Conductivity reconstruction on Riemannian manifolds from power densities

Weakly electric fish such as the elephantnose fish have for millions of years used electric signals for navigation. Elephantnose fish create an electric field around their body to detect, localise, and analyse objects within this field. This allows them to access the conducting properties of unknown objects. Knowing the conducting properties of objects is not only useful for elephantnose fish, but is also useful for human society. In medical imaging for instance, information about the conducting properties inside the body can indicate the presence of a tumor. However, it is not an easy task to access the conducting properties of organs from outside the body. The elephantnose fish is a showcase that can do all at once: Taking exterior measurements while perturbing its surroundings with the electric field and afterwards interpreting these measurements to understand the electric properties of objects in its surroundings. In order for scientists to accomplish similar results they need a thorough understanding of the physics involved: How the conductivity inside an object connects to currents and potentials on the outside of the object. Furthermore, they need an idea what measurements on the outside give suitable information of the interior, so that they can recover the conductivity. There are certain imaging modalities that have accomplished and build on these insights such as acousto-electric tomography, magnetic resonance electrical impedance tomography or current density imaging. What these methods have in common is that they combine measurements of the electric current at the boundary of the object with either ultrasound-induced deformations or measurements of the magnetic field to obtain internal information linked to the conductivity of the object. From this internal information, power or current densities, it is then possible to recover the conductivity in the interior. We address the problem of recovering the conductivity from the internal information obtained through exterior measurements. We investigate under which mathematical circumstances one is successful with recovering the conductivity. Here we consider different settings: A limited view setting, where one can only access the object from a part of the boundary or recovering the conductivity in a region on a surface. The former setting is inspired by applications such as breast cancer screening, where one cannot access the boundary of the breast from all angles. The latter setting is inspired by scenarios such as a pipe through a wall that can only be accessed from each side of the wall. The conducting properties of the inaccessible region on the pipe in the wall could give information about possible cracks. We build the theoretical foundation for problems that are difficult to address in medical imaging or for crack detection. This is only a small step in the complex reconstruction problem and requires modifications in order to be used in practical applications. It is impressive that evolution has helped to create a species of fish that can solve the whole problem, while it requires a tremendous effort to understand and solve the problem from a mathematical perspective.