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Quantitative thermo-acoustics and related problems

Thermo-acoustic tomography is a hybrid medical imaging modality that aims to combine the good optical contrast observed in tissues with the good resolution properties of ultrasound. Thermo-acoustic imaging may be decomposed into two steps. The first step aims at reconstructing an amount of electromagnetic radiation absorbed by tissues from boundary measurements of ultrasound generated by the heating caused by these radiations. We assume that this first step has been performed. Quantitative thermo-acoustics then consists of reconstructing the conductivity coefficient in the equation modeling radiation from the now known absorbed radiation. This second step is the problem of interest in this paper. Mathematically, quantitative thermo-acoustics consists of reconstructing the conductivity in Maxwell's equations from available internal data that are linear in the conductivity and quadratic in the electric field. We consider several inverse problems of this type with applications in thermo-acoustics as well as in acousto-optics. In this framework, we obtain uniqueness and stability results under a smallness constraint on the conductivity. This smallness constraint is removed in the specific case of a scalar model for electromagnetic wave propagation for appropriate illuminations constructed by the method of complex geometric optics solutions.