Mean and variance of single photon counting with deadtime

Abstract

The statistics of photon counting by systems affected by deadtime are potentially important for statistical image reconstruction methods. We present a new way of analysing the moments of the counting process for a counter system affected by various models of deadtime related to PET and SPECT imaging. We derive simple and exact expressions for the first and second moments of the number of recorded events under various models. From our mean expression for a SPECT deadtime model, we derive a simple estimator for the actual intensity of the underlying Poisson process; simulations show that our estimator is unbiased even for extremely high count rates. From this analysis, we study the suitability of the Poisson statistical model assumed in most statistical image reconstruction algorithms. For systems containing 'modules' with several detector elements, where each element can cause deadtime losses for the entire module, such as block PET detectors or Anger cameras, the Poisson statistical model appears to be adequate even in the presence of deadtime losses.

Dead time effect on the Brewer measurements: correction and estimated uncertainties

“…(11) the result converges to the corrected signal N 9 I . Given the Poisson nature of photon statistics, there are two formulas that are commonly used to calculate DT (Schätzel, 1986;Yu and Fessler, 2000), which depend on the nature of the counting system. For Brewers, the relevant algorithm is based on the assumption that all photons, either recorded by the counting system or lost, trigger a new dead time period (paralyzable system) and the extended formula (Eq.…”

“…8) is used. If it is assumed that the dead time is triggered only by the photons that are recorded by the counting system (nonparalyzable system) then the following, non-extended, formula applies (Schätzel, 1986;Yu and Fessler, 2000):…”

“…(8) by assuming a very small value of τ and by replacing the exponential term with its Taylor expansion. Though, it describes more accurately the effect of dead time on non-paralyzable systems (Schätzel, 1986;Yu and Fessler, 2000). Subsequently, a new formula corresponding to Eq.…”

“…In the beginning of the 1980s, the increased concern for the stratospheric ozone depletion (Farman et al, 1985) and its effects on surface UV levels (Kerr and McElroy, 1993;Zerefos, 2002) stimulated the deployment of the first Brewer ozone spectrophotometers. Until 1996 Brewer instruments were manufactured by SCI-TEC Instruments Inc. in Canada.…”

“…The Brewer network provides a variety of products such as the total columns of ozone (TOC) (Kerr et al, 1981), SO 2 (Cappellani and Bielli, 1995) and NO 2 (Cede et al, 2006;Diémoz et al, 2014), the aerosol optical depth (AOD) Gröbner and Meleti, 2004;Meleti and Cappellani, 2000) as well as global and direct irradiance spectra (Bais et al, 1993(Bais et al, , 1996. These measurements have supported scientific research for more than 30 years, enabling the investigation of their shortand long-term variability (Glandorf et al, 2005;Weatherhead et al, 1998;Zerefos, 2002) and interactions among them and among other atmospheric constituents (Bernhard et al, 2007). Additionally, good-quality ground-based measurements are very useful for the validation of satellite products which, under specific conditions, may be highly uncertain (Fioletov et al, 2002).…”