Hybrid inverse problems and internal functionals

Summary

This paper reviews recent results on hybrid inverse problems, which are also called coupled-physics inverse problems of multiwave inverse problems. Inverse problems tend to be most useful in, e.g., medical and geophysical imaging, when they combine high contrast with high resolution. In some settings, a single modality displays either high contrast or high resolution but not both. In favorable situations, physical effects couple one modality with high contrast with another modality with high resolution. The mathematical analysis of such couplings forms the class of hybrid inverse problems.

Hybrid inverse problems typically involve two steps. In a first step, a well-posed problem involving the high-resolution low-contrast modality is solved from knowledge of boundary measurements. In a second step, a quantitative reconstruction of the parameters of interest is performed from knowledge of the point-wise, internal, functionals of the parameters reconstructed during the first step. This paper reviews mathematical techniques that have been developed in recent years to address the second step.

Mathematically, many hybrid inverse problems find interpretations in terms of linear and nonlinear (systems of) equations. In the analysis of such equations, one often needs to verify that qualitative properties of solutions to elliptic linear equations are satisfied, for instance the absence of any critical points. This paper reviews several methods to prove that such qualitative properties hold, including the method based on the construction of complex geometric optics solutions.

The success of most medical imaging modalities rests on their high, typically submillimeter, resolution. Computerized tomography (CT), magnetic resonance imaging (MRI), and ultrasound imaging (UI) are typical examples of such modalities. In some situations, these modalities fail to exhibit a sufficient contrast between different types of tissues, whereas other modalities, for example based on the optical, elastic, or electrical properties of these tissues, do display such high contrast. Unfortunately, the latter modalities, such as optical tomography (OT), electrical impedance tomography (EIT) and elastographic imaging (EI), involve a highly smoothing measurement operator and are thus typically low-resolution as stand-alone modalities.