Theoretical and Distributional Aspects of Shape Analysis

The concept of general shape is introduced and particular cases for point sets in ℝk are considered. General shape spaces are orbit spaces and the principal groups of interest are the Euclidean similarity group and the isometry group. Distributional results for Gaussian configurations provide the main focus of the paper. Recent developments are reviewed, including using the QR decomposition, Bartlett’s decomposition and integration of the orthogonal group 0(K). Connections are made with existing work, especially with distributional results for the important K=2 dimensional case. Finally, possible extensions of all the work are considered in a discussion.

[1]  M. Bartlett.  XX.—On the Theory of Statistical Regression. , 1934 .

[2]  M. A. Girshick,et al.  Some Extensions of the Wishart Distribution , 1944 .

[3]  A. Kshirsagar.  Effect of non-centrality on the Bartlett decomposition of a Wishart matrix , 1962 .

[4]  A. James.  Distributions of Matrix Variates and Latent Roots Derived from Normal Samples , 1964 .

[5]  K. Mardia.  Statistics of Directional Data , 1972 .

[6]  K. Mardia,et al.  The von Mises–Fisher Matrix Distribution in Orientation Statistics , 1977 .

[7]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[8]  D. Kendall.  SHAPE MANIFOLDS, PROCRUSTEAN METRICS, AND COMPLEX PROJECTIVE SPACES , 1984 .

[9]  F. Bookstein.  Size and Shape Spaces for Landmark Data in Two Dimensions , 1986 .

[10]  D. Kendall,et al.  The structure and explicit determination of convex-polygonally generated shape-densities , 1987, Advances in Applied Probability.

[11]  W. Kendall.  SYMBOLIC COMPUTATION AND THE DIFFUSION OF SHAPES OF TRIADS , 1988 .

[12]  Christopher G. Small,et al.  Techniques of Shape Analysis on Sets of Points , 1988 .

[13]  Random spherical triangles II: Shape densities , 1989, Advances in Applied Probability.

[14]  K. Mardia,et al.  Shape distributions for landmark data , 1989, Advances in Applied Probability.

[15]  K. Mardia,et al.  The statistical analysis of shape data , 1989 .

[16]  Kanti V. Mardia,et al.  Shape Analysis of Triangles Through Directional Techniques , 1989 .

[17]  D. Kendall.  A Survey of the Statistical Theory of Shape , 1989 .

[18]  F. Bookstein,et al.  Statistical models for geometric components of shape change , 1990 .

[19]  T. K. Carne.  The Geometry of Shape Spaces , 1990 .

[20]  K. Mardia,et al.  General shape distributions in a plane , 1991, Advances in Applied Probability.

[21]  C. Goodall.  Procrustes methods in the statistical analysis of shape , 1991 .

[22]  K. Mardia,et al.  Size and shape analysis of landmark data , 1992 .