Generalized Projection-Slice Theorem for Fan Beam Diffraction Tomography

Abstract

A generalized projection-slice theorem is derived for transmission fan beam diffraction tomography within the Born or Rytov approximations. The development is based on the use of the so-called paraxial approximation which requires that the object being probed subtend a small angle relative to the source point and to the measurement plane. Within this approximation it is shown that the transmitted field measured over a plane surface located on the opposite side of the object from the insonifying point source determines the three-dimensional spatial Fourier transform of the object profile over the surface of an ellipsoid of revolution in Fourier space. In the special case where the point source is in the far field of the object the semiaxes of the ellipsoid become equal and the surface degenerates to a sphere and the result reduces to the usual projection-slice theorem of plane beam diffraction tomography.