On Artifacts in Limited Data Spherical Radon Transform: Flat Observation Surfaces

In this paper, we characterize the strength of the reconstructed singularities and artifacts in a reconstruction formula for limited data spherical Radon transform. Namely, we assume that the data is available only on a closed subset $\Gamma$ of a hyperplane in $\mathbb{R}^n$ ($n=2,3$). We consider a reconstruction formula studied in some previous works, under the assumption that the data is only smoothed out to a finite order $k$ near the boundary. For the problem in two-dimensional space when $\Gamma$ is a line segment, the artifacts are generated by rotating a boundary singularity along a circle centered at an end point of $\Gamma$. We show that the artifacts are $k$ orders smoother than the original singularity. For the problem in three-dimensional space when $\Gamma$ is a rectangle, the artifacts are generated by rotating a boundary singularity around either a vertex or an edge of $\Gamma$. The artifacts obtained by a rotation around a vertex are $2k$ orders smoother than the original singularity. Me...