UWB Beam-Based Local Diffraction Tomography—Part I: Phase-Space Processing and Physical Interpretation

In this two-part article, we study the ultrawideband beam-based schemes for tomographic inverse scattering. The theory utilizes discrete phase-space sets of beam waves, which constitute overcomplete frames everywhere in the propagation domain, and thus can be considered as a local alternative to the conventional plane-wave or Green's function integrals used in conventional diffraction tomography. Specifically, we formulate two inversion schemes, a multi-frequency domain scheme, and a time domain scheme. The former utilizes isodiffracting Gaussian beams, while the latter utilizes isodiffracting pulsed beams. Both schemes consist of two phases: In the preprocessing phase, the scattering data are expanded as a sum of beams whose amplitudes, referred to as the “beam-domain data,” are extracted from the data using local beam-based transforms. In the imaging phase, the beam data are backpropagated and used for local reconstruction. In this Part I we discuss the preprocessing phase. We define the beam-based transforms, and then use the Born approximation to establish a cogent physical interpretation of the beam-domain data. Specifically, we show that these data are related to the local Radon transform of the medium, which is interpreted physically as a local Snell's law. This relation will be used in Part II to reconstruct the medium.