Localization of Passive 3-D Coils as an Inverse Problem: Theoretical Analysis and a Numerical Method

This article provides both a theoretical analysis and a numerical method for the inverse source problem of locating multiple passive 3-D coils based on measurements of their superposed magnetic fields. In our context, a 3-D coil consists of three concentric circular coils being mutually perpendicular, and the term “passive” means that these coils are not connected to an active power source. Instead, their current is induced externally by a low-frequency alternating magnetic field which is generated by a closed exciter wire. The underlying inductively coupled system is modeled with regard to the 3-D coils as magnetic dipoles and their localization is formulated as an inverse problem. Since its ill posedness mainly arises from strong sensitivity to observational noise, an approximate upper bound for the localization error is derived mathematically by linearization. The Levenberg-Marquardt algorithm is applied as a method for localization and modified for better performance as well as the ability to estimate the number of 3-D coils in the localization area. Our method is tested with simulated and real data in order to confirm its capability of locating up to three passive 3-D coils within the front of a wooden shelf surrounded by the exciter wire and eight rectangular loop antennas.