This paper analyzes the reconstruction of diffusion and absorption parameters in an elliptic equation from knowledge of internal data. In the application of photoacoustics, the internal data are the amount of thermal energy deposited by high frequency radiation propagating inside a domain of interest. These data are obtained by solving an inverse wave equation, which is well-studied in the literature.We show that knowledge of two internal data based on well-chosen boundary conditions uniquely determines two constitutive parameters in diffusion and Schrödinger equations. Stability of the reconstruction is guaranteed under additional geometric constraints of strict convexity. No geometric constraints are necessary when 2n internal data for well-chosen boundary conditions are available, where n is spatial dimension. The set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics (CGO) solutions.

Photoacoustic tomography (PAT) is a novel hybrid medical imaging technique that aims to combine the large contrast of optical coefficients with the highresolution capabilities of ultrasound. We assume that the first step of PAT, namely the reconstruction of a map of absorbed radiation from ultrasound boundary measurement, has been done. We focus on quantitative photoacoustic tomography, which aims at quantitatively reconstructing the optical coefficients from knowledge of the absorbed radiation map. We present a non-iterative procedure to reconstruct such optical coefficients, namely the diffusion and absorption coefficients, and the Grüneisen coefficient when the propagation of radiation is modeled by a second-order elliptic equation. We show that PAT measurements allow us to uniquely reconstruct only two out of the above three coefficients, even when data are collected using an arbitrary number of radiation illuminations. We present uniqueness and stability results for the reconstructions of two such parameters and demonstrate the accuracy of the reconstruction algorithm with numerical reconstructions from two-dimensional synthetic data.

Inverse transport consists of reconstructing the optical properties of a domain from measurements performed at the domain's boundary. This paper concerns several types of measurements: time dependent, time independent, angularly resolved and angularly averaged measurements. We review recent results on the reconstruction of the optical parameters from such measurements and the stability of such reconstructions. Inverse transport finds applications e.g. in medical imaging (optical tomography, optical molecular imaging) and in geophysical imaging (remote sensing in the Earth atmosphere).