What Can We Hear From an Echo? Qualitative Methods for Inverse Scattering and Imaging

A dolphin hunting in murky water cannot see its prey, but from the echo of its clicks it can tell where a fish is, how big it is, and even what it is made of. Medical imaging, sonar, and nondestructive testing all ask the same question: what can we learn about a hidden object using only the waves it sends back? Mathematically, these are inverse problems. They are typically nonlinear and ill-posed, so classical iterative methods for solving them require a good initial guess, which is rarely available in practice.

In this talk, I will present qualitative reconstruction methods that recover the location and shape of hidden objects with little or no prior knowledge of what they are made of or how many there are. I will focus on scatterers with a thin conductive coating, modeled by two boundary parameters. For these, I will show how a factorization of the far-field operator leads to a direct sampling method that is fast and provably stable under noise. I will then discuss transmission eigenvalues, the special frequencies at which an object produces no echo, and how they serve as a fingerprint of the object’s material and coating. Finally, I will describe how these ideas extend to new settings, including bending waves in thin plates and electrical impedance tomography for detecting corrosion.