[Paper Review] On the Separability of Targets Using Binary Proximity Sensors

Krishnan Balasubramaniam, Animesh Kumar+2 more|arXiv (Cornell University)|Nov 19, 2012

Indoor and Outdoor Localization Technologies10 references3 citations

TL;DR

This paper investigates the separability of an asymptotically large number of static target locations using binary proximity sensors in two models: deterministic grid locations and i.i.d. uniform random locations. It derives order-optimal scaling laws for the number of sensors required to achieve full or partial separability, showing that full separability requires Θ(n log n) sensors for grid targets and ω(n² log n) for random targets, while partial separability requires only Θ(n) sensors in both cases. The results hold even under adversarial sensor behavior.

ABSTRACT

We consider the problem where a network of sensors has to detect the presence of targets at any of $n$ possible locations in a finite region. All such locations may not be occupied by a target. The data from sensors is fused to determine the set of locations that have targets. We term this the separability problem. In this paper, we address the separability of an asymptotically large number of static target locations by using binary proximity sensors. Two models for target locations are considered: (i) when target locations lie on a uniformly spaced grid; and, (ii) when target locations are i.i.d. uniformly distributed in the area. Sensor locations are i.i.d uniformly distributed in the same finite region, independent of target locations. We derive conditions on the sensing radius and the number of sensors required to achieve separability. Order-optimal scaling laws, on the number of sensors as a function of the number of target locations, for two types of separability requirements are derived. The robustness or security aspects of the above problem is also addressed. It is shown that in the presence of adversarial sensors, which toggle their sensed reading and inject binary noise, the scaling laws for separability remain unaffected.

Motivation & Objective

• To determine the minimum number of binary proximity sensors required to reliably detect and distinguish the presence of targets at n possible locations in a finite region.
• To analyze separability under two target location models: uniformly spaced grid points and i.i.d. uniform random placements.
• To derive order-optimal scaling laws for sensor count m(n) and sensing radius r(n) that ensure high-probability identification of target configurations.
• To evaluate robustness against adversarial sensors that inject binary noise or toggle readings.
• To establish conditions under which full separability (all n targets identified) and partial separability (fraction α of targets identified) are achievable with high probability.

Proposed method

• Models target locations as either a uniform grid or i.i.d. uniform random variables over a finite region I.
• Assumes sensor locations are i.i.d. uniform over the same region, independent of target locations, to model deployment uncertainty.
• Uses binary proximity sensors that output 1 if any target lies within sensing radius r(n), 0 otherwise.
• Applies Poisson approximation and moment generating function (MGF) analysis to model the number of sensors detecting each target location.
• Employs Chernoff bounds on the MGF of the difference between active and inactive sensor counts to derive probabilistic bounds on separability.
• Derives scaling laws by analyzing the probability that each target location is detectable (i.e., has at least one sensor detecting it), using the parameter γ = P(target present) and its complement.

Experimental results

Research questions

• RQ1 What is the minimum number of binary proximity sensors required to achieve full separability (correct identification of all target configurations) with high probability as n → ∞?
• RQ2 How does the required number of sensors scale for partial separability, where at least a fraction α of target locations must be correctly identified with probability ≥ β?
• RQ3 How do the scaling laws for separability differ between deterministic grid target locations and stochastically distributed random target locations?
• RQ4 What is the impact of adversarial sensors—those that toggle readings or inject noise—on the required sensor count for separability?
• RQ5 How does the sensing radius r(n) influence the separability performance, and what are the optimal choices for r(n) in different models?

Key findings

• For deterministic grid targets, full separability requires m(n) = Θ(n log n) sensors, and this scaling is order-optimal.
• For random target locations, full separability requires m(n) = ω(n² log n) sensors, indicating a significantly higher sensor cost than in the grid case.
• For partial separability, both grid and random target models require m(n) = Θ(n) sensors, with the threshold depending on α and β through the expression log(1/((1−α)(1−β))).
• The scaling laws for separability remain unchanged even when adversarial sensors are present, as long as they do not alter the overall detection statistics.
• The probability of full separability approaches 1 as n → ∞ when m(n)/n log n exceeds a threshold dependent on the target density γ.
• The analysis shows that the critical parameter for separability is 1 − 2√(γ(1−γ)), which governs the decay rate of misidentification probability and determines the required sensor density.