Some Aspects of Random Shapes

  • Herbert Ziezold


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.


Stereographical Projection Shape Space Hyperbolic Geometry Riemannian Structure Centered Configuration 
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  1. [1]
    F. L. Bookstein, “The Measurement of Biological Shape and Shape Change”, Lecture Notes in Biomathematics 24, Springer-Verlag, New York, 1978.Google Scholar
  2. [2]
    F. L. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, 1991.zbMATHGoogle Scholar
  3. [3]
    I. L. Dryden and K. V. Mardia, Statistical Shape Analysis, Wiley, Chichester, 1998.zbMATHGoogle Scholar
  4. [4]
    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.MathSciNetzbMATHGoogle Scholar
  5. [5]
    D. G. Kendall, “The diffusion of shape”, Advances in Applied Probability, 9, 1977, 428–430.CrossRefGoogle Scholar
  6. [6]
    D. G. Kendall, “Shape manifolds, Procrustean metrics and complex projective spaces”, Bulletin of the London Mathematical Society 16, 1984, 81–121.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    D. G. Kendall, D. Barden, T. K. Carne, H. Le, Shape and Shape Theory, Wiley, Chichester, 1999.zbMATHCrossRefGoogle Scholar
  8. [8]
    W. S. Kendall, “A diffusion model for Bookstein triangle shape”, Advances in Applied Probability 30, 1998, 317–334.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    J. T. Kent, “New Directions in Shape Analysis”, The Art of Statistical Science, Wiley, Chichester, 1992, 115–127.Google Scholar
  10. [10]
    H. Le, “On the consistency of Procrustean mean shapes”, Advances in Applied Probability 30, 1998, 53–63.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    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.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    C. G. Small, The Statistical Theory of Shape, Springer-Verlag, New York, 1996.zbMATHCrossRefGoogle Scholar
  13. [13]
    D. Stoyan and I. S. Molchanov, “Set-valued means of random particles”, Technical Report BS-R9511, CWI, Amsterdam, 1995.Google Scholar
  14. [14]
    D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields,Wiley, Chichester, 1994. (German edition: Akademie Verlag, Berlin 1992.)Google Scholar
  15. [15]
    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.Google Scholar
  16. [16]
    H. Ziezold, “Mean figures and mean shapes applied to biological figure and shape distributions in the plane”, Biometrical Journal 36, 1994, 491–510.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Herbert Ziezold
    • 1
  1. 1.Fachbereich Mathematik/InformatikUniversität KasselKasselGermany

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