Convex optimization of measurement allocation for magnetic tracking systems

Abstract

Magnetic tracking is a popular technique that exploits static and lowfrequency magnetic fields for positioning of quasi-stationary objects. One important system design aspect, which substantially influences the performance of the tracking system, is how to collect as much information as possible with a given number of measurements. In this work, we optimize the allocation of measurements given a large number of possible measurements of a generic magnetic tracking system that exploits time-division multiplexing. We exploit performance metrics based on the Fisher information matrix. In particular, the performance metrics measure worstcase or average performance in a measurement domain, i.e. the domain where the tracking is to be performed. An optimization problem with integer variables is formulated. By relaxing the constraint that the variables should be integer, a convex optimization problem is obtained. The two performance metrics are compared for several realistic measurement scenarios with planar transmitter constellations. The results show that the worst performance is obtained in the most distant parts of the measurement domain. Furthermore, measurement allocations optimized for worstcase performance require measurements in a larger area than measurement allocations optimized for average performance.

Analysis and optimization of surgical electromagnetic tracking systems by using magnetic field gradients

“…The gradient of the voltage induced by the i-th transmitting coil with respect to the position 𝒑 𝒔 of the sensor (i.e., its spatial gradient) can be expressed by [21]:…”

“…• 𝐹 2 is based on the computation of the spatial gradients expressed by (8). We follow an approach similar to [21], where the authors optimized the allocation of measurements for an EMTS based on TDM (time-division multiplexing) by using a cost function based on the Fisher Information Matrix to select a subset of a large number (at least 1089) of fixed coplanar sensors, with dipole moments all oriented along z-axis. They considered a constant 𝜎 𝑣 2 in the whole test volume.…”

“…is the Fisher Information Matrix (FIM), ∇ (𝒑 𝒔 ) 𝑣 𝑖𝑗 is the spatial gradient of 𝑣 𝑖𝑗 , defined by expression (8), and 𝜎 𝑖𝑗 2 is the variance of the voltage noise at the j-th point and related to the i-th transmitting coil. Under the assumption that the noise terms are independent, it follows from the Cramér-Rao inequality that maximizing the FIM (in some sense) minimizes the attainable covariance of the estimated position, providing a lower bound [21], [29]:…”

“…Since it is generally not possible to find an optimal FIM, the real-valued function 𝐹 2 can be chosen instead [21], [30]. Minimizing 𝐹 2 is related to the approach followed in Section 3.1 by using expression (12): in fact both the covariance (12) and the inverse of the FIM can be interpreted as the inverse of the Hessian of the weighted least-squares criterion [30].…”

“…In this paper we consider the same measurement volume as in [12], and we argue that the FG configuration can be optimized by maximizing the gradients of the induced voltage with respect to sensor position (as highlighted from preliminary experimental results), rather than maximizing the induced voltage as done in [12]; therefore, we follow an approach similar to [21], where the authors defined a metric based on the Fisher information matrix. Moreover, positional RMSE in [12] depended on the chosen reconstruction algorithm, whereas our proposed metric is valid regardless of the position reconstruction algorithms.…”