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

Abstract

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.

Sexual dimorphism using elliptical Fourier analysis: shape differences in the craniofacial complex

“…The process involved two steps: (1) each image was rotated (positional orientation) so that the major axis of the first ellipse was made horizontal to the x-axis; and (2) the bounded area was scaled up or down (size-standardization) so that it was equal to the square root of 10000 units. This 00B-300AD n = 32 females, n = 32 males Kamakura area-standardization approach utilized here is different from the one advocated by Kuhl and Giardina (1982), which sets the length of the semi-major axis of the first ellipse to 1.0 as the size standardization (see Lestrel, 1989a, b for discussion of these normalizations).…”

“…The EFF equations were originally developed by Kuhl and Giardina (1982). Details and applications of EFFs can be found in Lestrel (1989aLestrel ( , b, 1997bLestrel ( , 2000.…”

Unsupervised model-based clustering for typological classification of Middle Bronze Age flanged axes

“…The greater the number of harmonics, the better the reconstruction of the original contour. These coefficients are commonly used as new variables to describe the shape (Kuhl and Giardina, 1982;Lestrel, 1989;Navarro et al, 2004). Normalisation of flanged axes was performed by the major axis of the first harmonic (Kuhl and Giardina, 1982;Rohlf and Archie, 1984;Furuta et al, 1995;Zhan and Wang, 2012), and coefficients were size-normalised, using the square root of the harmonic amplitudes.…”

Taxonomy and biostratigraphy of the Late AlbianActinoceramus sulcatuslineage (Early Cretaceous Bivalvia, Inoceramidae)

“…The outlines were automatically smoothed (12 times) in order to eliminate pixel "noise" and insignificant outline details resulting from minor shell breakage. The numbers of harmonics that sufficiently de− scribe the outlines were evaluated from amplitude vs. har− monic number plots according to Lestrel (1989). These plots indicated that at least seven Fourier harmonics were necessary for the analysis of Arcomytilus, eight for Iso− gnomon, and nine for Eomiodon.…”