Use of a double Fourier series for three-dimensional shape representation

Abstract

The representation of three-dimensional star-shaped objects by the double Fourier series (DFS) coefficients of their boundary function is considered. An analogue of the convolution theorem for a DFS on a sphere is developed. It is then used to calculate the moments of an object directly from the DFS coefficients, without an intermediate reconstruction step. The complexity of computing the moments from the DFS coefficients is O(N 2 log N), where N is the maximum order of coefficients retained in the expansion, while the complexity of computing the moments from the spherical harmonic representation is O(N 2 log 2 N). It is shown that under sufficient conditions, the moments and surface area corresponding to the truncated DFS converge to the true moments and area of an object. A new kind of DFS—the double Fourier sine series—is proposed which has better convergence properties than the previously used kinds and spherical harmonics in the case of objects with a sharp point above the pole of the spherical domain.