Download Generalized DQE analysis of radiographic and dual‐energy imaging

Generalized DQE analysis of radiographic and dual-energy imaging
using flat-panel detectors
S. Richard
Department of Medical Biophysics, University of Toronto, Ontario, Canada M5G 2M9
J. H. Siewerdsena兲 and D. A. Jaffray
Ontario Cancer Institute, Princess Margaret Hospital, Ontario, Canada M5G 2M9
Departments of Medical Biophysics and Radiation Oncology, University of Toronto, Ontario,
Canada M5G 2M9
D. J. Moseley and B. Bakhtiar
Ontario Cancer Institute, Princess Margaret Hospital, Ontario, Canada M5G 2M9
共Received 22 October 2004; revised 31 January 2005; accepted for publication 16 March 2005;
published 26 April 2005兲
Analysis of detective quantum efficiency 共DQE兲 is an important component of the investigation of
imaging performance for flat-panel detectors 共FPDs兲. Conventional descriptions of DQE are limited, however, in that they take no account of anatomical noise 共i.e., image fluctuations caused by
overlying anatomy兲, even though such noise can be the most significant limitation to detectability,
often outweighing quantum or electronic noise. We incorporate anatomical noise in experimental
and theoretical descriptions of the “generalized DQE” by including a spatial-frequency-dependent
noise-power term, SB, corresponding to background anatomical fluctuations. Cascaded systems
analysis 共CSA兲 of the generalized DQE reveals tradeoffs between anatomical noise and the factors
that govern quantum noise. We extend such analysis to dual-energy 共DE兲 imaging, in which the
overlying anatomical structure is selectively removed in image reconstructions by combining projections acquired at low and high kVp. The effectiveness of DE imaging in removing anatomical
noise is quantified by measurement of SB in an anthropomorphic phantom. Combining the generalized DQE with an idealized task function to yield the detectability index, we show that anatomical
noise dramatically influences task-based performance, system design, and optimization. For the
case of radiography, the analysis resolves a fundamental and illustrative quandary: The effect of
kVp on imaging performance, which is poorly described by conventional DQE analysis but is
clarified by consideration of the generalized DQE. For the case of DE imaging, extension of a
generalized CSA methodology reveals a potentially powerful guide to system optimization through
the optimal selection of the tissue cancellation parameter. Generalized task-based analysis for DE
imaging shows an improvement in the detectability index by more than a factor of 2 compared to
conventional radiography for idealized detection tasks. © 2005 American Association of Physicists
in Medicine. 关DOI: 10.1118/1.1901203兴
Key words: dual-energy imaging, anatomical noise, flat-panel detectors, detective quantum
efficiency, noise-power spectrum, detectability index, generalized DQE, thoracic imaging, lung
nodules, imaging performance
I. INTRODUCTION
Lung cancer is the leading cause of cancer death, more lethal
than the next three most common cancers combined.1–4 The
key to survival is early detection—catching the disease at an
early stage, when nodules are typically less than 3 cm in
diameter and have not metastasized. Traditional chest radiography performs poorly in the detection of lung nodules,
missing approximately 30% on first read and rarely identifying tumors less than 16 mm in diameter.5 The main reason
for radiographic insensitivity is the lack of nodule conspicuity caused by overlying anatomical structures in the twodimensional 共2D兲 projection image.6 Imaging strategies for
improving conspicuity include depth discrimination 共e.g.,
tomosynthesis7,8 or computed tomography兲9 and tissue discrimination 关e.g., dual-energy 共DE兲 imaging兴.10–13 The work
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Med. Phys. 32 „5…, May 2005
herein considers the latter, advancing concepts central to understanding and improving imaging performance in advanced imaging techniques.
A practical prevalent approach to imaging performance
characterization focuses on the measurement and modeling
of spatial resolution and noise in terms of the modulation
transfer function 共MTF兲, noise-power spectrum 共NPS兲,
noise-equivalent quanta 共NEQ兲, and detective quantum efficiency 共DQE兲.14 Such metrics provide a valuable approach to
system characterization and optimization, with widespread
application in the development of novel imaging systems.
One limitation of this conventional approach, however, is
that the resulting metrics for imaging performance 共viz.,
NEQ and DQE兲 do not account for “anatomical noise” resulting from background and overlying anatomy. Such effects can be incorporated as a spatial-frequency-dependent
0094-2405/2005/32„5…/1397/17/$22.50
© 2005 Am. Assoc. Phys. Med.
