Mathematical models for bio-inspired imaging and plasmonics

Abstract

This thesis consists of two main parts. The first part is devoted to the study of the echo- and electro-sensing inverse problem. It provides mathematical and computational frameworks to explain how bats and weakly electric fish might identify and classify multiple targets. For the echo-sensing problem, we model the physical situation of a bat identifying and classifying a target. We focus on the frequency- and time-dependent setting. In the frequency domain, we provide and numerically test in the presence of noise a dictionary matching procedure for target classification based on comparing frequency-dependent distribution descriptors with precomputed ones in a dictionary of learned distributions. In the time domain, we introduce the concept of time-dependent polarization tensors for the wave equations associated to targets with constitutive parameters different from those of the background and size smaller than the operating wavelength. We show that these tensors are promising for performing imaging. For the electro-sensing problem, we draw inspiration from the biological behavior of the weakly electric fish, which is able to retrieve much more information about the shape and material of the targets when approaching them. We provide a new classification method which takes advantage of the multi-scale configuration. The method is based on a family of transform-invariant shape descriptors reconstructed at multiple scales. The evidence provided by the different descriptors at each scale is fused using the Dempster-Shafer Theory. The second part of the thesis is devoted to the study of modal expansions for non-Hermitian systems. We focus in particular on the analysis of the electromagnetic field scattered by a plasmonic nanoparticle with dispersive material parameters in a resonant regime. We provide a modal approximation of the low-frequency part of the scattered field in the time domain as a finite sum of modes oscillating at complex resonant frequencies.