On the exactness of the universal backprojection formula for the spherical means Radon transform. (1st March 2023)

• Record Type: Journal Article
• Authors: Agranovsky, M; Kunyansky, L
• Is Part Of: Inverse problems. Volume 39:Number 3(2023)
• Publication Date: 2023-03-01
• Subjects: universal backprojection formula -- thermoacoustic tomography -- explicit inversion formula -- spherical means; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/acb2ee
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: The spherical means Radon transform is defined by the integral of a function f in R n over the sphere S ( x, r ) of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data where x belongs to a hypersurface Γ ⊂ R n and r ∈ ( 0, ∞ ) has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When Γ coincides with the boundary ∂ Ω of a bounded (convex) domain Ω ⊂ R n, a function supported within Ω can be uniquely recovered from its spherical means known on Γ. We are interested in explicit inversion formulas for such a reconstruction. If Γ = ∂ Ω, such formulas are only known for the case when Γ is an ellipsoid (or one of its partial cases). This gives rise to a question: can explicit inversion formulas be found for other closed hypersurfaces Γ? In this article we prove, for the so-called 'universal backprojection inversion formulas', that their extension to non-ellipsoidal domains Ω is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.