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

• Published in: Medical physics (Lancaster) Vol. 43; no. 3; pp. 1295 - 1298
• Main Authors: Roessl, Ewald, Daerr, Heiner, Proksa, Roland
• Format: Journal Article
• Language: English
• Published: United States American Association of Physicists in Medicine 01.03.2016
• Subjects: Biological material, e.g. blood, urine; Haemocytometers; Comparators; Computed tomography; Computerised tomographs; computerised tomography; Devices sensitive to infra‐red, visible or ultra‐violet radiation; Devices sensitive to very short wavelength, e.g. x‐rays, gamma‐rays or corpuscular radiation; Digital computing or data processing equipment or methods, specially adapted for specific applications; Fourier Analysis; Fourier transforms; Image data processing or generation, in general; level‐crossing; medical image processing; Monte Carlo Method; Monte Carlo methods; photodetectors; Photodetectors (including infrared and CCD detectors); Photometry, e.g. photographic exposure meter; photon counting; Photons; photon‐counting detectors; Poisson's equation; Probability theory, stochastic processes, and statistics; pulse pile‐up; Radiometry - instrumentation; shot noise; stochastic processes; Transforming x‐rays; X-Rays; X‐ and γ‐ray sources, mirrors, gratings, and detectors; X‐ray absorption spectra; X‐ray detection; X‐ray detectors; X‐ray technique; shot noise; level-crossing; photon-counting detectors; pulse pile-up
• ISSN: 0094-2405; 2473-4209; 2473-4209
• DOI: 10.1118/1.4941743

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.