A comparative analysis of OTF, NPS, and DQE in energy integrating and photon counting digital x-ray detectors

Abstract

Purpose: One of the benefits of photon counting (PC) detectors over energy integrating (EI) detectors is the absence of many additive noise sources, such as electronic noise and secondary quantum noise. The purpose of this work is to demonstrate that thresholding voltage gains to detect individual x rays actually generates an unexpected source of white noise in photon counters.

Methods: To distinguish the two detector types, their point spread function (PSF) is interpreted differently. The PSF of the energy integrating detector is treated as a weighting function for counting x rays, while the PSF of the photon counting detector is interpreted as a probability. Although this model ignores some subtleties of real imaging systems, such as scatter and the energy-dependent amplification of secondary quanta in indirect-converting detectors, it is useful for demonstrating fundamental differences between the two detector types. From first principles, the optical transfer function (OTF) is calculated as the continuous Fourier transform of the PSF, the noise power spectra (NPS) is determined by the discrete space Fourier transform (DSFT) of the autocovariance of signal intensity, and the detective quantum efficiency (DQE) is found from combined knowledge of the OTF and NPS. To illustrate the calculation of the transfer functions, the PSF is modeled as the convolution of a Gaussian with the product of rect functions. The Gaussian reflects the blurring of the x-ray converter, while the rect functions model the sampling of the detector.

Results: The transfer functions are first calculated assuming outside noise sources such as electronic noise and secondary quantum noise are negligible. It is demonstrated that while OTF is the same for two detector types possessing an equivalent PSF, a frequency-independent (i.e., "white") difference in their NPS exists such that NPS(PC) > or = NPS(EI) and hence DQE(PC) < or = DQE(EI). The necessary and sufficient condition for equality is that the PSF is a binary function given as zero or unity everywhere. In analyzing the model detector with Gaussian blurring, the difference in NPS and DQE between the two detector types is found to increase with the blurring of the x-ray converter. Ultimately, the expression for the additive white noise of the photon counter is compared against the expression for electronic noise and secondary quantum noise in an energy integrator. Thus, a method is provided to determine the average secondary quanta that the energy integrator must produce for each x ray to have superior DQE to a photon counter with the same PSF.

Conclusions: This article develops analytical models of OTF, NPS, and DQE for energy integrating and photon counting digital x-ray detectors. While many subtleties of real imaging systems have not been modeled, this work is illustrative in demonstrating an additive source of white noise in photon counting detectors which has not yet been described in the literature. One benefit of this analysis is a framework for determining the average secondary quanta that an energy integrating detector must produce for each x ray to have superior DQE to competing photon counting technology.

Figure 1

A schematic diagram of the electrical circuit for processing current in the photodiode of an energy integrating detector is shown.

Figure 2

(a) The energy integrating circuit of Fig. 1 sums the current from each individual x ray and (b) integrates the net current over time to increase the charge and hence voltage across a storage capacitor. The output voltage per pixel is determined by the maximum potential difference (V_max) across the capacitor. The two subplots (a) and (b) are matched to their respective points in the circuit of Fig. 1.

Figure 3

A schematic diagram of the electrical circuit for processing current in the photodiode of a photon counting detector is shown.

Figure 4

In the photon counting circuit of Fig. 3, (a) the voltage gains exceeding the threshold established by the potentiometer are counted as representative of one x ray and (b) the total signal per pixel is found by summing these counts. The two subplots (a) and (b) are matched to their respective points in the circuit of Fig. 3.

Figure 5

Cross sections of the PSF surface are plotted versus position for two polar angles of the position vector (α=0° and 45°) and four blurring parameters (σ), assuming that the pixel is square with sides of length l and that the entire pixel is sensitive to the detection of x rays. The PSF is interpreted as a weighting function for detecting x rays in an energy integrator and as a probability function for detecting x rays in a photon counter.

Figure 6

The MTF is plotted versus frequency at two polar angles of the frequency vector (α=0° and 45°), assuming that the entire pixel is sensitive to the detection of x rays.

Figure 7

The variance of the two detector types is plotted versus the blurring of the x-ray converter for three pixel sensitivity areas.

Figure 8

The NPS is plotted versus the frequency, assuming that the entire pixel is sensitive to the detection of x rays for (a) an energy integrator and (b) a photon counter.

Figure 9

The NPS difference between the two detector types is shown to increase with the blurring of the x-ray converter for three pixel sensitivity areas.

Figure 10

The DQE is plotted versus the frequency, assuming that the entire pixel is sensitive to the detection of x rays for (a) an energy integrator and (b) a photon counter. In (c), the DQE difference between the two detector types is shown to be frequency dependent and to increase with the blurring of the x-ray converter. Subplots (a)–(c) implicitly share a common legend. In (d), DQE(0) is plotted versus the blurring of the x-ray converter for three pixel sensitivity areas.

Figure 11

For equivalent NPS and DQE between the two detector types, the average number of secondary quanta (m) that must be produced for each incident x ray in the energy integrating detector is plotted versus the blurring of the x-ray converter. The figure assumes 1000 x rays per pixel and electronic noise power (W_E) of zero, four, and eight x rays per pixel.