Series Expansions of the Reconstruction Kernel of the Radon Transform over a Cormack-Type Family of Curves with Applications in Tomography

Abstract

This paper is concerned with the Radon transform over a family of Cormack-type curves and provides an exact inversion formula. The studied family of curves, called $C_1$, appeared in previous works as a suitable manifold for modeling imaging concepts in conventional and Compton scattering tomography (CST). More specifically, the straight line, integral support of the classical Radon transform used in computed tomography (CT) belongs to $C_1$. In conventional tomography, many reconstruction techniques compute the derivative of the data with the aim of reducing the order of singularity of the reconstruction kernel associated here to the Radon transform in two dimensions. However, differentiating data requires a regularization step (for instance, convolution with a smooth function) which reduces the resolution of reconstructed images. Here, the proposed analytical inversion formula recovers the circular harmonic components of the sought object without differentiation of the data, which leads to an improvement of the final resolution. Furthermore, we deal with the singularity issue of the reconstruction kernel by applying a range property of the Radon transform. Since theoretical results are developed in a quite general context of inverse problems for Radon transforms over $C_1$, the potential applications of our algorithm appear to be numerous in the field of CT and CST. Numerical results in the framework of CT and of one modality on CST reveal the strength of this algorithm in terms of accuracy and stability in comparison with the well-known filtered back-projection.