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Richard et al.: Generalized DQE analysis
noise term in the “generalized DQE” as described by Barrett
et al.15,16 共who defined the generalized NEQ兲, providing a
more complete description of image quality by quantifying
the quantum noise, electronic noise, and anatomical noise in
performance descriptors. 共Note that the term “generalized”
here refers specifically to inclusion of anatomical background noise in the DQE.兲 Analysis of the generalized DQE
offers an understanding of the physical factors limiting the
detector signal and noise transfer characteristics in relation to
the spatial frequencies constituting the background noise and
can be extended to task-based performance measures, such
as the detectability index.17 Analysis of background noise in
mammography18–20 and more recently thoracic imaging,6
provides a valuable starting point for considering the imaging performance of the advanced applications of flat-panel
detectors 共FPDs兲, such as DE imaging, tomosynthesis, and
cone-beam computerized tomography 共CT兲.21
DE imaging offers a sensitive, specific, cost-effective, and
seemingly underutilized modality for thoracic imaging.
While not a new technique, DE imaging improves conspicuity and reduces anatomical noise by processing two images
acquired at two beam energies to reconstruct an image in
which soft tissue or bone is selectively cancelled, thus removing overlying anatomical structures from the image.
Studies have shown that DE imaging increases sensitivity
and specificity in detection of small noncalcified pulmonary
nodules and provides excellent characterization of calcified
nodules, an important indicator of nodule benignancy.12,13,22,23 FPDs are well suited to DE imaging, since
they are intrinsically digital and offer fast readout capabilities, high DQE, and superior signal-to-noise ratio performance in the processed DE images.24
In the sections below, we first advance a theoretical methodology for the incorporation of anatomical background
noise in cascaded systems analysis 共CSA兲 of the generalized
DQE. Second, we extend theoretical CSA modeling25 to DE
imaging, modeling weighted log-subtraction DE reconstruction as a deterministic operation performed on two independent projections and governed by the tissue cancellation parameter, wt. Third, we extend the CSA model for DE imaging
to include residual anatomical noise, yielding the generalized
DQE for a DE imaging system. These theoretical methods
for the determination of DQE are in turn extended to the
evaluation of task-based figures of merit—specifically, the
detectability index17—by considering a simple spatialfrequency-dependent task function in combination with the
generalized DQE.
II. THEORETICAL METHODS
A. Cascaded systems analysis
CSA provides a powerful analytical tool that describes the
signal and noise transfer characteristics of imaging systems
in a manner that is physically intuitive and identifies the
factors that limit imaging performance.25 Investigators have
employed such analysis extensively for FPDs,26,27 and other
imaging systems,28–31 showing that the NPS and DQE can be
predicted analytically over a broad range of detector configuMedical Physics, Vol. 32, No. 5, May 2005
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rations and imaging conditions. Assumptions inherent to
CSA include linearity 共mean response is linear as a function
of input quanta and obeys the superposition of two or more
inputs兲, shift invariance 共image is independent of the location
of the input兲, and stationarity 共first- and second-order
statistics—e.g., the mean, variance, and NPS—are spatially
and temporally invariant兲. Although real physical imaging
systems never completely satisfy these requirements, they
are typically assumed to hold over a range of relevant conditions. FPDs can be considered linear over small ranges in
exposure variation and are known to be highly linear32 over
broad exposure ranges up to at least 50% of pixel saturation.
While such systems are inherently shift variant to some degree, shift invariance in detector response is improved by
high pixel fill factor and presampling blur in the x-ray converter, both of which are the case for the indirect-detection
FPD employed below. Furthermore, Cunningham et al.33
demonstrated the applicability of CSA in describing signal
and noise performance in systems such as FPDs under conditions of wide-sense cyclostationarity 共i.e., system response
invariant under periodic shifts in the spatial domain, as with
a periodic pixel matrix兲.
