An unexpected role of transmission eigenvalues in imaging algorithms

Houssem Haddar (INRIA Paris, France)

Transmission eigenvalues are frequencies related to resonances inside scatterers and by duality to non scattering for an incident field being an associated eigenvector. Appearing naturally in the study of inverse scattering problems for inhomogeneous media, the associated spectral problem has a deceptively simple formulation but presents a puzzling mathematical structure, in particular it is a non-self-adjoint eigenvalue problem. It triggered a rich literature with a variety of theoretical results on the structure of the spectrum and also on applications for uniqueness results [1].
For inverse shape problems, these special frequencies were first considered as bad values (for some imaging algorithms, e.g., sampling methods) as they are associated with non injectivity of the measurement operator. It later turned out, as proposed in [2], that transmission eigenvalues can be used in the design of an imaging algorithm capable of revealing density of cracks in highly fractured domains, thus exceeding the capabilities of traditional approaches to address this problem. This new imaging concept has been further developed to produce average properties of highly heterogeneous scattering media at a fixed frequency (not necessarily a transmission eigenvalue) by encoding a special spectral parameter in the background that acts as transmission eigenvalues [3].
While targeting this unexpected additional value of transmission eigenvalues in imaging algorithms, the talk will also provide an opportunity to highlight some key results and open problems related to this active research area.

[1] F. Cakoni, D. Colton, H. Haddar. Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF, 98, 2022.
[2] L. Audibert, L. Chesnel, H. Haddar, K. Napal. Qualitative indicator functions for imaging crack networks using acoustic waves. SIAM Journal on Scientific Computing, 2021.
[3] L. Audibert, H. Haddar, F. Pourre. Reconstruction of average indicators for highly heteregenous scatterers. Preprint, 2023.