Method for analyzing complex two-dimensional forms: Elliptical Fourier functions

A generalized procedure, elliptical Fourier analysis, for accurately characterizing the shape of complex morphological forms of the type commonly encountered in the biological sciences, is described. Elliptical Fourier functions are derived as a parametric formulation from conventional Fourier analysis, i.e., as a pair of equations that are functions of a third variable. The use of elliptical Fourier functions circumvents three restrictions that have limited conventional Fourier analysis to certain classes of shapes. These restrictions are (1) equal divisions over the interval or period; (2) dependency on the coordinate system, i.e., conventional Fourier functions are not "coordinate free"; and (3) the presence of shapes with outlines that curve back on themselves, a common occurrence. These three limitations are effectively removed with the utilization of elliptical Fourier functions, facilitating the analysis of a much larger class of two-dimensional forms.