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

Abstract

The spherical means Radon transform f(x, r) is defined by the integral of a function f in ℝn 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 f(x, r) where x belongs to a hypersurface Γ⊆ℝn and r ∈ (0, ∞) has important applications in modern imaging modalities, such as photo- and thermoacoustic tomography. When Γ coincides with the boundary ∂Ω of a bounded (convex) domain Ω⊆ℝn, 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 the natural 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.

Inversion of the two‐data circular Radon transform centered on a curve on C(R2)${\cal C}(\mathbf {R}^2)$

“…Taking into account (8), 𝑃 ′ 𝑠 = (𝑋 ′ 𝑠 , 𝑌 ′ 𝑠 ) = (1, 0) and 𝑝(𝑠) = 𝜋∕2 for the derivatives of both sides of (9) with respect to 𝑠 and 𝑡, respectively, I obtain (using Green's theorem)…”

“…□ I have proved Theorem 1 directly. It is known that the inversion of SRT is equivalent (via the Kirchhoff-Poisson formula) to the following boundary-value problems for the wave equation 2,[6][7][8][9] { 𝑢(𝐱, 𝑡) ′′ 𝑡𝑡 − Δ 𝐱 𝑢(𝐱, 𝑡) = 0, (𝐱, 𝑡) ∈ 𝐑 2 × (0, ∞), 𝑢(𝐱, 0) = 𝑓(𝐱), 𝑢(𝐱, 0) ′ 𝑡 = 0,…”

“…[2][3][4][5] More often, in the mathematical literature, the injectivity of SRT for compactly supported functions is considered. [6][7][8][9]13,18 Problem 1. Let 𝛾 ⊂ 𝐑 2 be a smooth simple curve on the plane.…”