Abstract
Given x l, ... , x k in R m, the shape of x = (x 1, ... , x k ) is the equivalence class of x modulo similarity transformations in R m. Several metrics on the shape spaces will be introduced. This gives the opportunity to work with mean shapes and to use multivariate statistics, e. g. multidimensional scaling, and nonparametric statistics, e. g. discriminance analysis, for data analysis. Some connections to differential geometry and diffusion processes are also given.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. L. Bookstein, “The Measurement of Biological Shape and Shape Change”, Lecture Notes in Biomathematics 24, Springer-Verlag, New York, 1978.
F. L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, 1991.
I. L. Dryden and K. V. Mardia, Statistical Shape Analysis, Wiley, Chichester, 1998.
C. R. Goodall, “Procrustes methods in the statistical analysis of shape (with discussion)”, Journal of the Royal Statistical Society, Series B, 53, 1991, 285–339.
D. G. Kendall, “The diffusion of shape”, Advances in Applied Probability, 9, 1977, 428–430.
D. G. Kendall, “Shape manifolds, Procrustean metrics and complex projective spaces”, Bulletin of the London Mathematical Society 16, 1984, 81–121.
D. G. Kendall, D. Barden, T. K. Carne, H. Le, Shape and Shape Theory, Wiley, Chichester, 1999.
W. S. Kendall, “A diffusion model for Bookstein triangle shape”, Advances in Applied Probability 30, 1998, 317–334.
J. T. Kent, “New Directions in Shape Analysis”, The Art of Statistical Science, Wiley, Chichester, 1992, 115–127.
H. Le, “On the consistency of Procrustean mean shapes”, Advances in Applied Probability 30, 1998, 53–63.
H. Le and D. G. Kendall, “The Riemannian structure of Euclidean shape spaces: a novel environment for statistics”, Annals of Statistics 21, 1993, 1225–1271.
C. G. Small, The Statistical Theory of Shape, Springer-Verlag, New York, 1996.
D. Stoyan and I. S. Molchanov, “Set-valued means of random particles”, Technical Report BS-R9511, CWI, Amsterdam, 1995.
D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields,Wiley, Chichester, 1994. (German edition: Akademie Verlag, Berlin 1992.)
H. Ziezold, “On expected figures and a strong law of large numbers for random elements in quasi-metric spaces”, Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, (Prague, 1974), Volumen A. Reidel, Dordrecht, 1977, 591–602.
H. Ziezold, “Mean figures and mean shapes applied to biological figure and shape distributions in the plane”, Biometrical Journal 36, 1994, 491–510.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media New York
About this chapter
Cite this chapter
Ziezold, H. (2000). Some Aspects of Random Shapes. In: Althöfer, I., et al. Numbers, Information and Complexity. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6048-4_42
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6048-4_42
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4967-7
Online ISBN: 978-1-4757-6048-4
eBook Packages: Springer Book Archive