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

In this article, we characterize the strength of reconstructed singularities and the artifacts in a reconstruction formula for limited data spherical Radon transform. Namely, assume that the data is only available on a subset Γ of a hyperplane in Rn (n = 2, 3). We consider a reconstruction formula studied in some previous works, under the assumption that the data is only smoothen out to order k near the boundary. For the problem in two dimensional space and Γ is a line segment, we show that the artifacts, propagating along the circles centered at the end points of Γ, are k+ 12 orders smoother than the original singularities. For the problem in three dimensional space and Γ is a rectangle, we describe that the artifacts are generated either by rotating along a vertex or an edge of Γ. The artifacts obtained by rotation around a vertex are 2k + 1 orders smoother than the original singularity. Meanwhile, the artifacts obtained by rotation along an edge are k + 12 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 in [FQ14]