Spherical Radon transforms with smoothly varying radii

TL;DR

This paper introduces a new spherical Radon transform with smoothly varying radii, establishes conditions for its stability and injectivity, and demonstrates applications in tomography with explicit inversion formulas and simulations.

Contribution

It develops a comprehensive theoretical framework for Radon transforms over spheres with variable radii, including stability, injectivity, and explicit inversion formulas for practical tomography applications.

Findings

Transform satisfies Bolker condition under certain radius functions

Explicit inversion formulas derived for specific applications

Stable reconstruction demonstrated through simulations

Abstract

We present an analysis of a novel spherical Radon transform, $R$, which defines the integrals of a function, $f$, in $\mathbb{R}^n$ over spheres with arbitrary center ($\mathbf{y}$) and radii, $r(\mathbf{y})$, which vary smoothly with $\mathbf{y}$. We first establish sufficient and necessary conditions on $r$ and $\text{supp}(f)$ so that $R$ satisfies the Bolker condition, and further conditions which allow $f$ to be recovered stably from $Rf$. We then apply this theory to a number of example applications in Compton Scatter Tomography (CST) and Ultrasound Reflection Tomography (URT). For each application considered, we also provide injectivity proofs and explicit inversion formulae, some of which are based on the generalized theory presented by Palamodov ("Palamodov, V. P. (2012). A uniform reconstruction formula in integral geometry. Inverse Problems, 28(6), 065014."). We then combine our microlocal theory and injectivity results to prove stability estimates for our transforms. In addition, to validate our theory, we provide simulated image reconstructions.