Maximum-likelihood detection of sources among Poissonian noise

A maximum likelihood (ML) technique for detecting compact sources in images of the x-ray sky is examined. Such images, in the relatively low exposure regime accessible to present x-ray observatories, exhibit Poissonian noise at background flux levels. A variety of source detection methods are compared via Monte Carlo, and the ML detection method is shown to compare favourably with the optimized-linear-filter (OLF) method when applied to a single image. Where detection proceeds in parallel on several images made in different energy bands, the ML method is shown to have some practical advantages which make it superior to the OLF method. Some criticisms of ML are discussed. Finally, a practical method of estimating the sensitivity of ML detection is presented, and is shown to be also applicable to sliding-box source detection.

💡 Research Summary

The paper addresses the problem of detecting compact astronomical sources in X‑ray images where the background is dominated by Poissonian noise, a situation typical for the low‑exposure observations of current X‑ray observatories such as Chandra and XMM‑Newton. Traditional detection techniques—sliding‑box (or “cell”) methods and the Optimized Linear Filter (OLF)—perform adequately when the signal‑to‑noise ratio is high, but their efficiency degrades sharply as the background count rate falls into the Poisson regime. The sliding‑box approach uses a fixed count threshold over a moving aperture, which leads to a high false‑alarm rate when the background fluctuations are large. OLF improves the signal‑to‑noise ratio by convolving the image with a filter matched to the expected source profile and the background statistics, yet it treats each energy band independently; consequently, it cannot exploit the complementary information that may be present across multiple bands.

To overcome these limitations, the authors develop a Maximum‑Likelihood (ML) detection framework. They model the observed count n_i in each pixel i as a Poisson random variable with mean λ_i = b_i + s_i, where b_i is the background level and s_i is the source contribution parameterized by a set of source parameters θ (position, flux, possibly extent). Two hypotheses are defined: H₀ (no source) and H₁ (source present). The likelihood for the whole image under each hypothesis is the product of the individual Poisson probabilities. The log‑likelihood ratio

L = ln