Calculating shape descriptors from Fourier analysis: shape analysis of Asterionella (Heterokontophyta, Bacillariophyceae)

Abstract

In our study, shape descriptors were calculated for specimens from the diatom genus Asterionella, using the method of arc lengths and tangent angles in Fourier analsyis. Asterionella species are character-poor, populations being distinguished mostly by subtle shape differences. Fourier analysis has been used in the past as an aid in taxonomy for many organisms. Fourier coefficients are least-squares estimators of best-fit planar closed curves that provide a quantitative measure of shape and can be useful shape descriptors in character-poor organisms. We determined the relation between number of x,y coordinates and the number of Fourier coefficients used in shape analysis fo rAsterionella. In a worked example, using a single extracted specimen outline, the method of calculating Fourier coefficients is demonstrated; coefficients were calculated for 100, 140, and 200 coordinates, in order to determine how many coordinates are necessary for effective shape analysis. Then, for each in a size range of Asterionella specimens, Fourier coefficients were calculated, using the same number of coordinates for each. Reconstructed outlines were compared graphically with the original. Statistical measures of average difference, variance, standard deviation, and coefficient of variation were calculated between x's, y's and Euclidean distance for the original and reconstructed outlines. From this, using 100 x,y coordinates, the number of Fourier coefficients necssary to give the best-fit outline over a size range from 30 to 95 µm is 22. Although we used Asterionella as an example, the method may be applied to any diatom valve outline.