CSA represents each physical process in the imaging
chain as a gain stage 共e.g., conversion of x rays into optical
photons兲, a spatial spreading 共blurring兲 stage 共e.g., spreading
of optical photons in the scintillator兲, or a sampling stage
共e.g., readout of the detector signal at locations according to
the pixel matrix兲. At each stage of the imaging process, the
output is determined as a function of the input by transfer
equations derived by Rabbani et al.34 under the assumption
that the detector pixels can be considered arbitrarily small
共i.e., in the limit that the image signal is a continuous function, with transfer functions determined by the Fourier transform兲. Similarly for discrete data and the discrete Fourier
transform,35,36 the noise at stage i can be described by the
NPS, Si共u , ␯兲, where u and ␯ are spatial frequency coordinates. The mean signal or fluence is denoted q̄i. CSA modeling of FPD performance has been reported for a variety of
detector designs and imaging applications.26–29,37–39 A brief
description of the CSA model employed herein is provided
below, with a more detailed description of each stage given
in Appendix A. The model is similar to previously reported
approaches, modified in two important respects 共account of
variation in optical gain in branches of the parallel cascade,
and extension to DE image reconstruction兲.
Stage 0 describes the mean fluence of quanta, q̄0, incident
on the detector. The incident x-ray energy spectrum, q0共E兲,
was generated for a given kVp 共50–150 kVp兲 and added
filtration using the Spektr40 toolkit, based on data from the
TASMIP method of Boone and Seibert.41 At Stage 0, the
fluence per unit exposure 共q̄0 / X兲 is computed by integrating
the energy-dependent fluence per unit exposure42 with the
normalized incident spectrum. Stage 1 involves the quantum
detection efficiency, ḡ1, describing the probability that an x
ray will interact in the scintillator or photoconductor 共indirect or direct detection, respectively; henceforth, we consider
the former.兲 Stage 2 represents the conversion of x rays to
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optical photons, characterized by a mean optical gain, ḡ2. K
fluorescence in the scintillator was included in the model by
means of parallel-cascaded systems analysis as described by
Yao and Cunningham.43 The description was extended in this
work to include variance in the conversion gain, as opposed
to a deterministic or Poisson-distributed gain.29,30,43 共See Appendix A for details.兲 Depth dependence of energy deposition was not accounted for, although previous authors have
done so,44–46 showing that such dependence tends to increase
the NPS. Depth-dependent effects are accommodated in part
in a semiempirical approach through the use of energyaveraged and depth-averaged quantities, such as the experimental MTF, optical gain, etc., and is the subject of ongoing
investigation in detector performance modeling. Stage 3 is a
stochastic spreading stage, characterized by the scintillator
MTF, T3共u , ␯兲, and describing the blur of optical photons in
the scintillator. Stage 4 represents the conversion of optical
photons to electrons in the photodiode, described by the
quantum efficiency, ḡ4. Stage 5 is a deterministic spreading
stage corresponding to the integration of quanta by the photodiode and described by the presampling pixel MTF,
T5共u , ␯兲. Stage 6 represents the sampling of the detector signal and is characterized by the pixel pitch, apix, photodiode
2
2
aperture, apd, and fill factor, ff= apd
/ apix
. Finally, Stage 7
represents electronic readout with an additive noise term due
to the pixel dark noise, amplifier noise, and digitization.
S共u, ␯兲det
S⬘共u, ␯兲det
=
.
S共u, ␯兲actual S⬘共u, ␯兲actual
DQE共u, ␯兲 =
共3兲
This form, which of course is equivalent to Eq. 共1兲, is used in
derivations below, where
S⬘共u, ␯兲det =
T2共u, ␯兲
q̄0
共4兲
.
Although the conventional definition of the DQE is effective in describing quantum noise in an imaging system, it is
limited in that it takes no account of the background anatomical noise. Anatomical noise may far outweigh the quantum noise and electronic noise in an image; therefore, anatomical noise can be the most important factor in limiting
detectability6—for example, in the detection of lung nodules
in the presence of overlying ribs in a chest radiograph. For
this reason, it becomes worthwhile to include an additional
NPS term, SB⬘ 共u , ␯兲, corresponding to the NPS of image fluctuations
associated
with
background
anatomical
structure.16,48,49 Again, the relative NPS due to anatomical
fluctuations was obtained by dividing the NPS of each region
of interest 共ROI兲 by the mean signal squared of each ROI. A
simple empirical form for the anatomical NPS models the
spatial frequency dependence in proportion to a 1 / f
characteristic:18,50–52
SB⬘ 共u, ␯兲 ⬵
K
,
f␤
共5兲
where f is a radial spatial frequency term:
f = 冑u2 + ␯2 ,
B. The generalized DQE
The DQE is an important figure of merit in imaging performance analysis, which describes the spatial-frequencydependent signal and noise transfer characteristics of an imager. It can be described as a function of the experimentally
determined MTF and NPS along with an estimate of the
incident quantum fluence:
DQE共u, ␯兲 =
T2共u, ␯兲
q̄0S⬘共u, ␯兲
,
共1兲
where S⬘共u , ␯兲 contains the quantum and electronic noise of
the system, T共u , ␯兲 is the system MTF, and q̄0 is the mean
fluence of quanta at the detector. The prime symbol on the
NPS signifies the relative NPS, i.e., the absolute NPS,
S共u , ␯兲, divided by the mean signal squared:
S⬘共u, ␯兲 =
S共u, ␯兲
2
signal
.
