Target Counting under Minimal Sensing: Complexity and Approximations

Abstract

We consider the problem of counting a set of discrete point targets using a network of sensors under a minimalistic model. Each sensor outputs a single integer, the number of distinct targets in its range, but targets are otherwise indistinguishable to sensors: no angles, distances, coordinates, or other target-identifying measurements are available. This minimalistic model serves to explore the fundamental performance limits of low-cost sensors for such surveillance tasks as estimating the number of people, vehicles or ships in a field of interest to first degree of approximation, to be followed by more expensive sensing and localization if needed. This simple abstract setting allows us to explore the intrinsic complexity of a fundamental problem, and derive rigorous worst-case performance bounds. We show that even in the 1-dimensional setting (for instance, sensors counting vehicles on a road), the problem is non-trivial: target count can be estimated within relative accuracy of factor $\sqrt{2}$ and this is the best possible in the worst-case. We then address additional questions related to constructing feasible target placements, and noisy counters. In two dimensions, the problem is considerably more complicated: a constant-factor approximation is impossible. Our algorithms and analysis can easily handle some of the non-idealities of real sensors, such as asymmetric ranges and non-exact target counts.

This research was supported in part by the National Science Foundation under grants CNS-0626954 and CCR-0514738.