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

PURPOSE: 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. METHODS: 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. RESULTS: 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. CONCLUSIONS: 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.