Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivity

TL;DR

This paper analyzes the microlocal properties and injectivity of novel ellipsoid and hyperboloid Radon transforms, with applications to ultrasound tomography, providing theoretical insights and practical reconstructions.

Contribution

It introduces a new Radon transform over ellipsoids and hyperboloids, proves its microlocal properties and injectivity under specific conditions, and applies these results to ultrasound reflection tomography.

Findings

R is a Fourier Integral Operator satisfying the Bolker condition under certain support conditions.

Injectivity results are established for cylindrical geometries relevant to Ultrasound Reflection Tomography.

Example reconstructions validate the microlocal theory and demonstrate practical applicability.

Abstract

We present novel microlocal and injectivity analyses of ellipsoid and hyperboloid Radon transforms. We introduce a new Radon transform, $R$, which defines the integrals of a compactly supported $L^2$ function, $f$, over ellipsoids and hyperboloids with centers on a smooth connected surface, $S$. $R$ is shown to be a Fourier Integral Operator (FIO) and in our main theorem we prove that $R$ satisfies the Bolker condition if the support of $f$ is connected and not intersected by any plane tangent to $S$. Under certain conditions, this is an equivalence. We give examples where our theory can be applied. Focusing specifically on a cylindrical geometry of interest in Ultrasound Reflection Tomography (URT), we prove injectivity results and investigate the visible singularities. In addition, we present example reconstructions of image phantoms in two-dimensions, and validate our microlocal theory.