An explicit spectral decomposition of the ADRT

TL;DR

This paper provides an explicit spectral decomposition of the approximate discrete Radon transform (ADRT), enabling a formal inversion with efficient complexity and demonstrating competitive accuracy compared to iterative methods.

Contribution

It introduces a novel spectral factorization of the ADRT, leading to an explicit inversion formula with reduced computational complexity.

Findings

Inverse transform accuracy is competitive with iterative algorithms.

Explicit spectral decomposition is derived for each ADRT factor.

Inversion complexity is reduced to O(N^2 log^2 N) for N x N images.

Abstract

The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit spectral decomposition of each factor. We further show that this implies -- for data lying in the range of the ADRT -- that the transform of an $N \times N$ image can be formally inverted with complexity $\mathcal{O}(N^2 \log^2 N)$. We numerically test the accuracy of the inverse on images of moderate size and find that it is competitive with existing iterative algorithms in this special regime.