A Survey of the Statistical Theory of Shape

TLDR

This review surveys the theory of shape, introduced in 1977, and defines shape for a set of k points in m dimensions. The authors aim to identify natural shape spaces, analyze the probability structures induced on them by Euclidean space, and formulate and solve statistical problems about shape characteristics of empirical point sets. They outline recent developments in size‑and‑shape analysis, shape theory in Riemannian spaces, and shape‑theoretic aspects of random Delaunay tessellations. Applications of the theory are briefly illustrated in archaeology, astronomy, geography, and physical chemistry.

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 $\mathbf{R}^m$. 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.