Decentralized human trajectories tracking using hodge decomposition in sensor networks

Abstract

With the recent development of localization and tracking systems for both indoor and outdoor settings, we consider the problem of sensing, representing and analyzing human movement trajectories that we expect to gather in the near future. In this paper, we propose to use the topological representation, which records how a target moves around the natural obstacles in the underlying environment. We demonstrate that the topological information can be sufficiently descriptive for many applications and efficient enough for storing, comparing and classifying these natural human trajectories. We pre-process the sensor network with a purely decentralized algorithm such that certain edges are given numerical weights. Then we can perform trajectory classification by simply summing up the edge weights along the trajectory. Our method supports real-time classification of trajectories with minimum communication cost. We test the effectiveness of our approach by showing how to classify randomly generated trajectories in a multi-level arts museum layout as well as how to distinguish real world taxi trajectories in a large city.

Adaptive Discrete Vector Field in Sensor Networks

“…Sensor networks benefit from other applications of homology groups’ computation such as tracking [ 13 ] or network management [ 31 ]. Moreover, discrete vector fields are also useful within the sheaf approach applied to networks [ 32 ] as this theory appears to be well-suited for the integration of heterogeneous and non-localized data [ 33 , 34 , 35 , 36 ].…”

“…Other solutions are proposed in the literature to get the generators of the homology groups. Most of them rely on the connection, due to Hodge decomposition, between harmonic forms that are the solutions of a Laplace equation and the homology groups [ 12 , 13 , 14 ]. This equation is solved by a distributed iterative differential equation solver.…”

Tracking COVID-19 by Tracking Infectious Trajectories

“…However, given the fact that GPS can be sometimes inaccurate (in the order of dozens of meters, especially in cities), other techniques, proposed in the literature, can be considered. Among these solutions, we mention, radio-frequency, cellular networks, sensor networks, or even Internet-based localization techniques [8] – [12] . These approaches can accurately estimate device locations (for both indoor and outdoor scales like buildings and cities, etc.…”

Topological signatures for fast mobility analysis

“…In past work, Hodge theory and relative homology from algebraic topology have been used for the purpose of topological classification [27,35]. However, these approaches are only applicable to specific start and end points of trajectories, and yield only discrete classifications by homotopy type, which is not general enough for use in further analysis.…”

“…Hodge decomposition of a randomly generated vector field has been used in [35] to classify paths into homotopy types. This approach relies on a numerical process, and only applies to trajectories with same start and end points.…”