Reconstructive tomography with diffracting wavefields

Abstract

A unified theoretical framework based on the scalar wave equation is presented for transmission tomography of weakly inhomogeneous objects embedded in a known uniform background medium. The developed formulation differs from the usual treatments of diffraction tomography in that it is not limited to applications employing plane-wave illumination and planar measurement boundaries and can be extended to weakly inhomogeneous objects embedded in non-uniform backgrounds and to the class of strongly scattering objects. The new formulation is based on the mathematical operations of propagation and backpropagation which are generalisations of the conventional tomographic operations of projection and backprojection to the case of diffracting wavefields. The projection and backprojection operations are then obtained from the generalised operations in the zero wavelength limit. A class of reconstruction algorithms are developed within the new formulation that are generalisations, to the case of diffraction tomography, of the class of iterative ART (algebraic reconstruction technique) algorithms of conventional (X-ray) tomography. The reconstruction algorithms developed (called generalised ART algorithms) are shown to be applicable to situations where the tomographic data are sparse (limited-view problem) and to the incorporation of a priori information into the reconstruction process.