Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivity

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 L2 function, f, over ellipsoids and hyperboloids with centers on a smooth connected surface, S. Our transform 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 contained in a connected open set that is 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.