Inversion formula for the conical Radon transform arising in a single first semicircle second Compton camera with rotation

Abstract

Since a Compton camera and its data type were introduced, several types of conical Radon transforms have been studied. To increase the accuracy, new Compton camera designs have been proposed (Smith, Technologies 3(4):219-237, 2015). Among these, we consider the Single First Semicircle Second (SFSS) camera design consisting of a first (scattering) detector element and a second (absorption) detector shaped as a semicircle. Here we introduce a new type of a conical Radon transform which this SFSS Compton camera design brings about. To obtain sufficient data, we rotate the SFSS camera around the object of interest. Also, we provide an inversion formula for this conical Radon transform and generalize this result to an n-dimensional conical Radon transform. To demonstrate our suggested algorithm, numerical simulations for the 3-dimensional case are presented.