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.
Subject
Applied Mathematics,Computer Science Applications,Mathematical Physics,Signal Processing,Theoretical Computer Science
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