Detection of Gaussian signals via hexagonal sensor networks

This paper considers a special case of the problem of identifying a static scalar signal, depending on the location, using a planar network of sensors in a distributed fashion. Motivated by the application to monitoring wild-fires spreading and pollutants dispersion, we assume the signal to be Gaussian in space. Using a network of sensors positioned to form a regular hexagonal tessellation, we prove that each node can estimate the parameters of the Gaussian from local measurements. Moreover, we study the sensitivity of these estimates to additive errors affecting the measurements. Finally, we show how a consensus algorithm can be designed to fuse the local estimates into a shared global estimate, effectively compensating the measurement errors.

💡 Research Summary

The paper addresses the problem of identifying a static scalar field that follows a Gaussian spatial profile using a planar sensor network that operates in a fully distributed manner. Motivated by real‑world monitoring tasks such as wildfire spread and pollutant dispersion, the authors assume the underlying signal can be modeled as

  f(p) = A · exp(−‖p − c‖² / (2σ²))

where A is the amplitude, c the two‑dimensional center, and σ the spatial spread. The key contribution is the demonstration that a regular hexagonal tessellation of sensors enables each node to estimate these three parameters locally, using only its own measurement and those of its six immediate neighbors.

Network geometry – A hexagonal lattice is chosen because it provides the densest uniform coverage in the plane while guaranteeing exactly six neighbors per node. This uniformity yields equal inter‑sensor distances and identical angular relationships, which simplifies the algebraic structure of the estimation problem.

Local parameter estimation – Each node collects seven measurements (its own plus six neighbors). By applying a logarithmic transformation the nonlinear Gaussian model becomes linear in the unknowns, leading to an over‑determined linear system of the form

  log m_j = log A − ‖p_j − c‖² / (2σ²) + η_j

where η_j is transformed noise. The authors prove that the resulting coefficient matrix is full rank for any non‑degenerate placement, guaranteeing a unique least‑squares solution for (A, c, σ). Closed‑form expressions are derived, and the computational burden per node is shown to be O(1), making the method suitable for low‑power embedded devices.

Sensitivity analysis – To quantify how measurement errors ε_j affect the estimated parameters, the Jacobian J = ∂f/∂θ (θ =