ELLIPTICAL DESCRIPTORS: SOME SIMPLIFIED MORPHOMETRIC PARAMETERS FOR THE QUANTIFICATION OF COMPLEX OUTLINES

Abstract

Among morphometric methods, a method called Elliptical Fourier Analysis, has been developed to decompose complex outlines into Fourier series. This elliptical Fourier formulation has been rarely applied to date, which is probably explainable because of the mathematical complexity and the difficulty of translating the Fourier coefficients into simple geometrical concepts. Utilizing elliptical analysis, a simplified geometrical approach to the Fourier decomposition is proposed in this study. We showed that the geometrical locus of the points associated with each harmonic used in the Fourier decomposition is an ellipse. The contribution of each harmonic was then characterized with four new geometrical parameters called elliptical descriptors. These are: the half-length of the major axis (LAj), the half-length of the minor axis (LBj), the orientation of the major axis, and the phase angle. These descriptors, in contrast to classical Fourier coefficients, possess geometrical significance, and allow for an estimate of each ellipse consisting of: (1) the size of the ellipse (proportional to the product LAB· LBj), (2) the anisotropy of the ellipse (characterized by the ratio LAB/LBj), and (3) the orientation of the ellipse given by the orientation of the elliptical axes. These parameters completely define the geometry of the ellipse associated with each harmonic, and provide an evaluation of the importance of the harmonic contribution in the description of the form studied. Using these elliptical descriptors, an outline can be described, as well as reconstructed. A methodology is then proposed to characterize and to compare complex outlines using these elliptical descriptors. This new methodology allows the quantification of any form, regardless of their degree of complexity, and allows the translation of the morphological differences into simple geometrical concepts, a procedure difficult to carry out with conventional Fourier coefficients.