Algorithms for Fast Convolutions on Motion Groups

AbstractIn this paper we apply techniques from noncommutative harmonic analysis to the development of fast algorithms for the computation of convolution integrals on motion groups. In particular, we focus on the group of rigid-body motions in 3-space, which is denoted here as SE(3). The general theory of irreducible unitary representations (IURs) of the 3D motion group is described briefly. Using IURs in operator form, we write the Fourier transform of functions on the motion group as an integral over the product space SE(3)×S2. The integral form of the Fourier transform matrix elements allows us to apply fast Fourier transform (FFT) methods developed previously for R3, S2, and SO(3) to speed up considerably the computation of convolutions of functions on SE(3). Such convolutions have been shown to play an important role in a number of engineering disciplines. An algorithm for the fast computation of the Fourier transform is given and its complexity is discussed. The Fourier transform for the 3D “discrete motion group” (semi-direct product of the icosahedral group with the translation group) is also developed and the computational complexity is discussed