Bounds on multiple sensor fusion

We consider the problem of fusing measurements in a sensor network, where the sensing regions overlap and data are nonnegative real numbers, possibly resulting from a count of indistinguishable discrete entities. Because of overlaps, it is generally impossible to fuse this information to arrive at an accurate value of the overall amount or count of material present in the union of the sensing regions. Here we study the computation of the range of overall values consistent with the data and provide several results. Posed as a linear programming problem, this leads to questions associated with the geometry of the sensor regions, specifically the arrangement of their nonempty intersections. We define a computational tool called the fusion polytope, based on the geometry of the sensing regions. Its properties are explored, and in particular, a topological necessary and sufficient condition for this to be in the positive orthant, a property that considerably simplifies calculations, is provided. We show that in two dimensions, inflated tiling schemes based on rectangular regions fail to satisfy this condition, whereas inflated tiling schemes based on hexagons do.