Abstract

We present novel microlocal results for generalized ellipsoid and hyperboloid Radon transforms with centers on surfaces in Euclidean Space, and we apply our results to Ultrasound Reflection Tomography (URT). We introduce a new Radon transform, $R$, which integrates compactly supported distributions over ellipsoids and hyperboloids with centers on a smooth 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 and only if the support of the function is in a connected set that is not intersected by any plane tangent to $S$ In this case, backprojection type reconstruction operators such as the normal operator $R^* R$ do not add artifacts to the reconstruction. We apply our results to a cylindrical geometry that could be used in URT . We investigate the visible singularities in this modality. In addition, we present reconstructions of image phantoms in two dimensions that illustrate our microlocal theory.