A method for the analysis and synthesis of closed curves in the plane is developed using the Fourier descriptors FD's of Cosgriff [1]. A curve is represented parametrically as a function of arc length by the accumulated change in direction of the curve since the starting point. This function is expanded in a Fourier series and the coefficients are arranged in the amplitude/phase-angle form. It is shown that the amplitudes are pure form invariants as well as are certain simple functions of phase angles. Rotational and axial symmetry are related directly to simple properties of the Fourier descriptors. An analysis of shape similarity or symmetry can be based on these relationships; also closed symmetric curves can be synthesized from almost arbitrary Fourier descriptors. It is established that the Fourier series expansion is optimal and unique with respect to obtaining coefficients insensitive to starting point. Several examples are provided to indicate the usefulness of Fourier descriptors as features for shape discrimination and a number of interesting symmetric curves are generated by computer and plotted out.

A family of graph-theoretical algorithms based on the minimal spanning tree are capable of detecting several kinds of cluster structure in arbitrary point sets; description of the detected clusters is possible in some cases by extensions of the method. Development of these clustering algorithms was based on examples from two-dimensional space because we wanted to copy the human perception of gestalts or point groupings. On the other hand, all the methods considered apply to higher dimensional spaces and even to general metric spaces. Advantages of these methods include determinacy, easy interpretation of the resulting clusters, conformity to gestalt principles of perceptual organization, and invariance of results under monotone transformations of interpoint distance. Brief discussion is made of the application of cluster detection to taxonomy and the selection of good feature spaces for pattern recognition. Detailed analyses of several planar cluster detection problems are illustrated by text and figures. The well-known Fisher iris data, in four-dimensional space, have been analyzed by these methods also. PL/1 programs to implement the minimal spanning tree methods have been fully debugged.