On the people counting problem in smart homes: undirected graphs and theoretical lower-bounds

Abstract

Smart homes of the future will have to deal with multi-occupancy scenarios. Multi-occupancy systems entail a preliminary and critical feature: the capability of counting people. This can be fulfilled by means of simple binary sensors, cheaper and more privacy preserving than other sensors, such as cameras. However, it is currently unclear how many people can be counted in a smart home, given the set of available sensors. In this paper, we propose a graph-based technique that allows to map a smart home to an undirected graph G and discover the lower-bound of certainly countable people, also defined as certain count. We prove that every independent set of n vertices of an undirected graph G represents a minimum count of n people. We also prove that the maximum number of certainly countable people corresponds to the maximum independent sets of G, and that the maximal independent sets of G provide every combination of active sensors that ensure different minimum count. Last, we show how to use this technique to identify and optimise suboptimal deployment of sensors, so that the assumptions can be tightened and the theoretical lower-bound improved.