Hybrid Inverse Problems and
Several recent coupled-physics medical imaging modalities aim to
combine a high-contrast, low-resolution, modality with a
high-resolution, low-contrast, modality and ideally offer
high-contrast, high-resolution, reconstructions. Mathematically,
these modalities involve the reconstruction of constitutive
parameters in partial differential equations from knowledge of
internal functionals of the parameters and solutions to said
equations. This recent field of research is often referred to as
Hybrid Inverse Problems.
This talk presents recent theoretical results of uniqueness,
stability and explicit reconstructions for several hybrid inverse
problems. We provide an explicit characterization of what can (and
cannot) be reconstructed in coupled-physics imaging modalities such
as Magnetic Resonance Elastography, Transient Elastography,
Photo-Acoustic Tomography, and Ultrasound Modulation Tomography.
Numerical simulations confirm the high-resolution, high-contrast,
potential of these novel modalities.