Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra

TL;DR

This paper develops explicit inversion formulas for reconstructing functions from their spherical or circular means with centers on polygonal and polyhedral boundaries, enabling exact interior reconstructions even with external sources.

Contribution

It introduces new explicit filtration/backprojection formulas for the spherical mean transform on specific polyhedral domains using wave equation potentials.

Findings

Exact reconstruction within the domain surrounded by the boundary surface.

Formulas valid for rectangles, certain triangles, cuboids, right prisms, and pyramids.

Reconstruction remains accurate despite external sources.

Abstract

We present explicit filtration/backprojection-type formulae for the inversion of the spherical (circular) mean transform with the centers lying on the boundary of some polyhedra (or polygons, in 2D). The formulae are derived using the double layer potentials for the wave equation, for the domains with certain symmetries. The formulae are valid for a rectangle and certain triangles in 2D, and for a cuboid, certain right prisms and a certain pyramid in 3D. All the present inversion formulae yield exact reconstruction within the domain surrounded by the acquisition surface even in the presence of exterior sources.