Close the conditioning gap at the digital-line ghost frequencies

Establish a lower bound of the correct order for the smallest singular value of the full approximate discrete Radon transform in the frequency regime near ω=2π/3, where integer-sampling aliasing creates the ghost modes and current continuum and per-quadrant estimates do not control the joint four-quadrant operator.

Background

The paper identifies a gap between two available estimates. Continuum comparisons provide a lower bound on the lower third of the frequency spectrum, while the single-quadrant pole estimate provides only an O(1) floor at frequencies at least 2π/3.

The poorly conditioned global modes are concentrated near ω=2π/3 and arise from aliasing in integer-sampled lines. Controlling the Gram matrices on the relevant low-horizontal-frequency subspaces, rather than using only their smallest per-frequency eigenvalues, is necessary to obtain a sharp global lower bound.

References

A rigorous a priori bound on $D$ that does not inflate the cost is equivalent to a lower bound on $\sigma_{\min}$ of the ADRT of the right order; the continuum comparison gives it on the lower third of the spectrum, and the pole bound $\sigma_{\min}(A_\omega)\ge(2\sin\frac\omega4)n\ge1$ for $|\omega|\ge2\pi/3$ gives an $O(1)$ floor per quadrant at the ghost frequencies, which is too weak by $\sigma_{\min}(Rn)2$. Closing this gap is the open problem.

— Explicit Oblique Inverses of the Approximate Discrete Radon Transform  (2610.23363 - Rim, 16 Oct 2026) in Section 6, subsection “Summary”