Numerical Analysis of Target Enumeration via Euler Characteristic Integrals

TL;DR

This paper analyzes how the Euler characteristic integral method estimates target counts in discrete sensor fields, identifying errors and deriving formulas to improve accuracy and understanding for large target numbers.

Contribution

It provides a detailed mathematical analysis of errors in discrete Euler integral target counting and introduces a new estimator with asymptotic insights.

Findings

Counts first- and second-order errors in target estimation

Derives a formula for higher-order errors

Provides an asymptotic behavior analysis for large target numbers

Abstract

Given a continuous sensor field, we can apply the Euler characteristic integral approach to count the number of targets in the sensor field. If the sensor field is discrete, the Euler integral approach introduces errors into our target count. In this paper, we study the behavior of the Euler integral when applied to discrete sensor fields. Under precise assumptions, we count the number of first- and second-order errors in target count, and discover a formula proportional to much higher order errors. This allows us to derive a point estimator for the number of targets in a discrete sensor field. Finally we derive an asymptotic result, providing insight into how the discrete Euler integral behaves for a large number of targets.