When Can a System of Subnetworks Be Registered Uniquely?

Abstract

Consider a network with N nodes in d dimensions, and M overlapping subsets P1, ⋯,PM (subnetworks). Assume that the nodes in a given Pi are observed in a local coordinate system. We wish to register the subnetworks using the knowledge of the observed coordinates. More precisely, we want to compute the positions of the N nodes in a global coordinate system, given P1, ⋯, PM and the corresponding local coordinates. Among other applications, this problem arises in divide-and-conquer algorithms for localization of adhoc sensor networks. The network is said to be uniquely registrable if the global coordinates can be computed uniquely (up to a rigid transform). Clearly, if the network is not uniquely registrable, then any registration algorithm whatsoever is bound to fail. We formulate a necessary and sufficient condition for uniquely registra-bility in arbitrary dimensions. This condition leads to a randomized polynomial-time test for unique registrability in arbitrary dimensions, and a combinatorial linear-time test in two dimensions.