The monotonicity principle for magnetic induction tomography

The inverse problem dealt with in this article consists of reconstructing the electrical conductivity from the free response of the system in the magneto-quasi-stationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the monotonicity principle (MP), that is a monotone relation connecting the unknown material property to the (measured) free-response. The MP is relevant as the basis of noniterative and real-time imaging methods. The Monotonicity Principle has been found in many different physical problems governed by diverse PDEs. Despite its rather general nature, each physical/mathematical context requires the proper operator showing the MP to be identified. In order to achieve this, it is necessary to develop ad hoc mathematical approaches tailored to the specific framework. In this article, we prove that (i) there exists a monotonic relationship between the electrical resistivity and the time constants characterizing the free response for MQS systems and (ii) the induced current density can be represented through a modal expansion. These results are based on the analysis of an elliptic eigenvalue problem obtained from the separation of variables.