Novel Resolution Analysis for the Radon Transform in \( R^2\) for Functions with Rough Edges

Let [Formula: see text] be a compactly supported, piecewise [Formula: see text] function in [Formula: see text] with a jump across a sufficiently smooth, non-self-intersecting curve [Formula: see text]. Consider a family of modified functions [Formula: see text] so that [Formula: see text] has a jump across a curve [Formula: see text]. Each [Formula: see text] is an [Formula: see text]-size perturbation of [Formula: see text], which scales like [Formula: see text] along [Formula: see text]. The functions [Formula: see text] are obtained by extending continuously the smooth components of [Formula: see text] on either side of [Formula: see text] all the way to [Formula: see text] so that the location of the jump shifts from [Formula: see text] to [Formula: see text]. By linearity of the Radon transform and its inversion formula, we can consider only the perturbation [Formula: see text]. Let [Formula: see text] be the reconstruction of [Formula: see text] from its discrete Radon transform data using a filtered backprojection inversion formula, where [Formula: see text] is the data sampling rate. A simple asymptotic (as [Formula: see text]) formula to approximate [Formula: see text] in any [Formula: see text]-size neighborhood of [Formula: see text] was derived heuristically in an earlier paper of the author. Numerical experiments revealed that the formula is highly accurate even for nonsmooth (i.e., only Hölder continuous) [Formula: see text]. In this paper we provide a full proof of this result, which says that the magnitude of the error between [Formula: see text] and its easily and explicitly computable approximation is [Formula: see text]. The main assumption is that the level sets of the function [Formula: see text], which parametrizes the perturbation [Formula: see text], are not too dense.