An explicit spectral decomposition of the ADRT

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.