Ultrasonic Reflectivity Imaging in Three Dimensions: Exact Inverse Scattering Solutions for Plane, Cylindrical, and Spherical Apertures

Abstract

The problem of reconstructing the reflectivity of a three-dimensional medium with density and compressibility variations is examined. For the special case of continuous-wave (CW) insonification, exact inversion formulas have recently been reported for recovering an unknown scattering parameter from scattering measurements. In this work, exact solutions, or inversion formulas, are obtained for the general case of arbitrary broad-band insonification where the incident wave is assumed to be a spherically diverging broad-bandwidth pulse of arbitrary shape. Solutions are derived under the assumption that the scattering is sufficiently weak for the Born approximation to hold. Exact inversion formulas are obtained for three aperture geometries: a plane, cylindrical, or spherical recording surface enclosing the scattering region. Under most practical conditions, the process of back projection and coherent summation over spherical surfaces in image space, without prior filtering, is shown to provide a close approximation to the exact inversion procedure. Finally, in the case of the spherical geometry, the mathematical equivalence between the three-dimensional inverse Radon transform and the far-field approximation to the exact solution is demonstrated.