ON ARTIFACTS IN LIMITED DATA SPHERICAL RADON TRANSFORM: FLAT OBSERVATION SURFACES

摘要

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 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 G 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. 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 Frikel and Quinto [SIAM J. Appl. Math., 75 (2015), pp. 703-725].