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Near-optimal Sensor Placement for Detecting Stochastic Target Trajectories in Barrier Coverage Systems

Mingyu Kim, Daniel J. Stilwell, Harun Yetkin, Jorge Jimenez

TL;DR

The paper tackles near-optimal sensor placement in a 2D barrier coverage system where targets follow a $LGCLP$ (log-Gaussian Cox line process). It transforms linear target trajectories into a representation space $\mathcal{C}$, estimates the intensity via INLA, and selects sensor locations to maximize the probability of detecting all targets by greedily thinning the intensity (void probability), with nonlinear refinements and a backward mapping to the inertial space. Key contributions include the representation-space formulation for Poisson line processes, a greedy-plus-refinement placement strategy, and validation on AIS ship data from the Hampton Roads area. The approach provides a practical, scalable framework for deploying a finite number of sensors under trajectory uncertainty, with potential applicability to other barrier-coverage problems and non-linear target paths.

Abstract

This paper addresses the deployment of sensors for a 2-D barrier coverage system. The challenge is to compute near-optimal sensor placements for detecting targets whose trajectories follow a log-Gaussian Cox line process. We explore sensor deployment in a transformed space, where linear target trajectories are represented as points. While this space simplifies handling the line process, the spatial functions representing sensor performance (i.e. probability of detection) become less intuitive. To illustrate our approach, we focus on positioning sensors of the barrier coverage system on the seafloor to detect passing ships. Through numerical experiments using historical ship data, we compute sensor locations that maximize the probability all ship passing over the barrier coverage system are detected.

Near-optimal Sensor Placement for Detecting Stochastic Target Trajectories in Barrier Coverage Systems

TL;DR

The paper tackles near-optimal sensor placement in a 2D barrier coverage system where targets follow a (log-Gaussian Cox line process). It transforms linear target trajectories into a representation space , estimates the intensity via INLA, and selects sensor locations to maximize the probability of detecting all targets by greedily thinning the intensity (void probability), with nonlinear refinements and a backward mapping to the inertial space. Key contributions include the representation-space formulation for Poisson line processes, a greedy-plus-refinement placement strategy, and validation on AIS ship data from the Hampton Roads area. The approach provides a practical, scalable framework for deploying a finite number of sensors under trajectory uncertainty, with potential applicability to other barrier-coverage problems and non-linear target paths.

Abstract

This paper addresses the deployment of sensors for a 2-D barrier coverage system. The challenge is to compute near-optimal sensor placements for detecting targets whose trajectories follow a log-Gaussian Cox line process. We explore sensor deployment in a transformed space, where linear target trajectories are represented as points. While this space simplifies handling the line process, the spatial functions representing sensor performance (i.e. probability of detection) become less intuitive. To illustrate our approach, we focus on positioning sensors of the barrier coverage system on the seafloor to detect passing ships. Through numerical experiments using historical ship data, we compute sensor locations that maximize the probability all ship passing over the barrier coverage system are detected.
Paper Structure (7 sections, 11 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 7 sections, 11 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation in a Representation Space $\mathcal{C}$
  3. Stochastic linear target trajectory model: log-Gaussian Cox line process (LGCLP)
  4. Sensor model
  5. Suboptimal Sensor Placement: Maximizing Void probability Approximation
  6. Numerical Result
  7. Conclusion

Figures (5)

  • Figure 1: (top) Linear trajectory $q$ is in the inertial space. (bottom) The corresponding mapped point $l$ in the representation space.
  • Figure 2: (left) Estimated linear target trajectories passing through an area of interest. (right) The corresponding mapped points and the area of interest in the representation space.
  • Figure 3: (top) Visualized sensor performance function $\gamma_A$ (probability of detection of a linear trajectory $q$ with respect to $\zeta$ and $a_i$). (bottom) The corresponding mapped sensor performance $\gamma_\mathcal{C}$ in the representation space.
  • Figure 4: Procedure of stochastic target trajectory estimation - (top) Heatmap of target traffic data marinecadastre.gov around an area of interest $O$. (bottom-left) Estimated linear target trajectories within an area of interest with greedily selected 5 sensor locations. (bottom-center) Mapped unique points into the representation space using the linear trajectories from the (bottom-left). (bottom-right) Estimated mean intensity function of the mapper point pattern of the (center) in the representation space.
  • Figure 5: (top) Estimated mean intensity function of LGCP modeled target trajectories. (bottom) Thinned the mean intensity function from (top) by placing a sensor performance.