A Local Approach to Resolution Analysis of Image Reconstruction in Tomography

Abstract

We propose a novel approach to analyzing resolution of tomographic reconstruction. Instead of following a conventional approach to obtain a global accuracy estimate, we investigate how the reconstructed function $f_\epsilon$ approximates the singularities of the original object $f$. The data is a discretized 2D Radon transform of $f$. The object could be static or change with time (dynamic tomography). Suppose the step-sizes along the angular and affine variables are $O(\epsilon)$. We pick a point $\bar x_0$, where $f$ has a jump singularity, and obtain the leading singular behavior of $f_\epsilon$ in an $O(\epsilon)$-neighborhood of $\bar x_0$ as $\epsilon\to0$. It turns out that the limiting behavior of $f_\epsilon$ depends only on the data microlocally near the singularity being reconstructed. This significantly simplifies the analysis and allows us to investigate complicated settings, e.g., dynamic tomography. Also, our resolution analysis is algorithm-specific---the same approach can be used for analyzing and optimizing various linear reconstruction algorithms. We present the results of numerical experiments in the static and dynamic cases. These results demonstrate an excellent agreement between predicted and actual behaviors of $f_\epsilon$ near a jump discontinuity of $f$.