Detective Quantum Efficiency (DQE)

• Detective Quantum Efficiency (DQE) is a metric measuring how effectively an imaging detector transfers the squared signal-to-noise ratio from input quanta to output data.
• DQE is commonly expressed through frequency-dependent formulations using the modulation transfer function (MTF) and normalized noise power spectrum (NNPS), and it adapts to different detector types.
• DQE analysis informs practical detector comparisons across applications by highlighting trade-offs such as low-frequency efficiency versus high-frequency noise and aliasing effects.

Searching arXiv for recent and relevant papers on Detective Quantum Efficiency across imaging modalities. Searching for "detective quantum efficiency arXiv imaging detector review electron x-ray". Detective quantum efficiency (DQE) is a signal-to-noise transfer metric that quantifies how efficiently a detector or imaging chain preserves the squared signal-to-noise ratio from input quanta to output data. In its standard imaging form, it is defined as $$\mathrm{DQE}=\mathrm{SNR}_{\mathrm{o}}^2/\mathrm{SNR}_{\mathrm{i}}^2$$, and in the frequency domain as $$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$; for linear, shift-invariant detectors with Poisson input statistics and appropriately normalized gain, it is commonly expressed through the modulation transfer function (MTF) and normalized noise power spectrum (NNPS) as $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$ (McMullan et al., 2014). In energy-resolving systems, the same principle is generalized to matrix-valued or task-dependent forms normalized to an ideal detector (Persson et al., 2018, Tanguay et al., 2022). In single-photon detector instrumentation, however, the same acronym can denote overall photon detection efficiency and be written as $\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$ (Miehling et al., 2023). The term therefore names a family of closely related, but not identical, efficiency concepts.

1. Formal definitions and domain-specific meanings

The conventional imaging definition is the ratio of output to input squared signal-to-noise ratio, either at zero spatial frequency or as a function of spatial frequency. In counting or integrating detector analyses, this is the primary definition, and the distinction between input quanta, output counts, and output noise is explicit [(McMullan et al., 2014); (Mir et al., 2017)].

A second, closely related quantity is quantum efficiency (QE). In the false-event analysis of radiation detectors, QE is the intrinsic efficiency of a noise-less detector in the absence of false events, whereas DQE is the realized efficiency in the presence of false events and intrinsic detector noise. In that treatment, false events generated by the detector itself and false events introduced by the detection process both depress DQE below QE, and the resulting DQE depends on the mean exposure $S_i$ (Zanella, 2012). For a counting detector with process-induced false-event factor $f$, detector-generated false events $f_a$, and detector noise variance $\sigma_d^2$, the synthesis provided for that work yields

$$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$

This formulation makes explicit that DQE is not merely a detection probability.

In energy-resolving photon-counting detectors and spectroscopic x-ray detectors, DQE is no longer a single scalar independent of task. The formalism is generalized to matrix-valued $\mathrm{DQE}(\mathbf{u},E,E')$ or to task-specific quantities for detection, quantification, or pseudo-monoenergetic imaging (Persson et al., 2018, Tanguay et al., 2022). In compressed sensing under constant illumination, DQE is further generalized from a detector metric to a recording-setup metric,

$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$0

where $$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$1 accounts for the different number of measurements in the recorded data versus the hypothetical reference (Broek et al., 2018).

The abbreviation also has a distinct usage in ALD-coated MCP-PMT work, where DQE is the overall single-photon detection efficiency and is defined as

$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$2

That literature explicitly rejects a subtractive form such as “DQE = QE - CE” as physically nonsensical in that context (Miehling et al., 2023).

Context	DQE form	Reference
Conventional imaging detector	$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$3 or $$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$4	(McMullan et al., 2014)
Energy-resolving detector	Matrix-valued or task-dependent DQE normalized to an ideal detector	(Persson et al., 2018, Tanguay et al., 2022)
MCP-PMT single-photon detector	$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$5	(Miehling et al., 2023)
Constant-illumination compressed sensing	$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$6	(Broek et al., 2018)

This multiplicity of definitions is not a contradiction; it reflects different choices of input reference, output observable, and task.

