  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.
