On Uniquely Registrable Networks
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Consider a network with N nodes in d-dimensional Euclidean space, and M subsets of these nodes P_1,...,P_M. Assume that the nodes in a given P_i are observed in a local coordinate system. The registration problem is to compute the coordinates of the N nodes in a global coordinate system, given the information about P_1,...,P_M and the corresponding local coordinates. The network is said to be uniquely registrable if the global coordinates can be computed uniquely (modulo Euclidean transforms). We formulate a necessary and sufficient condition for a network to be uniquely registrable in terms of rigidity of the body graph of the network. A particularly simple characterization of unique registrability is obtained for planar networks. Further, we show that k-vertex-connectivity of the body graph is equivalent to quasi k-connectivity of the bipartite correspondence graph of the network. Along with results from rigidity theory, this helps us resolve a recent conjecture due to Sanyal et al. (IEEE TSP, 2017) that quasi 3-connectivity of the correspondence graph is both necessary and sufficient for unique registrability in two dimensions. We present counterexamples demonstrating that while quasi (d+1)-connectivity is necessary for unique registrability in any dimension, it fails to be sufficient in three and higher dimensions.
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