Analysis of resolution of tomographic-type reconstruction from discrete data for a class of distributions

Abstract

Let $f(x)$, $x\in\mathbb R^2$, be a piecewise smooth function with a jump discontinuity across a smooth surface $\mathcal S$. Let $f_{\Lambda\epsilon}$ denote the Lambda tomography (LT) reconstruction of $f$ from its discrete Radon data $\hat f(\alpha_k,p_j)$. The sampling rate along each variable is $\sim\epsilon$. First, we compute the limit $f_0(\check x)=\lim_{\epsilon\to0}\epsilon f_{\Lambda\epsilon}(x_0+\epsilon\check x)$ for a generic $x_0\in\mathcal S$. Once the limiting function $f_0(\check x)$ is known (which we call the discrete transition behavior, or DTB for short), the resolution of reconstruction can be easily found. Next, we show that straight segments of $\mathcal S$ lead to non-local artifacts in $f_{\Lambda\epsilon}$, and that these artifacts are of the same strength as the useful singularities of $f_{\Lambda\epsilon}$. We also show that $f_{\Lambda\epsilon}(x)$ does not converge to its continuous analogue $f_\Lambda=(-\Delta)^{1/2}f$ as $\epsilon\to0$ even if $x\not\in\mathcal S$. Results of numerical experiments presented in the paper confirm these conclusions. We also consider a class of Fourier integral operators $\mathcal{B}$ with the same canonical relation as the classical Radon transform adjoint, and a class of distributions $g\in\mathcal{E}'(Z_n)$, $Z_n:=S^{n-1}\times\mathbb R$, and obtain easy to use formulas for the DTB when $\mathcal{B} g$ is computed from discrete data $g(\alpha_{\vec k},p_j)$. Exact and LT reconstructions are particlular cases of this more general theory.