Explicit inversion of spherical Radon transforms in odd dimensions with partial radial data

TL;DR

The paper addresses reconstructing a compactly supported function from the spherical mean transform in odd dimensions with partial radial data ($0<t<1$), answering Rubin's question with an explicit, ODE-based inversion. The authors decompose the problem into radial and angular components, defining $k=\frac{n-3}{2}$ and $h_k(t)$ to derive an explicit differential equation $\left[\frac{\mathrm{d}}{\mathrm{d} t}D^{2k}h_k\right](t)$ whose right-hand side involves derivatives of $f$ at $1-t$, thereby bypassing Volterra integral equation methods. They extend the approach to general functions via spherical harmonics, producing per-harmonic ODEs for $\widetilde{f}_{q,s}$, with special-case analytical solutions in dimensions $n=3,5,7$. Numerical simulations with synthetic data and a publicly available code repository demonstrate the method's practicality and stability, confirming the theoretical results and illustrating its potential for thermoacoustic tomography with partial data. The work offers a simpler, explicit inversion framework in odd dimensions that improves computational efficiency and applicability when radial data are incomplete.

Abstract

We derive an explicit inversion algorithm for the spherical Radon transform in odd dimensions with partial radial data. We prove that the reconstruction of the unknown function can be reduced to solving ordinary differential equations, thereby providing a more explicit approach in odd dimensions than solving Volterra integral equation of the first kind established in prior works. We also provide analytical solutions in some special cases. Finally, we present numerical simulations validating our theoretical results. Our work answers a question posed by Rubin in ``Inversion formulae for the spherical mean in odd dimensions and the Euler-Poisson-Darboux equation,'' Inverse Problems 24 (2008), no. 2, 025021, 10 pp.

Figures (9)

• Figure 1: Dimension $n=3$, $f(x)=\frac{1}{2}e^{-\frac{{(x-.5)^2}}{2(.05)^2}}$ on $[0.0001,1]$, 150 node points. In this case, there is no associated ODE; instead, the solution is given by an explicit formula. This enables us to reconstruct the function in a neighborhood arbitrarily close to the origin.
• Figure 2: Dimension $n=3$, $f(x)= x^2(1-x)^2\text{if } 0.3<x<0.6,0\text{else}$ on $[0.001,1]$, 100 node points. Here again, there is no ODE, but we consider a discontinuous radial function.
• Figure 3: Dimension $n=5$, $f(x)=\frac{1}{2}e^{-\frac{{(x-.6)^2}}{2(.05)^2}}$ on $[0.05,0.99]$, 300 node points. Due to the singularity of the ODE at the origin, reconstruction of the function near the origin is inaccurate.
• Figure 4: Dimension $n=5$, $f(x)= 4x-1\text{if } 1/4<x<1/2,3-4x\text{if } 1/2\leq x <3/4,0\text{else}$ on $[0.15,.95]$, 100 node points. Due to the singularity of the ODE at the origin and the resulting lack of smoothness of the function, the reconstruction is inaccurate near the origin. However, we are able to reconstruct the function over more than 70% of its domain.
• Figure 5: Dimension $n=5$, $f(x)= x^2(1-x)^2\text{if } 0.3<x<0.60\text{else}$ on $[0.1,.95]$, 100 node points. Due to the singularity of the ODE at the origin and the resulting lack of smoothness of the function, the reconstruction is inaccurate near the origin. However, we are able to reconstruct the function over more than 80% of its domain.

Theorems & Definitions (21)

• Theorem 2.1: Explicit inversion for radial functions
• Remark 2.2
• Theorem 2.3: Explicit inversion for general functions
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 3.5