A Survey of the Statistical Theory of Shape

1989/05/01 by David G. Kendall · 8 citations
Mathematics · Environmental Science · Computer Science · #Morphological variations and asymmetry #Soil Geostatistics and Mapping #Geochemistry and Geologic Mapping

paper · pdf · doi:10.1214/ss/1177012582

openalex publication_date 1989/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

This is a review of the current state of the "theory of shape" introduced by the author in 1977. It starts with a definition of "shape" for a set of k points in m dimensions. The first task is to identify the shape spaces in which such objects naturally live, and then to examine the probability structures induced on such a shape space by corresponding structures in R^m. Against this theoretical background one formulates and solves statistical problems concerned with shape characteristics of empirical sets of points. Some applications (briefly sketched here) are to archeology, astronomy, geography and physical chemistry. We also outline more recent work on "size-and-shape," on shapes of sets of points in riemannian spaces, and on shape-theoretic aspects of random Delaunay tessellations.

Citations

• Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces
• The fundamental equations of a submersion.

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• Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements
• Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging
• Point cloud registration: matching a maximal common subset on pointclouds with noise (with 2D implementation)
• Hypoelliptic Diffusion Maps I: Tangent Bundles
• Ley line [wikipedia]