Abstract
We present an overview of a beam-based approach to ultra-wide band (UWB) tomographic inverse scattering, where beam-waves are used for local data-processing and local imaging, as an alternative to the conventional plane-wave and Green’s function approaches. Specifically, the method utilizes a phase–space set of iso-diffracting beam-waves that emerge from a discrete set of points and directions in the source domain. It is shown that with a proper choice of parameters, this set constitutes a frame (an overcomplete generalization of a basis), termed “beam frame”, over the entire propagation domain. An important feature of these beam frames is that they need to be calculated once and then used for all frequencies, hence the method can be implemented either in the multi-frequency domain (FD), or directly in the time domain (TD). The algorithm consists of two phases: in the processing phase, the scattering data is transformed to the beam domain using windowed phase–space transformations, while in the imaging phase, the beams are backpropagated to the target domain to form the image. The beam-domain data is not only localized and compressed, but it is also physically related to the local Radon transform (RT) of the scatterer via a local Snell’s reflection of the beam-waves. This expresses the imaging as an inverse local RT that can be applied to any local domain of interest (DoI). In previous publications, the emphasis has been set on TD data processing using a special class of localized space–time beam-waves (wave-packets). The goal of the present paper is to present the imaging scheme in the UWB FD, utilizing simpler Fourier-based data-processing tools in the space and time domains.
Highlights
Inverse scattering deals with determining the shape and the composition of an unknown object from measurements of the scattering field data due to a known illumination.This area has a wide range of medical, geophysical, oceanographical, industrial, etc., applications, using electromagnetic, acoustic, elastic, or seismic waves [1–5]
Practical algorithms rely on linearized weak scattering formulations using the Born, Rytov, Physical optics, or other single scattering approximations [2,5] which linearize the relation between the target and the field and provide the basis for diffraction tomography (DT) reconstruction [6]
The other class addresses radiation from extended sources, where the field is expressed as a sum of beam propagators that emerge from a discrete phase–space lattice of points and directions in the aperture
Summary
Inverse scattering deals with determining the shape and the composition of an unknown object from measurements of the scattering field data due to a known illumination. The other class addresses radiation from extended (aperture) sources, where the field is expressed as a sum of beam propagators that emerge from a discrete phase–space lattice of points and directions in the aperture These formulations utilize local window (e.g., Gaussian) functions to transform the data to the beam domain, and propagate the data using beams. The theory is structured on a frequency-independent phase–space sets of beams that constitute frames everywhere in the propagation domain This beam frame formulation enables the expansion of both the medium inhomogeneities and the scattering data with the same set of beam-basis functions, enabling a direct inversion over the beam domain.
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