2. Frequency-domain structure: MTF, NPS, NNPS, and aliasing

In conventional detector characterization, DQE is tied to spatial transfer and noise transfer. For direct electron detectors at 300 keV, the frequency-dependent form used is

$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$7

with the MTF modeled as the product of pixel modulation and an intrinsic detector term,

$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$8

and the intrinsic component fitted as a normalized sum of Gaussians corresponding to a circularly symmetric point-spread function (McMullan et al., 2014).

That same study treats aliasing explicitly. Because the Fourier transform of the point-spread function extends beyond Nyquist, aliased components contribute to the measured band-limited spectrum. The normalized noise power spectrum is then written as a sum over aliased orders,

$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$9

with alias indices $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$0 in Nyquist units (McMullan et al., 2014). Near Nyquist, the DQE of all detectors falls because aliased noise contributions increase.

A central consequence is that high MTF alone does not guarantee high DQE. At 300 keV, the K2 Summit had the best DQE at low spatial frequencies, but with increasing frequency its DQE fell below that of the Falcon II, even though the K2 super-resolution MTF at Nyquist was nearly $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$1 Falcon II’s. The reason given is the behavior of NNPS: the K2 has essentially flat NNPS and therefore DQE that follows $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$2, whereas the Falcon II has lower MTF but NNPS that tracks $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$3, keeping DQE relatively flat out to about $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$4 Nyquist and making it superior beyond about $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$5 Nyquist (McMullan et al., 2014).

Several detector studies use an ideal square-pixel or sampling-limited reference. For Medipix3, the ideal detector benchmark is

$$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$6

so that at Nyquist $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$7 and $$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$8 (Mir et al., 2016). Timepix4 similarly uses

$$\mathrm{DQE}(f)\approx \mathrm{MTF}(f)^2/\mathrm{NNPS}(f)$$9

with $\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$0 and $\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$1 (Ding et al., 6 Mar 2026).

This structure also clarifies why DQE may exceed an aperture-limited ideal reference in thresholded counting detectors. In Medipix3 and GaAs:Cr Medipix3 studies, values near to, or even exceeding, those for an ideal detector were reported under particular threshold settings because thresholding reduced effective pixel size or suppressed NPS more strongly than MTF over parts of the band (Mir et al., 2016, Paton et al., 2020).

3. Task-dependent and matrix-valued DQE in energy-resolving systems

For energy-resolving photon-counting detectors, a scalar DQE is insufficient because both signal and noise are multivariate across energy channels. The generalized formalism introduces a matrix-valued NEQ and DQE. In the energy basis,

$\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$2

and the associated DQE matrix is

$\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$3

For an ideal energy-resolving detector, $\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$4 (Persson et al., 2018).

Off-diagonal elements of the NEQ and DQE matrices are directly linked to imperfect energy resolution. The cited analysis states that they are related to loss of energy information due to imperfect energy resolution arising from mechanisms such as charge sharing, fluorescence escape, Compton scatter inside the detector, and pulse pile-up (Persson et al., 2018). This makes the matrix structure physically interpretable: diagonal elements quantify dose efficiency for single-energy perturbations, whereas off-diagonal elements quantify constructive or destructive coupling between energies.

The spectroscopic x-ray detector normalization paper replaces the generic ideal reference by a formally defined ideal SXD: one with an infinite number of infinitesimal energy bins, no spatial or energy distortion, and no decrease in frequency-dependent SNR of the incident quanta (Tanguay et al., 2022). Under small-signal linearization and Beer–Lambert modeling, a single matrix

$\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$5

governs ideal detection and quantification noise. Detection-task DQE is then normalized as

$\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$6

while quantification DQE is defined componentwise from the ratio of ideal to measured covariance (Tanguay et al., 2022).