共2兲
This experimental description of the DQE can be reformulated in a stochastic form47 well suited to CSA, in which
the actual NPS, S共u , ␯兲兩actual, is related to the deterministic
NPS, S共u , ␯兲兩det, 共i.e., the NPS for an idealized system that is
identical to the real system except that each stage is deterministic and adds no noise in the course of image propagation兲:
Medical Physics, Vol. 32, No. 5, May 2005
共6兲
and K and ␤ quantify the magnitude and frequency dependence of the anatomical noise, respectively. It should be
pointed out that 1 / f noise is a ubiquitous purely empirical
model drawn from various fields of science to describe various types of stochastic phenomena and is not specific to
medical imaging; it provides a useful model to quantify anatomical noise that gives an excellent fit to the measured results. Hence, it becomes possible to define a figure of merit
that takes into account both the quantum noise and the anatomical noise. The generalized DQE, denoted GDQE, is defined by reformulating the conventional DQE:
DQE共u, ␯兲 =
T2共u, ␯兲
共7兲
⬘ 共u, ␯兲 + Sadd
⬘ 兲
q̄0共SQ
in a generalized form that includes background anatomical
noise SB⬘ 共u , ␯兲,
GDQE共u, ␯兲 =
T2共u, ␯兲
⬘ 共u, ␯兲 + SB⬘ 共u, ␯兲T2共u, ␯兲 + Sadd
⬘ 兲
q̄0共SQ
,
共8兲
where SQ
⬘ 共u , ␯兲 is the quantum noise, SB⬘ 共u , ␯兲 is the background anatomical noise6 关e.g., empirically determined as in
Eq. 共5兲兴, and Sadd
⬘ is the electronic noise. In the denominator,
T共u , ␯兲 is written separate to SB⬘ 共u , ␯兲 so that the latter describes the frequency content of structures in the object,
taken as a source of stochastic variation independent of the
imaging system. This generalization includes the attenuation
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and spatial modulation of the incident quanta by incorporating the spatial-frequency-dependent anatomical noise term.
The present analysis does not account for x-ray scatter, although other work has begun to incorporate such in the
DQE.53,54
C. Extension to DE imaging
DE imaging is a technique in which two images acquired
at different energies are processed to reconstruct images in
which bone or soft tissue are selectively canceled. It exploits
differences in the probability of photoelectric and Compton
interactions in the object as a function of x-ray energy and
atomic number, with the photoelectric cross section exhibiting a stronger energy and Z dependence. Therefore, an image
acquired at lower energy 共e.g., 60 kVp兲 will have higher
bone contrast than an image acquired at a higher energy
共e.g., 120 kVp兲 due to calcium in the bone. A common algorithm for DE image reconstruction, derived from the straightforward manipulation of Beer’s law, is weighted log subtraction. While somewhat more sophisticated reconstruction
techniques are available, such as basis decomposition,11 log
subtraction is considered in the present analysis because it
lends itself well to CSA and has been shown to provide
roughly equivalent DE image quality to basis
decomposition.55 It can be shown that a soft-tissue-only image can be generated from a low- and high-energy image
as:55
IDE共x,y兲 =
IL 共x,y兲
wt
IH共x,y兲
,
共9兲
where DE, H , L denote the “soft-tissue” dual-energy, highenergy, and low-energy images, respectively. The tissue cancellation parameter, wt, is a freely variable factor that
weights the contribution of the high- and low-energy images
in the reconstruction. A similar expression can be written for
the “bone-only” DE image. From the flood-field NPS of the
low- and high-energy images, SL⬘ 共u , ␯兲 and SH
⬘ 共u , ␯兲, respectively, the flood-field NPS of the DE image is given by:
⬘ 共u, ␯兲 = w2t SL⬘ 共u, ␯兲 + SH⬘ 共u, ␯兲,
SDE
共10兲
as shown in Appendix B and illustrated in terms of CSA in
Fig. 1.
