruence, in Annual meeting of the psychometric society (Monterey, California, 1979)
B.F. Green, The orthogonal approximation of an oblique structure in factor analysis. Psychometrika 17(4), 429–440 (1952)
M. Gulliksson. The partial procrustes problem: a first look. Technical Report UMINF 95.11., (Department of Computing Science, Umeå University, Sweden, 1995)
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)
P.W. Holland, R.E. Welsch, Robust regression using iteratively reweighted least-squares. Commun. Stat.-Theory Methods 6(9), 813–827 (1977)
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)
D.G. Kendall, Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)
K. Kenobi, I.L. Dryden, Bayesian matching of unlabeled point sets using procrustes and configuration models. Bayesian Anal. 7(3), 547–566 (2012)
J.T. Kent, The complex bingham distribution and shape analysis. J. R. Stat. Society. Ser. B (Methodol.) 56(2), 285–299 (1994)
M.A. Koschat, D.F. Swayne, A weighted procrustes criterion. Psychometrika 56(2), 229–239 (1991)
S.P. Langron, A.J. Collins, Perturbation theory for generalized procrustes analysis. J. R. Stat. Society. Ser. B (Methodol.), pp. 277–284 (1985)
R. Larsen, Functional 2d procrustes shape analysis, in Scandinavian Conference on Image Analysis (Springer, 2005), pp. 205–213
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)
F.T. Luk, Oblique procrustes rotations in factor analysis. SIAM J. Sci. Stat. Comput. 5(4), 764–770 (1984)
K.V. Mardia, R. Edwards, M.L. Puri, Analysis of central place theory. Bull. Int. Stat. Inst. 47(2), 93–110 (1977)
E. Maset, F. Crosilla, A. Fusiello, Errors-in-variables anisotropic extended orthogonal procrustes analysis. IEEE Geosci. Remote. Sens. Lett. 14(1), 57–61 (2017)
C.I. Mosier, Determining a simple structure when loadings for certain tests are known. Psychometrika 4(2), 149–162 (1939)
H. Park, A parallel algorithm for the unbalanced orthogonal procrustes problem. Parallel Comput. 17(8), 913–923 (1991)
E.M. Phelps, A critique of the principle of the horizontal plane of the skull. Am. J. Phys. Anthropol. 17(1), 71–98 (1932)
P. Schönemann, A generalized solution of the orthogonal procrustes problem. Psychometrika 31(1), 1–10 (1966)
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)
R. Sibson, Studies in the robustness of multidimensional scaling: perturbational analysis of classical scaling. J. R. Stat. Society. Ser. B (Methodol.), 217–229 (1979)
R. Sibson, Studies in the robustness of multidimensional scaling: procrustes statistics. J. R. Stat. Society. Ser. B (Methodol.), 234–238 (1978)
C.G. Small, The Statistical Theory of Shape. Springer Science & Business Media (2012)
J.M.F. ten Berge, Orthogonal procrustes rotation for two or more matrices. Psychometrika 42(2), 267–276 (1977)
J.M.F. ten Berge, The rigid orthogonal procrustes rotation problem. Psychometrika 71(1), 201–205 (2006)
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)
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)
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)
G. Wahba, A Least Squares Estimate of Satellite Attitude. SIAM Review 7(3), (1965)
C. Wang, S. Mahadevan, Manifold alignment using procrustes analysis, in Proceedings of the 25th International Conference on Machine Learning (ACM, 2008), pp. 1120–1127
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 CISM International Centre for Mechanical Sciences
About this chapter
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)