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Concept

Sensor network

A field of many simple detectors spread over a region, each reporting a small local reading such as how many targets it can sense. The question is what can be recovered from the readings together, like the number of targets, without knowing which target each sensor sees.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Counting overlapping targets from sensor counts alone, by integrating against χ. A 36 by 32 grid of sensor counts from 7 overlapping discs; the sum of the Euler characteristics of the level sets is 7.

Counting targets by their holes

The Euler characteristic adds up like an area: glue two shapes together and it is the sum of theirs minus that of their overlap. So it can be used to measure, and measuring with it does something no area can. A field of sensors that each report only how many targets they detect — not which, not where — can have its readings added up, weighted by the Euler characteristics of the regions where the count is high, and the answer is exactly the number of targets.

topology · Euler characteristic Counting targets from sensor readings with mistakes in them. Three 80 by 72 sensor fields over 8 targets: true counts (integral 8), counts with 415 mistaken readings (integral 160), and the mistaken counts smoothed (rounded persistence count 8).

Counting targets when the sensors make mistakes

Add up the Euler characteristics of the regions where sensor counts are high and the answer is exactly the number of targets — until one sensor misreads. Each wrong reading is a new piece or a new hole, so the error grows with the number of sensors instead of averaging away. Smooth the readings and count each piece and hole by how long it survives as the threshold is lowered, and the count comes back exactly, with half the readings wrong, at the price of having to know how small a real feature can be.

topology · Euler characteristic

Named alongside it

The objects these essays reach for when they reach for this one.

Euler characteristicIntegralTopological invariantCounting argumentInclusion exclusionMeasurePersistenceRobustnessSmoothing

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