Fourier Descriptors for Plane Closed Curves

TLDR

Curves are parametrized by arc length through accumulated directional change from the starting point. The paper develops a method for analyzing and synthesizing planar closed curves using Fourier descriptors. The method expands the accumulated directional change function in a Fourier series, arranging coefficients in amplitude/phase‑angle form to analyze and synthesize closed curves. The study shows that Fourier descriptor amplitudes are form invariants linked to simple phase‑angle functions, enabling direct detection of rotational and axial symmetry, optimal and unique series expansion insensitive to starting point, and effective shape discrimination and synthesis of symmetric curves, as illustrated by several computer‑generated examples.

Abstract

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.