Fast and accurate false-alarm probabilities for linear detection statistics in Poisson noise
Abstract
Detecting weak signals in noisy data often requires calculating the statistical significance, or false- alarm probability, of a detection statistic. When the underlying noise follows Poisson statistics, as in low-count measurements such as photon-starved astronomical images, this calculation becomes non- trivial, and existing approaches typically rely on extensive simulations to estimate detection thresholds. We derive an accurate and efficient solution for computing false-alarm probabilities for arbitrary linear detection statistics under Poisson noise. We show that the detection statistic is a compound Poisson random variable, which unlocks efficient implementations based on the Fast Fourier Transform (FFT) and the Panjer recursion. The method is exact up to numerical discretization, accelerates computation by several orders of magnitude compared with simulations, and enables evaluation of false-alarm probabilities computationally inaccessible through simulation. Although motivated by low-count astronomical imaging, the framework applies broadly to detection problems involving Poisson-distributed events. It transforms the computation of detection significance under Poisson noise from a simulation problem into a deterministic numerical calculation. We provide open-source Python and MATLAB implementations.
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BibTeX
@misc{soumagnac2026fast,
title = {{Fast and accurate false-alarm probabilities for linear detection statistics in Poisson noise}},
author = {Maayane T. Soumagnac and Eran O. Ofek and Shachar Israeli and David Glukhov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05510},
eprint = {2611.05510},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05510},
}
RIS
TY - UNPB
TI - Fast and accurate false-alarm probabilities for linear detection statistics in Poisson noise
AU - Maayane T. Soumagnac
AU - Eran O. Ofek
AU - Shachar Israeli
AU - David Glukhov
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05510
UR - https://arxiv.org/abs/2611.05510
AB - Detecting weak signals in noisy data often requires calculating the statistical significance, or false- alarm probability, of a detection statistic. When the underlying noise follows Poisson statistics, as in low-count measurements such as photon-starved astronomical images, this calculation becomes non- trivial, and existing approaches typically rely on extensive simulations to estimate detection thresholds. We derive an accurate and efficient solution for computing false-alarm probabilities for arbitrary linear detection statistics under Poisson noise. We show that the detection statistic is a compound Poisson random variable, which unlocks efficient implementations based on the Fast Fourier Transform (FFT) and the Panjer recursion. The method is exact up to numerical discretization, accelerates computation by several orders of magnitude compared with simulations, and enables evaluation of false-alarm probabilities computationally inaccessible through simulation. Although motivated by low-count astronomical imaging, the framework applies broadly to detection problems involving Poisson-distributed events. It transforms the computation of detection significance under Poisson noise from a simulation problem into a deterministic numerical calculation. We provide open-source Python and MATLAB implementations.
ER -