Fast Convolutions on $\mathbb{Z}^2\backslash SE(2)$ via Radial Translational Dependence and Classical FFT

Abstract

Let $\mathbb{Z}^2\backslash SE(2)$ denote the right coset space of the subgroup consisting of translational isometries of the orthogonal lattice $\mathbb{Z}^2$ in the non-Abelian group of planar motions $SE(2)$. This paper develops a fast and accurate numerical scheme for approximation of functions on $\mathbb{Z}^2\backslash SE(2)$. We address finite Fourier series of functions on the right coset space $\mathbb{Z}^2\backslash SE(2)$ using finite Fourier coefficients. The convergence/error analysis of finite Fourier coefficients are investigated. Conditions are established for the finite Fourier coefficients to converge to the Fourier coefficients. The matrix forms of the finite transforms are discussed. The implementation of the discrete method to compute numerical approximation of $SE(2)$-convolutions with functions which are radial in translations are considered. The paper is concluded by discussing capability of the numerical scheme to develop fast algorithms for approximating multiple convolutions with functions which are radial in translations.