Resolution of 2D reconstruction of functions with nonsmooth edges from discrete Radon transform data

Abstract

Let $f$ be an unknown function in $\mathbb R^2$, and $f_\epsilon$ be its reconstruction from discrete Radon transform data, where $\epsilon$ is the data sampling rate. We study the resolution of reconstruction when $f$ has a jump discontinuity along a nonsmooth curve $\mathcal S_\epsilon$. The assumptions are that (a) $\mathcal S_\epsilon$ is an $O(\epsilon)$-size perturbation of a smooth curve $\mathcal S$, and (b) $\mathcal S_\epsilon$ is Holder continuous with some exponent $\gamma\in(0,1]$. We compute the Discrete Transition Behavior (or, DTB) defined as the limit $\text{DTB}(\check x):=\lim_{\epsilon\to0}f_\epsilon(x_0+\epsilon\check x)$, where $x_0$ is generic. We illustrate the DTB by two sets of numerical experiments. In the first set, the perturbation is a smooth, rapidly oscillating sinusoid, and in the second - a fractal curve. The experiments reveal that the match between the DTB and reconstruction is worse as $\mathcal S_\epsilon$ gets more rough. This is in agreement with the proof of the DTB, which suggests that the rate of convergence to the limit is $O(\epsilon^{\gamma/2})$. We then propose a new DTB, which exhibits an excellent agreement with reconstructions. Investigation of this phenomenon requires computing the rate of convergence for the new DTB. This, in turn, requires completely new approaches. We obtain a partial result along these lines and formulate a conjecture that the rate of convergence of the new DTB is $O(\epsilon^{1/2}\ln(1/\epsilon))$.