An Inversion Method for the Exponential Radon Transform Based on the Harmonic Analysis of the Euclidean Motion Group

This paper presents a new method for exponential Radon transform inversion based on the harmonic analysis of the Euclidean motion group of the plane. The proposed inversion method is based on the observation that the exponential Radon transform can be modified to obtain a new transform, defined as the modified exponential Radon transform, that can be expressed as a convolution on the Euclidean motion group. The convolution representation of the modified exponential Radon transform is block diagonalized in the Euclidean motion group Fourier domain. Further analysis of the block diagonal representation provides a class of relationships between the spherical harmonic decompositions of the Fourier transforms of the function and its exponential Radon transform. The block diagonal representation provides a method to simultaneously compute all these relationships. The proposed algorithm is implemented using the fast implementation of the Euclidean motion group Fourier transform and its performances is demonstrated in numerical simulations.