Inversion of Circular Averages using the Funk Transform

In radar, when the wavelength of the transmitted electromagnetic wave is considerably larger than the dimension of the antenna, the received signal is modeled as the integral of the mean reflectivity function of the illuminated scene over the intersection of spheres centered at the transmitter location and the surface topography. When the surface topography is flat the received signal becomes integral of the mean reflectivity function over circles which is also referred to as circular averages of the mean reflectivity function. Thus, reconstruction of the ground reflectivity from synthetic aperture radar data requires inversion of the circular averages. Apart from radar, circular averages inversion also arises in thermo-acoustic tomography and sonar. In this paper, we present a new inversion method for the circular averages that uses the relationship between the circular averages, hyperbolic X-ray and the Funk transforms. The method is exact and numerically efficient as compared to standard filtered backprojection algorithms. Numerical simulations demonstrate that the approach is practical.