Optimal sensor placement for intruder detection: New design principles

We consider the centralized detection of an intruder, whose location is modeled as uniform across a specified set of points, using an optimally placed team of sensors. These sensors make conditionally independent observations. The local detectors at the sensors are also assumed to be identical, with detection probability (P_D) and false alarm probability (P_F). We formulate the problem as an N-ary hypothesis testing problem, jointly optimizing the sensor placement and detection policies at the fusion center. We prove that uniform sensor placement is never strictly optimal when the number of sensors (M) equals the number of placement points (N). We prove that for N₂ >; N₁ >; M, where N₁, N₂ are number of placement points, the framework utilizing M sensors and N₁ placement points has the same optimal placement structure as the one utilizing M sensors and N₂ placement points. For M ≤ 6 and for fixed P_D (or P_F), increasing P_F (or P_D) leads to optimal placements that are higher in the majorization-based partial order. For M >; 6, this result does not necessarily hold and we provide a simple counterexample. It is conjectured that the set of optimal placements for a given (M, N) can always be placed on a majorization-based partial order.