Spherical Radon transforms with smoothly varying radii

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.