Euclidean motion group representations and the singular value decomposition of the Radon transform

Abstract

The matrix elements of the unitary irreducible representations of the Euclidean motion group are closely related to the Bessel and Gegenbauer functions. These special functions also arise in the singular functions of the singular value decomposition (SVD) of the Radon transform. In this paper, our objective is to study the Radon transform using the harmonic analysis over the Euclidean motion group and explain the origin of the special functions present in the SVD of the Radon transform from the perspective of group representation theory. Starting with a convolution representation of the Radon transform over the Euclidean motion group, we derive a method of inversion for the Radon transform using the harmonic analysis over the Euclidean motion group. We show that this inversion formula leads to an alternative derivation of the SVD of the Radon transform. This derivation reveals the origin of the special functions present in the SVD of the Radon transform. The derivation of the SVD of the Radon transform is a special case of a general result developed in this paper. This result shows that an integral transform with a convolution kernel is separable if the matrix elements of the irreducible unitary representations of the underlying group are separable.