Stealthy Optimal Range-Sensor Placement for Target Localization

TL;DR

We study stealthy range-sensor placement to maximize the localization information for targets while limiting the information leakage to the targets, formulated as a min-max Fisher Information Matrix (FIM) problem under a D-optimality criterion. The work provides a complete solution for the 2 sensors and 2 targets case, characterizing global optima via cyclic configurations on a common circle and parallelogram geometry under stealth constraints, and derives tight analytic lower and upper bounds for the case of arbitrarily many sensors with two targets. Extensions to $m\ge 3$ sensors show that optimal configurations exist on a circle at infinity when unconstrained, while constrained placements between targets can be analyzed via epigraph formulations and yield computable bounds, with the gap between bounds remaining small in numerical studies. The results offer design guidance for stealthy distributed sensing and secure collaborative sensing applications, highlighting geometric structures that maximize localization while limiting adversarial leakage.

Abstract

We study a stealthy range-sensor placement problem where a set of range sensors are to be placed with respect to targets to effectively localize them while maintaining a degree of stealthiness from the targets. This is an open and challenging problem since two competing objectives must be balanced: (a) optimally placing the sensors to maximize their ability to localize the targets and (b) minimizing the information the targets gather regarding the sensors. We provide analytical solutions in 2D for the case of any number of sensors that localize two targets.

Figures (9)

• Figure 1: The problem setup in this paper. A set of $m$ sensors (in green) are to be placed such that their ability to localize a set of $n$ targets (in red) is maximized while the targets' ability to localize the sensors is limited.
• Figure 2: Left: Shaded region that must contain $t_1$ and $t_2$ (relative to $s_1$ and $s_2$) so that the objective \ref{['opt:obj']} is at least $\eta^2$. The region is formed by two intersecting circles ($\eta=0.7$ shown). The dotted circle shows $\eta=1$. Right: Shaded region that must contain $s_1$ and $s_2$ (relative to $t_1$ and $t_2$) so that information leakage level is at most $\gamma$ ($\gamma=0.7$ shown).
• Figure 3: The seven possible configurations for two sensors and two targets. Each case is characterized by different constraints relating $\theta_1$, $\theta_2$, $\beta_1$ and $\beta_2$ which are given in \ref{['prop:cases']}. If sensors $s_1$ and $s_2$ are interchangeable then $\mathcal{C}_4 = \mathcal{C}_5$ and $\mathcal{C}_6 = \mathcal{C}_7$ and there are only five distinct cases.
• Figure 4: Examples of optimal sensor configurations when positions are unconstrained (\ref{['theorem 1']}). The solid circles delineate the feasible set (see \ref{['fig: level sets']}). The dotted circle is any larger circle passing through $t_1$ and $t_2$. A configuration is optimal if and only if the sensors $s_1$ and $s_2$ lie on this larger circle and are diametrically opposed. This can happen with alternating sensors and targets (left) or with both sensors on the same side (right).
• Figure 5: Example of an optimal sensor configuration when sensors are constrained to lie in the region between both targets (\ref{['theorem 2']}). Solid circles delineate the feasible set (see \ref{['fig: level sets']}). A configuration is optimal if $s_1$ and $s_2$ lie on opposite arcs and $t_1$, $s_1$, $t_2$ and $s_2$ form a parallelogram.

Theorems & Definitions (8)

• Remark 1
• Proposition 1
• Lemma 1
• Theorem 1
• proof
• Theorem 2
• proof
• Remark 2