Task dependence is not a formal refinement only; it changes comparative rankings. In a Monte Carlo CdTe example, the matrix-valued framework predicted a zero-frequency dose efficiency relative to an ideal detector of $\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$7 for detecting water and $\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$8 for detecting bone, but only $\mathrm{DQE}=\mathrm{QE}\cdot \mathrm{CE}$9 for quantifying water and $S_i$0 for quantifying bone (Persson et al., 2018). In a CT simulation study, CdTe systems outperformed Si systems for detection tasks in the low-count-rate regime, while silicon outperformed one or both CdTe systems for material decomposition (Persson et al., 2020). DQE therefore depends on whether the relevant task is detectability, basis separation, or parameter estimation.

4. Physical determinants of DQE

The physical factors that govern DQE differ by detector class, but several recurring mechanisms dominate: counting versus integrating operation, charge sharing, carrier diffusion, energy straggling, backscatter, pile-up, and coherence losses.

In direct electron detection at 300 keV, counting mode improves low-frequency DQE by eliminating readout noise and removing intrinsic energy-loss variability (“straggling”) at low frequencies. The same study shows, however, that coincidence loss depresses both gain and DQE. For an average $S_i$1 electrons per pixel per frame in a counting detector,

$S_i$2

and

$S_i$3

At about $S_i$4 e/pixel/s, corresponding to $S_i$5 at $S_i$6 fps, this predicts about $S_i$7 count loss and about $S_i$8 drop in $S_i$9 (McMullan et al., 2014). Integrating detectors, by contrast, retain high $f$0 except at extremely low exposure rates, but are limited by straggling and by the point-spread function associated with carrier diffusion and epilayer thickness (McMullan et al., 2014).

In hybrid pixel detectors, sensor material and thickness strongly shape DQE. A Medipix3 study comparing Si and GaAs:Cr sensors found that the performance of the GaAs:Cr device was markedly superior to that of the Si device for high-energy electrons. At 200 keV, GaAs:Cr exceeded Si in both MTF and DQE; at 300 keV its performance degraded but remained comparable to Si at 200 keV (Paton et al., 2020). The same work attributes the advantage to shorter electron ranges and reduced lateral spread in the higher-$f$1 sensor. Earlier Medipix3 measurements at 60 and 80 keV showed that charge summing mode can deliver simultaneous, near-ideal values of both MTF and DQE, whereas single-pixel mode forces a threshold-dependent trade-off between the two (Mir et al., 2016).

Timepix4 provides a different illustration of the same physics. In raw event data mode, $f$2 exceeded $f$3 at both 100 kV and 200 kV, but at Nyquist the DQE remained above $f$4 at 100 kV and dropped close to zero at 200 kV. The reported explanation is that higher-energy electrons travel further in silicon, generate larger charge-sharing clusters, broaden the point-spread function, and drive the MTF sharply down; the lower high-frequency NNPS at 200 kV does not compensate for the squared MTF loss (Ding et al., 6 Mar 2026).

In wave-imaging comparisons, DQE is also controlled by coherence. For weak-phase imaging, an ideal Zernike phase-contrast microscope yields

$f$5

and reaches unity for perfect coherence and a $f$6 phase shift. In-focus single-sideband ptychography, by contrast, is bounded by

$f$7

with a peak of about $f$8. The same paper shows that ptychography is more robust to partial coherence, especially temporal incoherence, than HRTEM (Bennemann et al., 15 Sep 2025).

These cases illustrate a general point: DQE is set by the entire chain of stochastic signal formation and noise formation, not by spatial transfer alone.

5. Measurement methodologies and sources of uncertainty

Because DQE is an SNR ratio, its measurement requires a consistent estimate of both signal transfer and noise transfer. In detector practice, this usually means independent measurement of MTF and NPS, plus a gain or fluence calibration.

