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.
KeywordsStereographical Projection Shape Space Hyperbolic Geometry Riemannian Structure Centered Configuration
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- F. L. Bookstein, “The Measurement of Biological Shape and Shape Change”, Lecture Notes in Biomathematics 24, Springer-Verlag, New York, 1978.Google Scholar
- J. T. Kent, “New Directions in Shape Analysis”, The Art of Statistical Science, Wiley, Chichester, 1992, 115–127.Google Scholar
- D. Stoyan and I. S. Molchanov, “Set-valued means of random particles”, Technical Report BS-R9511, CWI, Amsterdam, 1995.Google Scholar
- D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields,Wiley, Chichester, 1994. (German edition: Akademie Verlag, Berlin 1992.)Google Scholar
- 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