Abstract
In problems of parameter estimation from sensor data, the Fisher information provides a measure of the performance of the sensor; effectively, in an infinitesimal sense, how much information about the parameters can be obtained from the measurements. From the geometric viewpoint, it is a Riemannian metric on the manifold of parameters of the observed system. In this paper, we consider the case of parameterized sensors and answer the question, “How best to reconfigure a sensor (vary the parameters of the sensor) to optimize the information collected?” A change in the sensor parameters results in a corresponding change to the metric. We show that the change in information due to reconfiguration exactly corresponds to the natural metric on the infinite-dimensional space of Riemannian metrics on the parameter manifold, restricted to finite-dimensional sub-manifold determined by the sensor parameters. The distance measure on this configuration manifold is shown to provide optimal, dynamic sensor reconfiguration based on an information criterion. Geodesics on the configuration manifold are shown to optimize the information gain but only if the change is made at a certain rate. An example of configuring two bearings-only sensors to optimally locate a target is developed in detail to illustrate the mathematical machinery, with Fast Marching methods employed to efficiently calculate the geodesics and illustrate the practicality of using this approach.
Highlights
This paper is an attempt to initiate the construction of an abstract theory of sensor management, in the hope that it will help to provide both a theoretical underpinning for the solution of practical problems and insights for future work
Our approach is set within the mathematical context of differential geometry and we aim to show that geodesics on a particular manifold provide the theoretical best-possible sensor reconfiguration in the same way that the Cramer–Rao lower bound provides a minimum variance for an estimator
We consider the problem of sensor management from an information-geometric standpoint [8,9]
Summary
This paper is an attempt to initiate the construction of an abstract theory of sensor management, in the hope that it will help to provide both a theoretical underpinning for the solution of practical problems and insights for future work. The Fisher information matrix defines a metric, the Fisher–Rao metric, over the physical space where the target resides, or in more generality a metric over the parameter space in which the estimation process takes place, and this metric is a function of the location of the sensors This observation permits analysis of the information content of the system, as a function of sensor parameters, in the framework of differential geometry (“information geometry”) [8,9,10,11]. The idea of regarding the Fisher–Rao metric as a measure of performance of a given sensor configuration and understanding variation in sensor configuration in terms of the manifold of such metrics is a powerful concept It permits questions concerning optimal target trajectories, as discussed here, to minimize information passed to the sensors and, as will be discussed in a subsequent paper, optimal sensor trajectories to maximize information gleaned about the target. Various properties of geodesics on this space are derived, and the mathematical machinery is demonstrated using concrete physical examples (Section 5)
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