Morse-Smale Decomposition, Cut Locus and Applications in Sensor Networks

We apply Morse theory in the study of sensor networks and distributed sensor data. Sensor nodes are deployed in a 2D region M with boundaries and possibly interior holes, and the sensor data samples a continuous real function f . We are interested in both the topology of the discrete sensor field in terms of the sensing holes (voids without sufficient sensors deployed), as well as the topology of the signal field in terms of its critical points. Towards this end, we extend the construction of the Morse-Smale complex in the setting of a domain with boundaries and develop distributed algorithms to construct the Morse-Smale decomposition. The sensor field is decomposed into simply-connected pieces, inside each of which the function is homogeneous, i.e., the integral lines flow uniformly from a local maximum to a local minimum. This compact structure captures the essential topological features of the signal field sampled by distributed sensors, and has numerous applications in sensor data aggregation and distributed data-guided navigation in the network. A major component of the result in this paper is to establish the equivalence of the Morse-Smale decomposition with the ‘cut locus’ of the flow, defined as the points through which the flow has different homotopy types (get around the holes in different ways), or different limit endpoints, with the flows in their neighborhood. Since the cut locus can be detected locally in a discrete network, this connection turns out to be the key in the robust detection of the saddle points and the Morse-Smale decomposition of the sensor field.