On the inversion of generalized Radon transforms over n-dimensional paraboloids

Abstract

In this paper, we investigate the problem of determining a function from its surface integrals of the second type over n-dimensional circular paraboloids in Rn+1 with fixed focal length and fixed direction of symmetry axis, without any restrictions on their vertices. We obtain some inversion formulas with a method based on the Fourier transform and reducing the reconstruction problem to solving a Volterra integral equation of the first kind. We present some examples on the direct numerical implementations of the resulting inversion formulas.