From an Elliptical Fourier Representation of a Shape Boundary to a Curvature Representation with Applications Using the Cranial Outline from Archaic Homo Species

摘要

The elliptical Fourier representation provides a powerful way to mathematically represent a shape boundary, with the goodness-of-fit only limited by the number of points used to represent the boundary and the number of terms included in the elliptical Fourier representation. The elliptical Fourier representation, though, has indeterminacies based on: (1) the choice of a Cartesian coordinate system for measuring boundary point locations, (2) a potentially infinite number of parameters for the elliptical Fourier representation, and, (3) parameters lacking meaningful biological interpretation~3. Methods used to reduce the dimensionality of the parameter space such as Principal Component Analysis (PCA) are not satisfactory as there is no assurance that just the first few principal components are biologically meaningful. One may, however, resolve the coordinate problem, the dimensionality problem, and the lack of straight-forward biological interpretation by the transformation of the elliptical Fourier representation into a curvature and (in the case of 3-D) torque function that uses a single parameter, s, the arc length distance of a boundary point from a fixed reference point (which may be a single landmark, homologous across different shapes to be compared) and a functional representation based on the curvature and torque of the boundary in 3-D at distance s from the reference point. In this paper only 2-D outlines are evaluated. In 2-D, the functional representation is determined by just the curvature of the boundary at distance s from the reference point. Unlike the parameters in the elliptical Fourier representation, the curvature of the boundary is biologically meaningful and allows for the introduction of landmarks such as the location of the maximum curvature along the boundary. Examples of this procedure, using the internal cranial (endocranial) outline from a number of archaic Homo species, are discussed.