Procrustes Statistics to Test for Significant OLS Similarity Transformation Parameters
Ordinary least squares (OLS) similarity transformations are often used in practice to perform datum conversions in a wide spectrum of geodetic and photogrammetric problems.
Traditionally, the estimation of the unknown parameters is computed by solving a non linear transformation model, build on a set of correspondence point coordinates known in both reference systems.
In the past many efforts have been made to provide a direct OLS solution to the parameter estimation problem. Among the solutions investigated (e.g. Sansò 1973), today very promising results are expected from some algorithms of the Procrustes Analysis (e.g. Crosilla 1999; Grafarend and Awange 2000, 2002).
These methods furnish the mathematical tools to perform a direct similarity transformation parameters estimation between two datums (Ordinary Procrustes Analysis), and among n 2 different datums by an iterative sequence of simultaneous direct solutions (Generalised Procrustes analysis).
In the paper we present a method, inspired to the perturbation theory of the Procrustes Analysis proposed by Langron and Collins (1985), aimed to evaluate the level of significance of the different parameters (translation, rotation and global dilatation) computed by the solution of a classical OLS similarity transformation problem. The procedure has been developed for both the Ordinary Procrustes problem and for the Generalised Procrustes problem.
KeywordsProcrustes statistics Procrustes analysis OLS similarity transformation
Unable to display preview. Download preview PDF.
- Beinat A and Crosilla F (2001) Generalised Procrustes Analysis for Size and Shape 3—D Object Reconstructions. In: Optical 3—D Measurement Techniques. Gruen and Kahmen ( Eds ), Wien, pp. 345–353.Google Scholar
- Borg I and Groenen P (1997) Modern Multidimensional Scaling. Theory and Applications. Springer-Verlag, New York.Google Scholar
- Commandeur JJF (1991) Matching configurations. DSWO Press, Leiden University, III, M & T series; 19.Google Scholar
- Crosilla F (1999) Procrustes Analysis and Geodetic Sciences. In: Quo vadis geodesia, Krumm and Schwarze (Eds), Festschrift to EW Grafarend on the occasion of his 60th bithday, pp 69–78.Google Scholar
- Crosilla F and Beinat A (2002) Use of Generalised Procrustes analysis for photogrammetric block adjustment by independent models. ISPRS Journal of Photogrammetry and Remote Sensing,56(3), pp. 195209.Google Scholar
- Goodall C (1991) Procrustes methods in the statistical analysis of shape. Journal Royal Stat. Soc., Part B 53, 2, pp 285–339.Google Scholar
- Grafarend EW and Awange JL (2000) Determination of vertical deflections by GPS/LPS measurements. Zeitschrift für Vermessungswesen, 125, pp 279–288.Google Scholar
- Grafarend EW and Awange JL (2002) Nonlinear analysis of the threedimensional datum transformation (conformal group C7(3)), Journal of Geodesy, in printGoogle Scholar
- Horn BKP (1987) Closed Form Solutions of Absolute Orientation Using Unit Quaternions. Journal of Optical Society of America, A-4(4), pp 629–642.Google Scholar
- Horn BKP, Hilden HM and Negandaripour S (1988) Closed Form Solution of Absolute Orientation Using Orthonormal Matrices. Journal of Optical Society of America, A-5(7), pp 1127–1135.Google Scholar
- Langron SP and Collins AJ (1985) Perturbation theory for Generalized Procrustes Analysis. Journal Royal Statistical Society, 47 (2), pp 277–284.Google Scholar
- Kristof W and Wingersky B (1971) Generalization of the orthogonal Procrustes rotation procedure to more than two matrices. Proc. of the 79-th Annual Cony. of the American Psychological Ass., 6, pp 81–90Google Scholar
- Schönemann PH (1966) A generalized solution of the orthogonal Procrustes problem. Psychometrika, 31 (1).Google Scholar
- Sibson R (1978) Studies in the Robustness of Multidimensional Scaling: Procrustes Statistics. J. Royal Statistical Society, 40 (2), pp 234–238.Google Scholar
- Sibson R (1979) Studies in the Robustness of Multidimensional Scaling: perturbational analysis of classical scaling. J. Royal Statistical Society, 41 (2), pp 217–229.Google Scholar
- Strang G and Borre K (1997) Linear Algebra, Geodesy, and GPS,Wellesley-Cambridge Press.Google Scholar
- Thompson EH (1958) On exact linear solution of the problem of absolute orientation. Photogrammetria, 13 (4), pp 163–178.Google Scholar