In this paper, we investigate a method of representing convex regions of the plane based on the sinusoidal transform and its associated Fourier descriptors. We analyze the derivatives of the sinusoidal transform and establish an inverse transform. We obtain a characterization of the set of periodic functions of one variable which are the sinusoidal transforms of 'well-behaved' convex regions of the plane. We derive negative results concerning the prospects for extending the sinusoidal representation scheme to non-convex regions. We show how various geometrical quantities and relations can be extracted from the sinusoidal transforms of regions, and hence from their sinusoidal Fourier descriptors.
"Shape representation using Fourier coefficients of the sinusoidal transform", Proc. SPIE 3168, Vision Geometry VI, (20 October 1997); doi: 10.1117/12.279658; https://doi.org/10.1117/12.279658