On Artifacts in Limited Data Spherical Radon Transform: Curved Observation Surface

TL;DR

This paper analyzes artifacts in limited data spherical Radon transform with curved observation surfaces, showing how artifacts are smoother than original singularities and introducing new proof techniques for 3D cases.

Contribution

It extends previous work to curved observation surfaces, characterizes artifact strength, and introduces a novel lifting method for 3D analysis.

Findings

Artifacts are $k$ orders smoother than original singularities in 2D.

If singularities are conormal, artifacts are $k+0.5$ orders smoother.

New proof technique involving space lifting for 3D case.

Abstract

In this article, we consider the limited data problem for spherical mean transform. We characterize the generation and strength of the artifacts in a reconstruction formula. In contrast to the third's author work [Ngu15b], the observation surface considered in this article is not flat. Our results are comparable to those obtained in [Ngu15b] for flat observation surface. For the two dimensional problem, we show that the artifacts are $k$ orders smoother than the original singularities, where $k$ is vanishing order of the smoothing function. Moreover, if the original singularity is conormal, then the artifacts are $k+\frac{1}{2}$ order smoother than the original singularity. We provide some numerical examples and discuss how the smoothing effects the artifacts visually. For three dimensional case, although the result is similar to that [Ngu15b], the proof is significantly different. We introduce a new idea of lifting the space.