An Analytical Model of NPS and DQE Comparing Photon Counting and Energy Integrating Detectors

Abstract

In this work, analytical models of the optical transfer function (OTF), noise power spectra (NPS), and detective quantum efficiency (DQE) are developed for two types of digital x-ray detectors. The two detector types are (1) energy integrating (EI), for which the point spread function (PSF) is interpreted as a weighting function for counting x-rays, and (2) photon counting (PC), for which the PSF is treated as a probability. The OTF is the Fourier transform of the PSF. The two detector types, having the same PSF, possess an equivalent OTF. NPS is the discrete space Fourier transform (DSFT) of the autocovariance of signal intensity. From first principles, it is shown that while covariance is equivalent for both detector types, variance is not. As a consequence, provided the two detector types have equivalent PSFs, a difference in NPS exists such that NPS_PC ≥ NPS_EI and hence DQE_PC ≤ DQE_EI. The necessary and sufficient condition for equality is that the PSF is either zero or unity everywhere. A PSF modeled as the convolution of a Lorentzian with a rect function is analyzed in order to illustrate the differences in NPS and DQE. The Lorentzian models the blurring of the x-ray converter, while the rect function reflects the sampling of the detector. The NPS difference between the two detector types is shown to increase with increasing PSF width. In conclusion, this work develops analytical models of OTF, NPS, and DQE for energy integrating and photon counting digital x-ray detectors.

Keywords: Energy integrating detector; detective quantum efficiency (DQE); modulation transfer function (MTF); noise power spectra (NPS); optical transfer function (OTF); photon counting detector; point spread function (PSF).

Fig. 1

Shown here are schematic diagrams of the electrical circuits for processing current in the photodiodes of (A) an energy integrating detector and (B) a photon counting detector.

Fig. 2

(A) The PSF is plotted versus position in increments of pixel length (l) for four different FWHMs (Γ) of the Lorentzian x-ray converter blurring function, assuming that the entire pixel is sensitive to the detection of x-rays (a = 1). Increasing Γ reduces the width of the plateau of the PSF and increases the spread of its tails. (B) The MTF is plotted versus spatial frequency, illustrating that increasing Γ worsens resolution. The two subfigures share a common legend.

Fig. 3

(A) The variance of the two detector types is plotted versus Γ for three pixel sensitivity lengths. (B) The relative NPS is plotted versus spatial frequency assuming that the entire pixel is sensitive to the detection of x-rays. (C) The NPS difference between the two detector types is shown to increase with increasing PSF width. (D) The DQE is plotted versus spatial frequency for the two detector types assuming that the entire pixel is sensitive to the detection of x-rays.

Fig. 4

(A) For both detector types, DQE(0) is plotted versus Γ for three pixel sensitivity lengths. (B) The DQE difference between the two detector types is spatial frequency dependent and increases with the blurring of the x-ray converter. This subplot implicitly assumes that the entire pixel is sensitive to the detection of x-rays.