Inversion of the circular averages transform using the Funk transform

Abstract

The integral of a function defined on the half-plane along the semi-circles centered on the boundary of the half-plane is known as the circular averages transform. Circular averages transform arises in many tomographic image reconstruction problems. In particular, in synthetic aperture radar (SAR) when the transmitting and receiving antennas are colocated, the received signal is modeled as the integral of the ground reflectivity function of the illuminated scene over the intersection of spheres centered at the antenna location and the surface topography. When the surface topography is flat the received signal becomes the circular averages transform of the ground reflectivity function. Thus, SAR image formation requires inversion of the circular averages transform. Apart from SAR, circular averages transform also arises in thermo-acoustic tomography and sonar inverse problems. In this paper, we present a new inversion method for the circular averages transform using the Funk transform. For a function defined on the unit sphere, its Funk transform is given by the integrals of the function along the great circles. We used hyperbolic geometry to establish a diffeomorphism between the circular averages transform, hyperbolic x-ray and Funk transforms. The method is exact and numerically efficient when fast Fourier transforms over the sphere are used. We present numerical simulations to demonstrate the performance of the inversion method.Dedicated to Dennis Healy, a friend of Applied Mathematics and Engineering.