For direct electron detectors and hybrid pixel detectors, MTF is typically measured by a knife-edge method. In the 300 keV direct detector comparison, MTF was measured from the shadow of sharp platinum rods, with careful verification of edge quality for the K2 Summit because of its high MTF and small pixels (McMullan et al., 2014). Medipix3 and GaAs:Cr Medipix3 studies likewise used slanted or inclined knife-edge methods, derived an edge-spread function, differentiated to obtain the line-spread function, and Fourier transformed to obtain MTF (Mir et al., 2017, Paton et al., 2020). Timepix4 used a slanted knife-edge method following Dimova et al. and reported DQE from raw event data without clustering (Ding et al., 6 Mar 2026).

NPS and NNPS are usually estimated from flat-field image stacks. The direct electron detector comparison scaled measured power spectra to unity at zero spatial frequency and used analytically computed alias corrections from the fitted MTF parameters (McMullan et al., 2014). Medipix3 and GaAs:Cr Medipix3 computed NPS from Fourier transforms of flat-field images under uniform illumination, then used separate low-frequency corrections based on binning analyses to estimate $f$9 or NPP (Mir et al., 2016, Paton et al., 2020). Timepix4 required a specific zero-frequency correction because subtraction of the frame mean removes DC and near-DC components; its $f_a$0 was therefore estimated from the variance of spatially binned difference images as the bin size increased (Ding et al., 6 Mar 2026).

Several works emphasize that zero-frequency DQE is especially sensitive to methodological details. In the 300 keV electron-detector comparison, apparent $f_a$1 values inferred from per-event signal distributions were $f_a$2 for DE-20, $f_a$3 for Falcon II, and $f_a$4 for K2, but the authors cautioned that these values were highly sensitive to systematic effects such as truncation, saturation, and event selection, especially for K2 (McMullan et al., 2014). In the false-event analysis, QE cannot be measured directly when process-induced false events are indistinguishable from true events, so QE must instead be computed analytically from the interaction physics (Zanella, 2012).

SEM work illustrates a different measurement philosophy. Instead of assuming Poisson-distributed secondary-electron emission and extracting DQE from image SNR, the histogram-based method directly counts the mean number of secondary electrons detected per pixel dwell and defines

$f_a$5

where $f_a$6 is beam current, $f_a$7 dwell time, $f_a$8 the electron charge, and $f_a$9 the total secondary-electron yield of the sample pixel (Agarwal et al., 2020). The paper argues that this avoids incorrect DQE values caused by the common Poisson approximation for secondary-electron emission.

For integrating neutron image plates, the methodology is fully cascaded-statistical rather than frequency-domain. The output is the number of electrons collected at the PMT anode during laser readout, and the DQE is the zero-frequency ratio of squared output to input SNR. The final expression contains neutron absorption efficiency, variance in the number of photostimulable centers, photon transport, photocathode efficiency, and PMT multiplication noise (1901.10380).

Across these modalities, the dominant uncertainties arise from fluence calibration, low-frequency noise estimation, threshold dependence, aliasing, and the treatment of correlated or false events.

6. Comparative performance and applications

At 300 keV in low-dose electron microscopy, all three direct backthinned CMOS detectors studied—DE-20, Falcon II, and K2 Summit—had higher DQE than Kodak SO-163 film across the full frequency range. The K2 Summit had the highest DQE at low spatial frequencies and was therefore advantageous for alignment and tomography, whereas Falcon II became superior at high spatial frequencies beyond about $\sigma_d^2$0 Nyquist (McMullan et al., 2014). This comparison is one of the clearest demonstrations that application-specific frequency bands matter more than a single headline detector ranking.

