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

In this article, 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 only available 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 smoothened out to a finite order $k$ near the boundary. For the problem in the two dimensional space and $\Gamma$ 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 the three dimensional space and $\Gamma$ is a rectangle, we describe that 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. Meanwhile, the artifacts obtained by a rotation around an edge are $k$ orders smoother than the original singularity. For both two and three dimensional problems, the visible singularities are reconstructed with the correct order. We, therefore, successfully quantify the geometric results obtained recently by J. Frikel and T. Quinto.