D. The generalized DE image DQE
From the DE NPS, it is possible to derive the generalized
DE DQE, denoted as GDQEDE.
FIG. 1. Flow-chart representation of NPS propagation for DE imaging. The
high- and low-energy NPS are computed independently and combined as a
function of the DE reconstruction tissue cancellation parameter, wt, to yield
the DE image NPS. The gray boxes correspond to the cascaded systems
processes in Fig. 14, computed for low- and high-energy images.
⬘ 共u, ␯兲det = w2t
SDE
TL2 共u, ␯兲
q̄0L
+
2
共u, ␯兲
TH
q̄0H
.
共12兲
Rearranging terms, we obtain:
⬘ 共u, ␯兲det =
SDE
2
共u, ␯兲
TH
q̄0DE
共13兲
,
where an “effective” DE image fluence is defined as:
1
q̄0DE共u, ␯兲 = q̄0H
1+
T2 共u, ␯兲 q̄0H
w2t 2L
TH共u, ␯兲 q̄0L
.
共14兲
This term is adopted for notational convenience and should
not be considered a fluence in the usual sense. It is, in general, spatial-frequency dependent, but offers a description of
the resultant fluence that is obtained when the low- and highenergy images are combined in DE image reconstruction.
The spatial-frequency-dependence in q̄0DE共u , ␯兲 arises due to
possible differences in the spatial-frequency transfer characteristics 共i.e., MTF兲 at low and high energy. Under the simplifying assumption that the MTF for the high- and lowenergy images are the same:
TL共u, ␯兲 = TH共u, ␯兲 = T共u, ␯兲,
共15兲
the effective DE fluence becomes:
1. The DE “deterministic” NPS
From Eq. 共10兲, for a DE soft-tissue-only image:
⬘ 共u, ␯兲det = w2t SL⬘ 共u, ␯兲det + SH⬘ 共u, ␯兲det .
SDE
共11兲
Substituting the low- and high-energy deterministic NPS
gives:
Medical Physics, Vol. 32, No. 5, May 2005
1
q̄0DE = q̄0H
1+
q̄0
w2t H
q̄0L
.
共16兲
Note that q̄0DE reduces to q̄0H in the special case wt = 0 关i.e.,
nullification of the low-energy component in IDE共x , y兲兴.
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2. The DE “actual” NPS
d⬘ =
The DE NPS can be written as:
⬘ 共u, ␯兲,
⬘ 共u, ␯兲actual = w2t SL⬘ 共u, ␯兲actual + SH⬘ 共u, ␯兲actual + SDE,B
SDE
共17兲
where S⬘共u , ␯兲兩actual contains both quantum and electronic
noise for the low- or high-energy image, and SDE,B
⬘ 共u , ␯兲 is
the residual background anatomical noise for the DE image.
If we assume that the additive electronic noise is the same in
both the high- and low-energy images,
⬘ 共u, ␯兲 + Sadd
⬘ 共u, ␯兲
⬘ 共u, ␯兲actual = w2t 共SL,Q
⬘ 共u, ␯兲兲 + 共SH,Q
SDE
⬘ 共u, ␯兲
⬘ 共u, ␯兲兲 + SDE,B
+ Sadd
⬘ 共u, ␯兲 + SDE,add
⬘
⬘ 共u, ␯兲,
= SDE,Q
共u, ␯兲 + SDE,B
共18兲
where we have grouped terms such that the DE NPS is expressed in terms of a DE quantum noise component:
⬘ 共u, ␯兲 = w2t SL,Q
⬘ 共u, ␯兲 + SH,Q
⬘ 共u, ␯兲,
SDE,Q
共19兲
and a DE additive noise component:
⬘
⬘ 共w2t + 1兲.
SDE,add
= Sadd
共20兲
3. The generalized DE DQE:
Combining the deterministic and actual DE NPS yields
the generalized DQE 关see Eq. 共3兲兴 for a DE soft-tissue image:
GDQEDE共u, ␯兲 =
T2共u, ␯兲
⬘ 共u, ␯兲 + SDE,B
⬘ 共u, ␯兲 + SDE,add
⬘
q̄0DE共SDE,Q
兲
.
