Super-resolution imaging through inverse scattering problems

Inverse scattering problems are concerned with recovering the physical properties of objects, such as their location, shape and medium inhomogeneity through the measurement of acoustic, elastic or electromagnetic waves. They play fundamental roles in a broad range of applications, including radar and sonar, geophysical exploration, non-destructive testing, medical imaging and near-field optics. A major challenge in inverse scattering problems is to break the diffraction limit, which asserts the highest resolution of the details for the recovered objects is limited by one half of the wavelength. Super-resolution imaging refers to the solution methods for which the diffraction limit is broken and the finest details of the reconstructed image is beyond one half of the wavelength. This project aims to build effective mathematical models and develop efficient numerical methods for super-resolution imaging associated with a few important inverse scattering problems.

In the first part of the project we propose a novel model and method for the imaging of infinite or closed rough surfaces. Instead of measuring the near-field data directly as in traditional techniques, we place a prism of negative-index metamaterial above the surface and measure the field on the farther end of the prism. Using a method recently developed by the PI and his collaborators, we found an infinite resolution is obtained with some ideal parameter values for the imaging of infinite rough surfaces. We shall continue investigating this promising model in more detail, in particular the effect on the resolution if the parameters are perturbed from the ideal values. We shall also generalize the method to more practical 3D problems and to closed surfaces such as bounded obstacles or cavities.

In the second part of the project we consider super-resolution imaging of obstacles by elastic waves, an important research area with limited results. We shall investigate both the direct scattering and the inverse scattering problems. For the direct scattering problem, we shall establish the existence and uniqueness by investigating both the original boundary value problem for the total displacement field and the coupled boundary value problem for the pressure and shear wave components. For the inverse scattering problem, we shall study the domain derivative of of the direct scattering operator and apply iterative optimization methods. We will also generalize the continuation method recently developed for inverse acoustic and electromagnetic scattering problems with multiple frequency data and adapt it to inverse elastic scattering problems