Topology Preserving Map to Physical Map - A Thin-Plate Spline Based Transform

A Topology Preserving Map (TPM) is an easily obtainable localization free connectivity based map that preserves physical layout features of 2D/3D sensor networks. This paper considers how to obtain physical maps and physical coordinates from a TPM when the physical locations of a subset of nodes are known. First, we present a General Procrustes Alignment (GPA) based solution, which is the optimal linear transformation solution to the problem. Second approach is based on thin-plate spline (TPS), which transforms the set of topology coordinates to physical coordinates using radial basis functions. Five representative 2D network topologies are used to evaluate and compare the TPS approach with GPA approach, and also with the existing distance vector-hop (DV-Hop) technique. Results are presented for the cases where the reference nodes are selected randomly from the entire network, or randomly from the inner and outer boundaries of the network. The results show that with less than 10% of nodes as reference nodes, a map with an average error less than 0.6 of radio range can be achieved with the TPS approach, which significantly outperforms both the GPA and the DV-Hop. TPS approach generalizes directly to 3-D networks as well.