Gauge-invariant registration in networks

Integration and exploitation of information collection across a distributed network of assets usually requires the establishment and maintenance of registration of coordinates across the nodes of the network. Here “registration” covers a range of possibilities, including clock synchronization and registration of frames of reference. The registration problem is posed in terms of network represented by a graph Γ with vertices corresponding to the sensors. Attached to the edges of the graph are noisy measurements of the “difference” between the two coordinate systems. This “difference” is expressed in terms of a member of a Lie group of coordinate transformations. Effectively, the registration problem is specified in terms of a connection on the edges, and becomes one of estimating a gauge transformation to align the coordinate systems across the network. The key descriptor of the difficulty of the estimation problem, the Fisher information, can be simply expressed in terms of the geometry of the situation and provides a link between the homological chains and cochains for the graph.