# On Uniquely Registrable Networks

## TL;DR

This paper establishes a comprehensive framework for understanding when a network's global node coordinates can be uniquely determined from local observations, linking rigidity theory and graph connectivity.

## Contribution

It provides a necessary and sufficient condition for unique registrability based on the rigidity of the body graph and clarifies the role of graph connectivity in different dimensions.

## Key findings

- Characterizes unique registrability via rigidity of the body graph.
- Shows equivalence between k-vertex-connectivity and quasi k-connectivity.
- Disproves sufficiency of quasi (d+1)-connectivity for dimensions three and higher.

## Abstract

Consider a network with $N$ nodes in $d$-dimensional Euclidean space, and $M$ subsets of these nodes $P_1,\cdots,P_M$. 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 $P_1,\cdots,P_M$ 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.