Elliptical Fourier descriptors for contours in three dimensions: a new tool for morphometrical analysis in biology

Elliptical Fourier descriptor analysis is a method for the morphometric study of curves. It has been used in the two-dimensional plane for closed contours, but rarely for lines in the three-dimensional space. The method consists of an expansion of a contour as a sum of ellipses. In this article, we study three-dimensional contours, i.e. lines embedded in the three-dimensional space. We compute for the first time the relations between the Fourier coefficients and its geometric parameters. We then use these relations for normalization and reorientation of three-dimensional contours. Such an algorithm can be used to perform inter-individual comparisons between contours, regardless of differences in viewpoint or global size. Human and small animal illustrative examples using biomedical X-ray CT imaging data of open bone structures demonstrate the interest and potential of the method for morphological analysis.