Microlocal analysis for spherical Radon transform: two nonstandard problems

Abstract

Assume that is a connected convex closed hyper-surface in . Let be the spherical Radon transform that sends a function f to its integrals on the set of spheres centered on . In this article, we consider a general family of filtered-backprojection reconstruction formulas for , in two non-standard cases: (i) f is supported outside of and (ii) f is supported inside and is non-smooth. For the first case, we show that the resulting operator is a pseudo-differential operator with a singular symbol. It is associated with two cleanly intersecting Lagrangians; one is the diagonal and one is a fold canonical relation obtained by reflection. For the second case, we restrict our attention to the two dimensional space and being a square. We show that generates artifacts along circles centered at the vertices of . This fact, in particular, implies that there are no exact filtered-backprojection reconstruction formulas in this case.