A Fourier approach to pulse pile-up in photon-counting x-ray detectors

An analytic Fourier approach to predict the expected number of counts registered in a photon-counting detector subject to pulse pile-up for arbitrary photon flux, detector response function, and pulse-shape is presented. The analysis provides a complete forward model for energy-sensitive, photon-counting x-ray detectors for spectral computed tomography.

The formalism of the stochastic theory of the expected frequency of level crossings of shot noise processes is applied to the pulse pile-up effect and build on a recently published analytic Fourier representation of the level crossing frequency of shot noise processes with piece-wise continuous kernels with jumps.

The general analytic result is validated by a Monte Carlo simulation for pulses of the form g(t) = e(-t/τ) (t > 0) and a Gaussian detector response function. The Monte Carlo simulations are in excellent agreement with the analytic predictions of photon counts within the numerical accuracy of the calculations.

The phenomenon of pulse pile-up is identified with the level-crossing problem of shot noise processes and an exact, analytic formula for the expected number of counts in energy-sensitive, photon-counting x-ray detectors for arbitrary photon flux, response function, and pulse-shapes is derived. The framework serves as a theoretical foundation for future works on pulse pile-up.