Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivity

We present novel microlocal and injectivity analyses of ellipsoid hyperboloid Radon transforms. introduce a new transform, R, which defines the integrals compactly supported L2 function, f, over ellipsoids hyperboloids with centers on smooth connected surface, S. Our transform is shown to be Fourier Integral Operator (FIO) in our main theorem we prove that R satisfies Bolker condition if support f contained open set not intersected by any plane tangent Under certain conditions, this an equivalence. give examples where theory can applied. Focusing specifically cylindrical geometry interest Ultrasound Reflection Tomography (URT), results investigate visible singularities. In addition, example reconstructions image phantoms two-dimensions validate theory.