Fourier Biometrics: Harmonic Amplitudes as Multivariate Shape Descriptors

Younker, J. L. and R. Ehrlich (Department of Geology, Indiana-Purdue University, Indianapolis, Indiana 46202) 1977. Fourier biometrics: Harmonic amplitudes as multivariate shape descriptors. Syst. Zool. 26:336-342. -Fourier analysis in closed form provides a highly efficient method for measuring overall morphological similarity and identifying specific types of morphological variation. In shape analysis of ostracode outlines, continuous variation representing phylogenetically induced changes as well as shape families reflecting taxonomic categories can be recognized. Two-dimensional shape is described by a series of terms in a Fourier expansion of periphery radius as a function of polar angle for members of the ostracode genus Rabilimis and for six genera of Hemicytheridae. Raw data, consisting of Cartesian coordinates of the two-dimensional outline are converted to polar coordinates using the center of gravity of the form as a reference point. Coefficients of each term in the expansion represent the contribution of a fundamental harmonic form to the total shape description. When morphological variability is partitioned across the harmonic amplitude spectrum so that individual coefficients represent separate sources of morphological variation, removal of specific harmonics allows comparison of variation resulting from genetic vs. nongenetic factors. [Morphological analysis; Fourier biometrics. ] Taxonomic criteria used in the classification of organisms are often the result of artificial dissection of overall form into characters that can be quantitatively or qualitatively expressed. Variation in morphology results from both genetic and nongenetic factors, with the strength of the nongenetic factors determining the ease of recognition of true taxonomic differences between taxa. The use of multivariate techniques such as principal components permits isolation of underlying axes of variation, but clearly the results obtained are only as good as the parameters chosen to represent morphological variation. Thompson (1942) suggested that morphological variation could be described by the use of radial coordinates and polar equations, expressing changes in growth directions and rates of growth by altering the coefficients of polar trigonometric equations. Sneath (1967) attempted to look at variation in overall form by means of trend surface analysis, basically an ex1 Present address: Department of Geological Sciences, Box 4348, University of Illinois-Chicago Circle, Chicago, Illinois 60680. tension of coordinate transformation as proposed by Thompson. The closed-form Fourier shape analysis technique, described by Ehrlich and Weinberg (1970), is a method capable of greatly increasing the efficacy of form analysis. Their initial application involved shape analysis of sand grains as a means for discrimination of distinct populations, and for monitoring gradual changes in grain shape. Applied to biological form, this method permits the study of morphological variation in the manner envisioned by Thompson. Closed-form Fourier shape analysis treats the total two-dimensional cross sectional view, producing a series of orthogonal terms which describe the outline as precisely as required. Each term represents the contribution of a known shape component (fundamental harmonic order) to the overall shape description. Individual terms can be used as multivariate data for taxonomic studies, and the overall shape spectrum provides a continuous variable which can be monitored temporally and spatially. Fourier analysis in Cartesian coordinates has been used in seismic data processing as a means whereby certain