Explicit Oblique Inverses of the Approximate Discrete Radon Transform

Abstract

The approximate discrete Radon transform (ADRT) is a fast, hierarchical approximation of the Radon transform whose levels have explicit spectral decompositions, but its Moore-Penrose inverse, needed for data outside the range, has no closed form. We construct explicit left inverses of the ADRT, oblique projections onto its range, that are exact for every choice of their parameters and amplify noise by a constant factor relative to the Moore-Penrose inverse. Realized with polynomial filters on the butterfly structure of the levels, they are fixed linear maps of cost $O(N^{5/2}\log^{3/2}N/σ_{\min})$ for an $N\times N$ image, where $σ_{\min}$ is the smallest singular value of the ADRT; with the measured growth $σ_{\min}\sim N^{1/4}$ this is $O(N^{2.25}\log^{3/2}N)$. A hybrid that keeps the explicit inverse of the cross-quadrant level is cheaper and closer to the Moore-Penrose inverse. In compiled implementations both outperform a polynomial solver of the normal equations from $N=256$ on, and their conditioning is traced to the aliasing of integer-sampled lines.