At lower electron energies, Medipix3 in charge summing mode approached ideal square-pixel behavior. At 60 keV, Nyquist MTF of about $\sigma_d^2$1 was reported at low $\sigma_d^2$2, close to the ideal $\sigma_d^2$3, and the DQE curves lay within about $\sigma_d^2$4 of the ideal $\sigma_d^2$5 reference across the band; at 80 keV, DQE remained within about $\sigma_d^2$6 of ideal (Mir et al., 2016). In a complementary Medipix3 study spanning 60 to 200 keV in single-pixel mode, the theoretical square-pixel DQE at Nyquist of about $\sigma_d^2$7 was experimentally achieved only for 60 keV electrons, with progressive degradation at higher energies (Mir et al., 2017).

Timepix4 extends the high-efficiency regime into event-driven hybrid-pixel TEM. In raw event readout mode, $\sigma_d^2$8 at 100 kV and $\sigma_d^2$9 at 200 kV, while $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$0 at 100 kV and $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$1 at 200 kV (Ding et al., 6 Mar 2026). The same work demonstrated that, despite near-zero real-space Nyquist DQE at 200 kV, parallel-beam diffraction still showed weak diffracted information beyond a $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$2 mrad half-angle because high $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$3 and favorable NNPS supported angularly averaged weak-signal detection (Ding et al., 6 Mar 2026).

In photon-counting CT, task dependence is decisive. For a 300 mm object at 120 kVp, CdTe detectors with $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$4 pixels had $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$5–$$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$6 higher DQE than the 60 mm Si system with tungsten for detection tasks, whereas for two-material decomposition the corresponding numbers ranged from $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$7 lower to $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$8 higher DQE; the $$\mathrm{DQE}=\frac{\mathrm{QE}}{1+f+\frac{f_a+\sigma_d^2}{\mathrm{QE}\,S_i}}.$$9 CdTe design was $\mathrm{DQE}(\mathbf{u},E,E')$0–$\mathrm{DQE}(\mathbf{u},E,E')$1 lower in DQE for two-material decomposition compared to Si (Persson et al., 2020). The Monte Carlo assessment of four photon-counting detector concepts reached a closely related conclusion: CdTe direct conversion gave strong non-spectral DQE, Si had an advantage for true spectral decomposition at low spatial frequencies, coincidence counters restored much of the spectral performance lost in plain CdTe, and LaBr$\mathrm{DQE}(\mathbf{u},E,E')$2 optical counting gave the strongest true spectral DQE among the systems studied at low spatial frequencies (Stierstorfer et al., 2024).

Outside x-ray and electron imaging, DQE remains a compact summary of stochastic efficiency. For a commercial neutron image plate BAS-IP ND 2025, the measured DQE was $\mathrm{DQE}(\mathbf{u},E,E')$3 at $\mathrm{DQE}(\mathbf{u},E,E')$4, while the simplified theoretical estimate was about $\mathrm{DQE}(\mathbf{u},E,E')$5, with the dominant limitation attributed to fluctuations in energy deposition from natural gadolinium capture products (1901.10380). For the latest ALD-coated MCP-PMTs, the reported single-photon DQE, defined as $\mathrm{DQE}(\mathbf{u},E,E')$6, reached about $\mathrm{DQE}(\mathbf{u},E,E')$7 in devices with QE near $\mathrm{DQE}(\mathbf{u},E,E')$8 and CE around $\mathrm{DQE}(\mathbf{u},E,E')$9–$$\mathrm{DQE}(f)=\mathrm{SNR}_{\mathrm{o}}^2(f)/\mathrm{SNR}_{\mathrm{i}}^2(f)$$00 (Miehling et al., 2023).

Taken together, these results show that DQE is best understood not as a single universal scalar, but as a rigorously normalized measure of how efficiently a specified detection chain converts incident quanta into usable information. In some settings it is a frequency-dependent imaging metric; in others it is a task-dependent multichannel information metric; in still others it is an overall single-photon detection probability. The common principle is unchanged: DQE measures how much of the ideal input SNR survives the combined effects of transfer, noise, and stochastic loss.