Diffraction Tomography with Arrays of Discrete Sources and Receivers

We present both theory and numerical simulations of diffraction tomography for arrays of line sources. The approach taken is applicable to a wide range of imaging problems where discrete sources and receivers are located near the object to be imaged rather than in its far field. As such, these new results tie into the vast body of research on inverse scattering, which to date is based largely on planewave sources. Our derivation implicitly includes a method for synthesizing plane waves; this method leads to inversion formulas based on the generalized projection-slice theorem. Although related techniues for synthesizing plane waves from discrete arrays are known, it is helpful to have available a theory that incorporates the synthesis directly into the scattering theory. The formulation is presented for propagating fields satisfying the Born and Rytov approximations in weakly inhomogeneous media and provides a convenient means for treating both forward and inverse scattering problems. Through a numerical example, we illustrate two important features of diffraction tomography inversion, i. e., 1) the effects of limited view and 2) results of probing with different signal frequencies. The example utilizes data generated by an exact forward modeling technique thus providing strong evidence supporting the usefulness of weak scattering approximations for inversion problems.