共21兲
This expression lends itself to experimental and/or theoretical analysis in a manner similar to the conventional or generalized DQE for radiography. Each of the terms may be
experimentally determined, as shown below, or estimated
theoretically via CSA 共or semiempirically, in the case of
SDE,B
⬘ 兲. Moreover, it is generalized by inclusion of anatomical
noise, which captures the improvements in imaging performance for a DE image.
冕 冕
uNyq
␯Nyq
−uNyq
−␯Nyq
DQE共u, ␯兲W2Task共u, ␯兲dud␯ .
共22兲
Written this way, the detectability index neatly separates
properties of the imaging system 共DQE兲 from aspects of the
imaging task 共WTask兲. The detectability index computed in
proportion to the integral over the DQE 共as opposed to the
NEQ兲 can be understood as a detectability index per unit
fluence. In the calculations below, the detectability index was
computed from the DQE and task function integrated along
the u = ␯ diagonal for calculational convenience and up to the
Nyquist frequency, f Nyq. This one-dimensional 共1D兲 integration is proportional to the 2D integration 关Eq. 共22兲兴 and
therefore does not change any of the results presented below.
Integration over the Nyquist region is appropriate in that the
imaging system is sampled, and the measured NPS includes
aliasing effects associated with “folding” of noise from
higher frequencies inside the Nyquist region.
A simple detection task was considered which weights all
spatial frequencies equally. While other frequency-dependent
task functions are certainly worth consideration 共e.g., a
Gaussian or Bessel function, associated with detection of a
Gaussian or circ object, respectively兲, task functions associated with real observer performance likely involve both low
and high frequencies 共associated with the size and edge of
the nodule, respectively兲. As a hypothesis-testing task, the
uniform task function corresponds to differentiation between
signal-present 共a delta function兲 and signal-absent 共noiseonly兲 cases,17 WTask共f兲 = constant. The simple constant task
function is chosen primarily for calculational convenience in
the current manuscript, and consideration of more complex
task functions is the subject of future work. In this simple
case, the detectability is proportional to the area under the
DQE. Two important applications were made of the detectability index in this work: The first was to compare the detectability index computed for chest radiography as a function of kVp using the conventional DQE and the generalized
DQE; the second application was in optimization of the tissue cancellation parameter in DE imaging of the chest.
III. EXPERIMENTAL METHODS
A. Measurement of ⌫, MTF, NPS, and DQE
1. Experimental setup
E. Task-based figure of merit: detectability index
The Fourier transform of an object relative to background
defines the imaging task in the form of idealized, modelobserver task functions conveying the spatial frequencies of
interest in performing a given task. For a given imaging task,
such as a detection or discrimination of a given structure
against a uniform or noisy background,17 a spatialfrequency-dependent task function, WTask共f兲, can be defined.
The DQE may in turn be combined with the task function to
yield the detectability index, d⬘, providing a practical metric
for task-based system evaluation, optimization, and investigation of model observer performance:21,54,56,57
Medical Physics, Vol. 32, No. 5, May 2005
An imaging bench was constructed 共See Fig. 2兲 as an
experimental platform for advanced applications of FPDs,
including DE imaging, tomosynthesis, and CBCT. The imaging bench consists of an x-ray tube 共Rad 94 in a Sapphire
housing; W target; 0.4–0.8 mm focal spot; 14° anode angle;
Varian Medical Systems, Salt Lake City UT兲 powered by a
constant potential generator 共CPX 380, EMD Inc., Montreal
QC兲. The FPD 共RID-1640A, PerkinElmer Optoelectronics,
Santa Clara CA兲 has a 1024⫻ 1024 共41⫻ 41 cm2兲 active matrix of a-Si: H photodiodes and thin-film transistors with a
400 ␮m pixel pitch, 80% fill factor, and a 250 mg/ cm2
CsI:TI x-ray converter. A computer control system 共3GHz
Pentium-4 processor with 2 GB random access memory兲
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4. NPS
FIG. 2. Experimental bench. 共1兲 Kilovoltage x-ray tube with collimator. 共2兲
Flat-panel detector. 共3兲 Adjustable framework and motion control system
mounted on an optical bench. The imaging bench was developed as an
experimental proving ground for imaging performance in DE imaging, tomosynthesis, and cone-beam CT.
provides synchronized x-ray exposure and FPD readout. A
motion control system 共6 K series with Gemini drives,
Parker Daedal, Harrison PA兲 allowed precise and reproducible adjustment of system geometry.
