Reconstruction of two-dimensional patterns from Fourier descriptors

Two kinds of Fourier shape descriptors (FDs) are considered in this article: ZR (defined by Zahn and Roskies) and G (defined by Granlund). The first section of this article studies ZR descriptors. Four modifications of ZR descriptors are proposed. These new descriptors are based on the step signature, smoothed signature, linearized smoothed signature, and the curvature function, respectively. The amplitudes of FDs are shown to be invariant under rotations, translations, changes in size, mirror reflections, and shifts in the starting point. In all cases the reconstruction accuracy in terms of the number of FDs is studied, resulting in approximation error bounds. An efficient reconstruction formula based on the composite rectangle integration method is proposed. This method also provides polygonal approximation for arbitrary contours. In the second section of this article, the theoretical results are illustrated with examples and verified in numerical experiments involving digitized patterns. In the same experiments the performances of ZR and G descriptors are compared.