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Shape distributions for landmark data

Published online by Cambridge University Press:  01 July 2016

K. V. Mardia  and
I. L. Dryden
K. V. Mardia*
Affiliation:
University of Leeds
I. L. Dryden*
Affiliation:
University of Leeds
*
Postal address for both authors: Department of Statistics, University of Leeds, Leeds LS2 9JT, UK.
Postal address for both authors: Department of Statistics, University of Leeds, Leeds LS2 9JT, UK.
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Abstract

The paper obtains the exact distribution of Bookstein's shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendall's uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Bookstein's result for three landmarks. The adequacy of these approximations is also studied.

Keywords

DISTRIBUTION ON TRIANGLESDISTRIBUTION ON QUADRILATERALSIMAGE ANALYSISINVARIANT SHAPE MEASURESLANDMARKSSHAPE SPACE
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 4 , December 1989 , pp. 742 - 755
Copyright
Copyright © Applied Probability Trust 1989 

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References

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Applied Mathematics Series 55, National Bureau of Standards, United States.Google Scholar
Bookstein, F. L. (1986) Size and shape spaces for landmark data in two dimensions. Statist. Sci. 1, 181–242.Google Scholar
Johnson, D. R., O'Higgins, P., Mcandrew, T. J., Adams, L. M. and Flinn, R. M. (1985) Measurement of biological shape: a general method applied to mouse vertebrae. J. Embryol. exp. Morph. 90, 363–377.Google Scholar
Johnson, N. L. and Kotz, S. (1972) Distributions in Statistics Vol. 4: Continuous Multivariate Distributions. Wiley, New York.Google Scholar
Jones, M. C. (1987) On moments of ratios of quadratic forms in normal variables. Statist. Prob. Lett. 6, 129–136 (Corrections 6, 369).CrossRefGoogle Scholar
Kendall, D. G. (1984) Shape manifolds, procrustean metrics and complex projective spaces. Bull. London. Math. Soc. 16, 81–121.Google Scholar
Kendall, D. G. (1986) In discussion to Bookstein (1986).CrossRefGoogle Scholar
Mäkeläinen, T., Schmidt, K. and Styan, G. P. H. (1981) On the existence and uniqueness of the maximum likelihood estimate of a vector-valued parameter in fixed-size samples. Ann. Statist. 9, 758–767.Google Scholar
Mardia, K. V. (1980) In discussion to ‘Simulating the ley hunter’ by S. Broadbent. J. R. Statist. Soc. A 143, 137.Google Scholar
Mardia, K. V. (1981) Recent directional distributions with applications. Statistical Distributions in Scientific Work, ed. Taillie, C., Patil, G. P. and Baldessari, B., 6, 1–19, Reidel, Dordrecht.Google Scholar
Mardia, K. V. (1987) In discussion to ‘Statistical models of chemical kinetics in liquids’ by P. Clifford, N. J. B. Green and M. J. Pilling. J. R. Statist. Soc. B 49, 290–291.Google Scholar
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979) Multivariate Analysis. Academic Press, London.Google Scholar