Near-Optimal Sensor Placement for Detection of Poisson Distributed Targets

Abstract

In this dissertation, we address the problem of sensor placement for detecting uncertain targets. We model target arrivals using a Poisson process to capture the inherent randomness of event occurrences and emphasize the importance of accounting for uncertainty in the sensor placement strategy. To tackle this, we propose a computationally efficient approximation method based on a lower bound derived from Jensen's inequality. This approach leverages the mean of the uncertain target model to yield a suboptimal yet tractable solution suitable for real-time applications. We evaluate the accuracy of this approximation by quantifying its deviation from the original formulation and providing an upper bound on the approximation error. While the initial framework is formulated in a 1-dimensional spatial domain along a line segment for simplicity, we extend it to a 2-dimensional setting to handle uncertain linear target trajectories using a log-Gaussian Cox line process. Furthermore, we develop an improved closed-form approximation that incorporates both the mean and variance of the target distribution using a second-order Taylor series expansion, offering increased accuracy and a tighter error bound. The effectiveness of our proposed methods is demonstrated using real-world ship traffic data from the Hampton Roads channels in Virginia, USA, obtained from the Office for Coastal Management and the Bureau of Ocean Energy.