STATISTICAL-MODELS FOR GEOMETRIC COMPONENTS OF SHAPE CHANGE

Data for studies of biological shape often consist of the locations of individually named points, landmarks considered to be homologous (to correspond biologically) from form to form. In 1917 D'Arcy Thompson introduced an elegant model of homology as deformation: the configuration of landmark locations for any one form is viewed as a finite sample from a smooth mapping representing its biological relationship to any other form of the data set. For data in two dimensions, multivariate statistical analysis of landmark locations may proceed unambiguously in terms of complex-valued shape coordinates (ε,v) = (C-A)/(B-A) for sets of landmark triangles ABC, These are the coordinates of one vertex/landmark after scaling so that the remaining two vertices are at (0,0) and (1,0). Expressed in this fashion, the biological interpretation of the statistical analysis as a homology mapping would appear to depend on the triangulation. This paper introduces an analysis of landmark data and homology mappings using a hierarchy of geometric components of shape difference or shape change. Each component is a smooth deformation taking the form of a bivariate polynomial in the shape coordinates and is estimated in a manner nearly invariant with respect to the choice of a triangulation. © 1990, Taylor & Francis Group, LLC. All rights reserved.