The representation of shapes by Fourier descriptors is a time-honored technique that has received relatively little attention lately. Nevertheless, it has its benefits and is suitable for describing a range of medical structures in two dimensions. Delineations in medical applications often consist of continuous outlines of structures, where no information of correspondence between samples exist. In this article, we discuss a Euclidean alignment method that works directly with the functional representation of Fourier descriptors, and that is optimal in a least-squares sense. With corresponding starting points, the alignment of one shape onto another consists of a single expression. If the starting points are arbitrary, we present a simple algorithm to bring a set of shapes into correspondence.
Results are given for three different data sets; 62 outlines of the corpus callosum brain structure, 61 outlines of the brain ventricles, and 50 outlines of the right lung. The results show that even though starting points, translations, rotations and scales have been randomized, the alignment succeeds in all cases.
As an application of the proposed method, we show how high-quality shape models represented by common landmarks can be constructed in an automatic fashion. If the aligned Fourier descriptors are inverse transformed from the frequency domain to the spatial domain, a set of roughly aligned landmarks are obtained. The positions of these are then adjusted along the contour of the objects using the minimum description length criterion, producing ample correspondences. Results on this are also presented for all three data sets.