Imaging methods of multiple scattering in isotropic point-like discrete random media [electronic resource]

During the last decades growing attention has been paid to the effect of propagation of waves in different types of randomly inhomogeneous media. This increasing interest is due, among other factors, to the novel methods that are being proposed for imaging small objects in cluttered environments and the need to have good solvers to correctly model fluctuations effects in these media. This is a major challenge in many research areas such as medical imaging, remote sensing, nondestructive testing and wireless communications, etc. In these applications, distinctly different mathematical models are used for imaging in continuous and discrete random media. The discrete random media are used to describe multiple scattering in continuous random media but it is unlikely that discrete inclusions are always adequate to represent inhomogeneities that have no clear boundaries between their different components. In imaging problems the emitted waves may be scattered multiple times by one or many inhomogeneities of the medium. Hence, the recorded waves include information about the medium through which they have propagated. The framework originally developed by Foldy, Lax, Twersky, and Keller together with the Lippmann-Schwinger equation is employed for the multiply scattered waves, in the frequency domain, in the case of an ensemble of randomly distributed point-like scatterers. The Foldy-Lax-Lippmann-Schwinger formalism is an exact method when scatterers are far from each other, and the scattering properties from a single isolated scatterer are known and easy to compute from the solution of a linear system. Thus, a very complicated scattering with multiples of all orders can be described in terms of a succession of interactions among multiply scattered waves in the cluttered media. Resonance is one of the most striking phenomena in multiple scattering. In a two-body system experiment, I show that resonances lead to sharp peaks in the spectrum as a function of energy when the scattering amplitudes of scatterers are not weak. The effects of resonances provide useful information about the image distortion arising from the underlying interactions of multiply scattered waves at resonance frequencies. I approach the problem of image distortion in randomly distributed point media by either removing the frequencies above the cutoff ratio or pruning the signals whose strength exceeds the threshold of permitted strength. The former is called the spectral cutoff method and the latter is the prune-and-average method. It should be emphasized that the scattering matrix of the Foldy-Lax-Lippmann-Schwinger formalism is a non-Hermitian Euclidean random matrix. Our heuristic formulation for the eigenvalue distribution of the scattering matrix can be expected to provide useful information about the choice of the cutoff parameter of the spectral-cutoff method. I apply the Foldy-Lax-Lippmann-Schwinger formulation to the correlation-based imaging in discrete random media. This approach seems relevant to seismology, nondestructive testing, structural health monitoring, and wireless sensor networks. I then use numerical simulation for different configurations to explore the limitations of this methodology.

Advisor: Papanicolaou George primary advisor. Thesis advisor
Farhat Charbel advisor. Thesis advisor
Ryzhik Leonid advisor. Thesis advisor