Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-08T14:45:16.233Z Has data issue: false hasContentIssue false
  • English
  • Français

Distributions of projective invariants and model-based machine vision

Part of: Sampling theory, sample surveys

Published online by Cambridge University Press:  01 July 2016

K. V. Mardia ,
Colin Goodall  and
Alistair Walder
K. V. Mardia*
Affiliation:
University of Leeds
Colin Goodall*
Affiliation:
Pennsylvania State University
Alistair Walder*
Affiliation:
University of Leeds
*
∗ Postal address: Department of Statistics, University of Leeds, Leeds LS2 9JT, UK.
∗∗ Postal address: Department of Statistics, Pennsylvania State University, University Park, PA 16802 USA.
∗ Postal address: Department of Statistics, University of Leeds, Leeds LS2 9JT, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

Keywords

MACHINE VISIONPROJECTIVE INVARIANTS

MSC classification

Primary: 62D05: Sampling theory, sample surveys

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 641 - 661
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ambartzumian, R. V. (1991) Factorization Calculus and Geometric Probabilities. Cambridge University Press, Cambridge.Google Scholar
Astrom, K. and Morin, L. (1992) Random cross ratios. Technical Report. RT 88 IMAG-14 Lifia.Google Scholar
Bookstein, F. L. (1986) Size and shape spaces for landmark data in two dimensions (with discussion). Statist. Sci. 1, 181–242.Google Scholar
Bookstein, F. L. (1991) Morphometric Tools for Landmark Data. Cambridge University Press, Cambridge.Google Scholar
Coelho, C., Heller, A., Mundy, J. L., Forsyth, D. A. and Zisserman, A. (1992) An experimental evaluation of projective invariants in geometric invariance in computer vision. In Geometric Invariance in Computer Vision. ed. Mundy, J. L. and Zisserman, A.. MIT Press, Boston, MA. pp. 87–104.Google Scholar
Farin, G. (1993) Curves and Surface for Computer Aided Geometric Design. Academic Press, London.Google Scholar
Goodall, C. R. (1991) Procrustes methods in the statistical analysis of shape (with discussion). J. R. Statist. Soc. B 53, 285–339.Google Scholar
Goodall, C. R. and Mardia, K. V. (1991) A geometrical derivation of the shape density. Adv. Appl. Prob. 23, 496–514.Google Scholar
Goodall, C. R. and Mardia, K. V. (1993) Multivariate aspects of shape theory. Ann. Statist. 21, 848–866.Google Scholar
Goodall, C. R. and Mardia, K. V. (1994) The statistical analysis of cross ratios with application to machine vision. To be submitted.Google Scholar
Hartshorne, R. (1977) Algebraic Geometry. Springer, Berlin.Google Scholar
Horadam, A. F. (1970) A Guide to Undergraduate Projective Geometry. Pergamon, Oxford.Google Scholar
Kendall, D. G. (1984) Shape manifolds, procrustean metrics and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121.CrossRefGoogle Scholar
Kendall, D. G. (1989) A survey of the statistical theory of shape (with discussion). Statist. Sci. 4, 87–120.Google Scholar
Kent, J. T. (1994) The complex Bingham distribution and shape analysis. J. R. Statist. Soc. B 56, 285–300.Google Scholar
Kent, J. T. and Mardia, K. V. (1994). Statistical shape methodology in image analysis. In Shape in Pictures ed. Ying-Lie, O.. Springer, Berlin. pp. 443–452.CrossRefGoogle Scholar
Mardia, K. V. (1970) The Families of Bivariate Distributions . Griffin, London.Google Scholar
Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, London.Google Scholar
Mardia, K. V. and Dryden, I. L. (1989a) Shape distributions for landmark data. Adv. Appl. Prob. 21, 742–755.Google Scholar
Mardia, K. V. and Dryden, I. L. (1989b) The statistical analysis of shape data. Biometrika 76, 271–81.Google Scholar
Mardia, K. V. and Hainsworth, T. J. (1993) Image warping and Bayesian reconstruction with grey level templates. In Statistics and Images, Vol. 1. ed. Mardia, K. V., and Kanji, G.. Carfax, Oxford. pp. 257–280.Google Scholar
Mardia, K. V., Rabe, S. and Kent, J. T. (1994) Statistics, shape and images. IMA Proc. on Complex Stochastic Systems and Engineering Applications. In press.Google Scholar
Maybank, S. J. (1993) Error tradeoff's for the cross-ratio in model based vision. Proc. Workshop in Computer Vision for Space Applications. Antibes, France.Google Scholar
Maybank, S. J. (1994) Classification based on the cross-ratio. In Application of Invariance in Computer Vision. ed. Mundy, J. L., Zisserman, A. and Forsyth, D.. Springer, Berlin. pp. 453–472.Google Scholar
Maybank, S. J. and Beardsley, P. A. (1994) Applications of invariants to model based vision. J. App. Statist. 21, 431–457.Google Scholar
Mundy, J. L. and Zisserman, A. (1992) Geometric Invariance in Computer Vision. MIT press, Boston, MA.Google Scholar
Stuart, A. and Ord, K. (1987) Kendall's Advanced Theory of Statistics , Vol. 1. Edward Arnold, London.Google Scholar