Non-Parametric Decompounding of Pulse Pile-Up Under Gaussian Noise With Finite Data Sets

A novel estimator is proposed for estimating the energy distribution of photons incident upon a detector in X-ray spectroscopic systems. It is specifically designed for count-rate regimes where pulse pile-up is an issue. A key step in the derivation of the estimator is the novel reformulation of the problem as a decompounding problem of a compound Poisson process. A non-parametric decompounding algorithm is proposed for pile-up correction with finite-length data sets. Non-parametric estimation typically includes appropriately choosing a 'kernel bandwidth'. Simulations demonstrate our data-driven bandwidth selection is close to optimal, and outperforms asymptotic-based selection in the typical regions of interest to spectroscopic applications. Non-parametric approaches are particularly useful when the shape of the detector response varies with each interaction. The method exhibits similar accuracy to other state-of-the-art non-parametric methods, while being much faster to compute.