TY - JOUR

T1 - On the exactness of the universal backprojection formula for the spherical means Radon transform

AU - Agranovsky, M.

AU - Kunyansky, L.

N1 - Funding Information:
The question answered in this paper was posed by Professor Haltmeier in a conversation with the first author at the 9th Conference ‘Inverse Problems, Modeling and Simulation’, IPMS 2018 held on Malta. The preparation of the present paper started in 2022, during the 10th occurrence of this conference. The authors thank Professor Haltmeier for the interesting question and the organizers of IPMS-2022 for creating a stimulating environment and excellent conditions for collaboration. The second author acknowledges support by the NSF, through the award NSF/DMS 1814592. Finally, the authors are thankful to anonymous referees for helpful suggestions that noticeably improved clarity of this paper
Publisher Copyright:
© 2023 IOP Publishing Ltd.

PY - 2023/3

Y1 - 2023/3

N2 - 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.

AB - 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.

KW - explicit inversion formula

KW - spherical means

KW - thermoacoustic tomography

KW - universal backprojection formula

UR - http://www.scopus.com/inward/record.url?scp=85147138888&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85147138888&partnerID=8YFLogxK

U2 - 10.1088/1361-6420/acb2ee

DO - 10.1088/1361-6420/acb2ee

M3 - Article

AN - SCOPUS:85147138888

SN - 0266-5611

VL - 39

JO - Inverse Problems

JF - Inverse Problems

IS - 3

M1 - 035002

ER -