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We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, landmarks) in a region using local counts performed by a network of sensors, each of which measures the number of targets nearby but neither their identities nor any positional information. We formulate and solve several such problems based on the types of sensors and mobility of the targets. The main contribution of this paper is the adaptation of a topological sheaf integration theory — integration with respect to Euler characteristic — to yield complete solutions to these problems. 1. Topological enumeration. 1.1. Sensors. Sensor networks are poised to impact society in fundamental ways anal- ogous to the impact of the personal computer and the internet. The rapid development of small-scale sensor devices coupled with wireless ad hoc networking capability is giving birth to a wide variety of sensor networks for applications to agriculture, defense, environmental monitoring, and more. At present, there are severe limits on sizes and types of implementable networks. In partic- ular, there are trade-offs between power consumption, sensing complexity (how much data is gathered and processed on-board), sensor size, sensor range (how large a neighborhood of the sensor is scanned), and communication bandwidth. Current implementations of sen- sor networks in environmental and agricultural monitoring are limited to dozens or at most hundreds of sensors (see e.g., (15) for a typical deployment). This constraint will not persist. Ubiquitous sensing — the saturation of physical environments with networked sensors — is a future scenario whose possibility is strongly suggested by Moore's law, by advances in micro- and nano-scale electronic device fabrication, and by advances in wireless networking capabilities (8). The potential to integrate sensors into building materials, roads, soil, and other media portend an extraordinary increase in the ability to monitor street traffic, crowd dynamics, wildlife habitats, and crop development. This paper initiates a mathematical (and in particular topological) approach to target- counting problems in sensor networks. Counting is a fundamental application of sensors, both in present and potential settings. Scenarios where counting is critical include: agricul- ture (crop/weed/insect populations, herd size); border security (people, vehicles, shipping containers); commerce (customers, stock, mail parcels); ecology (wildlife population, partic- ulate pollutants); situational awareness (troops, vehicles, artillery); traffic control (vehicles, pedestrians); and more. Note that objects to be counted span several orders of magnitude in scale, depending on the particular application domain. In many instances, targets may be in motion.

Target Enumeration via Euler Characteristic Integrals