A New Approach to Imaging by Waves

In many areas of national importance, such as the design and manufacturing of advanced and novel materials, public safety, medical imaging, and underground exploration, it is essential to be able to image and perform nondestructive testing of materials using electromagnetic, sound, or elastic waves. Unfortunately, effective methods for testing complicated materials for structural defects or for identifying unknown targets in an efficient way with little a priori information are still in a state of infancy.

This is particularly the case for material that exhibits directional properties, nonlinear interaction with interrogating waves, or peculiar geometric structures, all of which are present in contemporary applications in the above areas. In this project, the investigator and her graduate students will develop entirely new techniques in inverse scattering theory that can handle the aforementioned imaging problems, in order to obtain reliable target signatures or usable information about objects being examined in computationally efficient ways.

The goal is to minimize dependence on a priori information describing the physics and/or geometry of unknown targets as well as of mathematical and computational complications arising from the complexity of the hosting background. This study will combine practical applications with the mathematical elegance of new imaging techniques that have recently led to establishing a new field in mathematics called the qualitative approach or direct imaging techniques.

This research is a multifaceted effort to develop fast imaging methods of advanced and novel materials (including nonlinear materials) based on generalized linear sampling methods and spectral parameters related to non-scattering phenomena. There are three main projects:1) A Spectral Approach to Imaging with Waves: Motivated by the theory of transmission eigenvalues, this project proposes to develop a general framework for modifying the scattering operator in order to provide new eigenvalue problems associated with the scattering by an inhomogeneity.

Particularly important is the determination of these (real or complex) eigenvalues from the scattering data, together with their relation to the material properties of the inhomogeneity. A major effort of the investigator is to apply this technique to image complex structures, including anisotropic/absorbing/dispersive media, thin layers, and meta-surfaces. 2) Interior Eigenvalues and Non-scattering Frequencies: A necessary condition for the non-scattering of a particular incident wave is that the wave number (or a specified parameter) is an eigenvalue of an interior eigenvalue problem defined on the support of the scatterer.

On the other hand, the converse is true if at an eigenvalue the corresponding eigenfunction is extendable outside the support as a solution to the Helmholtz equation. Currently, only the case of scatterers with a corner is understood. The investigator states a more general conjecture and lays out a proposed approach to prove it.

This project is theoretical in nature, but if successful, it would lead to a more general uniqueness result for the support of the scattering object with one incident wave. 3) Initiate the Development of the Qualitative Imaging Approach for Nonlinear Inhomogeneities. Despite the great interest and extensive research on the theory and computation of qualitative (direct) inversion methods, nothing has been done in connection with imaging media that exhibit nonlinear interaction with interrogating waves, which is the case for many contemporary engineered materials. This project aims to pioneer such a study.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.