[Paper Review] Inversion of the Spherical Mean Transform with Sources on a Hyperplane

Aleksei Beltukov|ArXiv.org|Oct 8, 2009

Numerical methods in inverse problems12 references19 citations

TL;DR

This paper presents a novel inversion formula for the spherical mean transform with sources on a hyperplane in n-dimensional upper half-space, using an intertwining operator that links it to a convolution with a paraboloid kernel. The key contribution is an explicit, stable reconstruction formula based on Fourier analysis and classical Radon theory, valid for all rapidly decreasing functions and applicable in any dimension.

ABSTRACT

The object of this study is an integral operator $\mathcal{S}$ which averages functions in the Euclidean upper half-space $\mathbb{R}_{+}^{n}$ over the half-spheres centered on the topological boundary $\partial \mathbb{R}_{+}^{n}$. By generalizing Norton's approach to the inversion of arc means in the upper half-plane, we intertwine $\mathcal{S}$ with a convolution operator $\mathcal{P}$. The latter integrates functions in $\mathbb{R}^{n}$ over the translates of a paraboloid of revolution. Our main result is a set of inversion formulas for $\mathcal{P}$ and $\mathcal{S}$ derived using a combination of Fourier analysis and classical Radon theory. These formulas appear to be new and are suitable for practical reconstructions.

Motivation & Objective

• To develop a new, explicit inversion formula for the spherical mean transform with sources on a hyperplane in the upper half-space ℝⁿ⁺¹.
• To generalize Norton’s approach from the 2D case to arbitrary dimensions n ≥ 1.
• To establish a direct connection between the spherical mean transform and the classical Radon transform via a convolution operator.
• To provide a stable, back-projection-type reconstruction algorithm suitable for tomographic applications.

Proposed method

• Introduce a pair of mappings that intertwine the spherical mean transform 𝒮 with a convolution operator 𝒫 acting on functions in ℝⁿ⁺¹.
• Define 𝒫 as a convolution operator integrating over translates of a paraboloid of revolution.
• Use Fourier analysis and distribution theory to derive the inverse of 𝒫, denoted 𝒫⁻¹.
• Establish the inverse of the spherical mean transform via the composition 𝒮⁻¹ = 𝒮* ∘ 𝒦_y ∘ 𝒮, where 𝒦_y is a one-dimensional operator related to the classical Λ-operator.
• Leverage classical Radon inversion theory to justify the structure and validity of the inversion formula.
• Verify the formula through differential geometric techniques, including pull-back of volume forms and determinant computation of a Jacobian matrix.

Experimental results

Research questions

• RQ1 Can the spherical mean transform with sources on a hyperplane be inverted using a method that avoids strong analytic constraints on the function domain?
• RQ2 How can the spherical mean transform be related to the classical Radon transform through a convolution operator?
• RQ3 Is it possible to derive an explicit, stable inversion formula for the spherical mean transform that applies in all dimensions n ≥ 1?
• RQ4 Can the inversion formula be expressed in a back-projection form involving the adjoint operator 𝒮* for improved numerical stability?
• RQ5 What is the precise geometric and analytic structure of the inverse operator, particularly in terms of the paraboloid convolution kernel?

Key findings

• The paper derives a new inversion formula for the spherical mean transform 𝒮 that is valid for all rapidly decreasing functions in the Schwartz space 𝒮(ℝⁿ⁺¹), without restrictive analytic assumptions.
• The inverse is expressed as 𝒮⁻¹(f) = 𝒮* ∘ 𝒦_y ∘ 𝒮(f), where 𝒦_y is a one-dimensional pull-back of the classical Λ-operator, ensuring simplicity and stability.
• The convolution operator 𝒫, which intertwines with 𝒮, is shown to be invertible via distribution theory, yielding an explicit formula for 𝒫⁻¹.
• The Jacobian determinant of the transformation used in the volume form pull-back is computed as 1/(1+|z|²)⁽ⁿ⁺¹⁾/², confirming the correct scaling for the inversion.
• The method establishes a direct and natural link between the spherical mean transform and the classical Radon transform, with the latter providing the theoretical foundation for the inversion.
• The proposed inversion formula is suitable for practical tomographic reconstructions due to its compositional structure and compatibility with Fourier-based numerical algorithms.