Resolution of 2 Dimensional Reconstruction of Functions with Nonsmooth Edges from Discrete Radon Transform Data

Let \(f_\epsilon\) , \(0\lt \epsilon \le \epsilon_0\) , be a family of functions in \(\mathbb R^2\) , and \(f_\epsilon ^{\text{rec}}\) be a reconstruction of \(f_\epsilon\) from its discrete Radon transform data. Here \(\epsilon\) is both the data sampling rate and the parameter of the family. We study the resolution of reconstruction when \(f_\epsilon\) has a jump discontinuity along a nonsmooth curve \(\mathcal S_\epsilon\) . The assumptions are that (a) \(\mathcal S_\epsilon\) is a family of \(O(\epsilon )\) -size perturbations of a smooth curve \(\mathcal S\) , and (b) \(\mathcal S_\epsilon\) is Hölder continuous with some exponent \(\gamma \in (0,1]\) . Thus the size of the perturbation \(\mathcal S\to \mathcal S_\epsilon\) is of the same order of magnitude as the data sampling rate. We compute the discrete transition behavior (DTB) defined as the limit \(\text{DTB}(\check x):=\lim_{\epsilon \to 0}f_\epsilon ^{\text{rec}}(x_0+\epsilon \check x)\) , 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 rougher. 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 \(O(\epsilon ^{1/2}\ln (1/\epsilon ))\) .