Sensing and Filtering : A Tutorial Based on Preimages and Information Spaces

This tutorial presents a fresh perspective on sensing uncertainty and filtering with the intention of understanding what information is minimally needed to achieve a specified task. The guiding principle is not to sense, represent, and encode more than is necessary. The concepts and tools are motivated by many tasks of current interest, such as tracking, monitoring, navigation, pursuit-evasion, exploration, and mapping. First, an overview of sensors that appear in numerous systems is presented. Following this, the notion of a virtual sensor is explained, which provides a mathematical way to model numerous sensors while abstracting away their particular physical implementation. Dozens of useful models are given, each as a mapping from the physical world to the set of possible sensor outputs. Preimages with respect to this mapping represent a fundamental source of uncertainty: These are equivalence classes of physical states that would produce the same sensor output. Pursuing this idea further, the powerful notion of a sensor lattice is introduced, in which all possible virtual sensors can be rigorously compared. The next part introduces filters that aggregate information from multiple sensor readings. The integration of information over space and time is considered. In the spatial setting, classical triangulation methods are expressed in terms of preimages. In the temporal setting, an information-space framework is introduced that encompasses familiar Kalman and Bayesian filters, but also introduces a novel family called combinatorial filters. Finally, the planning problem is presented in terms of filters and information spaces. The tutorial concludes with some discussion about connections to many related research fields and numerous open problems and future research directions.