Spatial frequency-dependent pulse-height spectrum and method for analyzing detector DQE(f) from ensembles of single X-ray images

Abstract

Purpose: Scintillators and photoconductors used in energy integrating detectors (EIDs) have inherent variations in their imaging response to single-detected X-rays due to variations in X-ray energy deposition and secondary quanta generation and transport, which degrades DQE(f). The imaging response of X-ray scintillators to single X-rays may be recorded and studied using single X-ray imaging (SXI) experiments; however, no method currently exists for relating SXI experimental results to EID DQE(f). This work proposes a general analytical framework for computing and analyzing the DQE(f) performance of EIDs from single X-ray image ensembles using a spatial frequency-dependent pulse-height spectrum.

Methods: A spatial frequency (f)-dependent gain, $\stackrel{\sim }{g}\left(f\right)$ , is defined as the Fourier transform of the imaging response of an EID to a single-detected X-ray. A f-dependent pulse-height spectrum, $Pr\left[\stackrel{\sim }{g}\left(f\right)\right]$ , is defined as the 2D probability density function of $\stackrel{\sim }{g}\left(f\right)$ over the complex plane. $Pr\left[\stackrel{\sim }{g}\left(f\right)\right]$ is used to define a f-dependent Swank factor, A_S (f), which fully characterizes the DQE(f) degradation due to single X-ray noise. A_S (f) is analyzed in terms of its degradation due to Swank noise, variations in the frequency-dependent attenuation of $|\stackrel{\sim }{g}\left(f\right)|$ , and noise in $arg\stackrel{\sim }{g}\left(f\right)$ which occurs due to variations in the asymmetry in each single X-ray's imaging response. Three example imaging systems are simulated to demonstrate the impact of depth-dependent variation in $\stackrel{\sim }{g}\left(f\right)$ , remote energy deposition, and a finite number of secondary quanta, on $Pr\left[\stackrel{\sim }{g}\left(f\right)\right]$ , A_S (f), MTF(f), and NPS(f)/NPS(0), which are computed from ensembles of single X-ray images. The same is also demonstrated by simulating a realistic imaging system; that is, a Gd₂ O₂ S-based EID. Using the latter imaging system, the convergence of A_S (f) estimates is investigated as a function of the number of detected X-rays per ensemble.

Results: Depth-dependent $\stackrel{\sim }{g}\left(f\right)$ variation resulted in A_S (f) degradation exclusively due to depth-dependent optical Swank noise and the Lubberts effect. Conversely, the majority of A_S (f) degradation caused by remote energy deposition and finite secondary quanta occurred due to variations in $arg\stackrel{\sim }{g}\left(f\right)$ . When using input X-ray energies below the K-edge of Gd, variations in the frequency-dependent attenuation of $|\stackrel{\sim }{g}\left(f\right)|$ accounted for the majority of A_S (f) degradation in the GOS-based EID, and very little Swank noise and variations in $arg\stackrel{\sim }{g}\left(f\right)$ were observed. Above the K-edge, however, A_S (f) degradation due to Swank noise and variations in $arg\stackrel{\sim }{g}\left(f\right)$ greatly increased. The convergence of A_S (f) was limited by variation in $arg\stackrel{\sim }{g}\left(f\right)$ ; imaging systems with more variation in $arg\stackrel{\sim }{g}\left(f\right)$ required more detected X-rays per ensemble.

Conclusions: An analytical framework is proposed that generalizes the pulse-height spectrum and Swank factor to arbitrary f. The impact of single X-ray noise sources, such as the Lubberts effect, remote energy deposition, and finite secondary quanta on detector performance, may be represented using $Pr\left[\stackrel{\sim }{g}\left(f\right)\right]$ , and quantified using A_S (f). The approach may be used to compute MTF(f), NPS(f), and DQE(f) from ensembles of single X-ray images and provides an additional tool to analyze proposed EID designs.

Keywords: detective quantum efficiency; energy integrating detector; modulation transfer function; noise power spectrum; pulse-height spectrum.