Guillaume Bal (University of Chicago, USA)
Title: Mathematical aspects of hybrid inverse problems
Short abstract: These lectures provide an introduction to hybrid, or coupled-physics, medical imaging modalities that aim to combine high contrast with high resolution. We focus on the modalities called photo-acoustic tomography and elastography. We describe the different inverse problems associated with these imaging methods and focus on their injectivity and stability properties.
Marc Bonnet (ENSTA Paris, France)
Title: The concept of topological derivative in qualitative inverse scattering
Abstract: In this lecture, we address qualitative inverse scattering approaches based on asymptotic approximations of the scattering of waves by inhomogeneities of small size. We show how such models can be obtained and justified by means of volume integral equation (VIE) approaches, and then incorporate them into imaging functionals based on the concept of topological derivative. We discuss mathematical arguments and results that support TD-based imaging, and present numerical experiments. We finally describe extensions of the foregoing approaches that exploit asymptotic models (i) of higher order or (ii) for surface-breaking defects.
Fioralba Cakoni (Rutgers University, USA)
Title: The transmission eigenvalue problem, non-scattering phenomena and inverse scattering for inhomogeneous media
Short abstract: This lecture is a continuation of the lecture given by Prof. Andreas Kirsch. The focus of this lecture is the role of the transmission eigenvalue problem for solving the inverse scattering problem for inhomogeneous media.
The transmission eigenvalues are related to non-scattering wave numbers for which there exists an incident wave that does not scatterer by the given medium, in other words the far field operator (or relative scattering operator) is not injective with not dense range. We will discuss questions central to the inverse scattering theory include discreteness of the spectrum that is closely related to the determination of the support of inhomogeneity from scattering data using linear sampling and factorization methods, location of transmission eigenvalues in the complex plane that is essential to the development of the time domain linear sampling method , and the existence of transmission eigenvalues as well as the accurate determination of real transmission eigenvalues from scattering data, which has become important since real transmission eigenvalues could be used to obtain information about the scattering media.
Andreas Kirsch (Karlsruhe Institute of Technology, Germany)
Title: An Introduction to Inverse Problems
Short abstract: In this series of talks a very first introduction into the mathematical theory of inverse problem is presented. We begin with some classical examples, such as the inverse heat equation, the problem of impedance tomography, and an inverse scattering problem. In the second part we sketch the idea of regularization and restrict ourselves to Tikhonov’s regularization method and Landweber’s method as an example of an iterative regularization scheme. We finish the second part with some comments on other regularization techniques. In the third part we consider the direct and inverse scattering problems for the scattering of time-harmonic plane waves by some inhomogeneous medium. We sketch an integral equation approach for the treatment of the direct problem, introduce the far field pattern, and present the idea of the Factorization Method for the determination of the support of the contrast.
Mourad Sini (RICAM, Austria)
Title: Inverse Problems in Imaging using Resonant Contrast Agents
Short abstract: Many of the traditional inverse problems use remotely measured responses to wave-based interrogations in order to reconstruct (few) material coefficients interring into the considered model. Such inverse problems are known to be highly unstable. In several imaging modalities, it is advised, in the engineering community, to first inject contrast agents into the domain of interests before performing the experiments. With such perturbations, one can enhance the contrast appearing in the domain and hence “see better”.
In these lectures, we propose an approach on how to model and understand such contrast agents. Most importantly, we show, by examples of imaging modalities, how the generated contrast can be used to extract the needed material coefficients.