2. The system gain, ⌫ „sensitivity…
The gain, ⌫, was determined by measuring the mean signal level per unit exposure in 50 “flood” images. A silicon
diode 共R100 detector with Barracuda exposure meter; RTI
Electronics, Molndal Sweden兲 placed on the detector 共SDD
= 144 cm兲 was used to measure the exposure. ⌫ was determined from linear fits to mean signal versus exposure. The
ROI was 128⫻ 128 pixels at the center of the panel, and fits
were performed on measurements up to 50% saturation level,
where the detector response is highly linear.
3. MTF
The MTF was measured by placing a precision-machined
straight Pb edge 共2 mm thick兲 directly on the detector at a
slight angle 共⬃5 ° 兲. Fifty projections at ⬃50% detector signal saturation were acquired and averaged to reduce the effect of x-ray quantum noise. Approximately 50 realizations
共10 rows⫻ 500 columns兲 were obtained in each measurement. For each realization, a shift and add technique was
performed to oversample the step function produced by the
edge, as described by Fujita et al.58 This technique generated
an oversampled edge-spread function from which the derivative yielded the line-spread function 共LSF兲. A median filter
共sliding window, finite impulse response filter兲 was applied
to smooth the tails of the LSF, removing some of the highfrequency noise introduced by differentiation. For each realization, the Fourier transform of the 1D LSF 共normalized to
unit area兲 yielded the MTF. The average of all the MTFs
共⬃50兲 for each realization gave the final measured MTF.
Medical Physics, Vol. 32, No. 5, May 2005
The NPS was measured from 100 “flood” images, which
were gain and offset corrected to account for stationary
variations by the mean of 50 flood and “dark” fields acquired
with and without the exposure of x rays to the FPD, respectively. The panel was read 15 times between flood projections in order to minimize correlations due to image lag.
Each nonoverlapping realization was 100⫻ 100 pixels in
size. Approximately 8000 realizations formed the ensemble.
The relative NPS was computed by normalizing the fast Fourier transform squared of each realization by the physical
area of the realization and dividing by the mean signal
squared.35,59 The mean of the 8000 NPS estimates gave the
final NPS, with error bars provided by the sample standard
deviation. The NPS was measured at 60, 80, 100, 120, and
140 kVp, with 2 mm Al and 0.6 mm Cu filtration. The mAs
共12.5, 2.5, 0.8, 0.5, and 0.4, respectively兲 was set so that
⬃50% signal level was observed on the detector. The R100
photodiode was placed directly on the detector 共SDD
= 144 cm兲 near the center of the panel and just above the
field of view for measurement of the exposure.
5. DQE
The measured DQE was determined from the measured
MTF and the measured relative NPS. The classical description of the DQE was used, as in Eq. 共1兲:
DQE共u, ␯兲 =
MTF2共u, ␯兲
q̄0NPS⬘共u, ␯兲
.
共23兲
The mean fluence was computed from the measured exposure and fluence per unit exposure, q0 / X, computed using
Spektr.40
B. Measurement of anatomical NPS and the
generalized DQE
The anatomical NPS was measured in a manner similar to
that of the flood-field NPS except that real patient or anthropomorphic phantom radiographs formed the image data.
Analysis was restricted to a region of interest in the lung.
The two sets of data included clinical patient data 共University Health Network, Toronto ON兲, where 90 patients were
chosen randomly from the clinical database. The chest radiographs were acquired at 140–150 kVp using a Siemens FD-X
digital chest unit 共⬃200 ␮m pixel pitch兲. A second set of
data was acquired on the imaging bench described above
using an anthropomorphic chest phantom 共Fig. 2兲.60 The
phantom consists of a modified Rando™ phantom with a
human skeleton, a custom-formulated lung material formed
of microbubble-infused polyurethane, and embedded spheres
of various diameter and composition 共intended to simulate
lung nodules兲. The anatomical NPS measured in the patient
population was compared to that in the chest phantom as a
means to evaluate the appropriateness of the phantom for
realistic anatomical NPS estimates. The use of the phantom
was mainly motivated by the flexibility offered in the wide
range of conditions available for anatomical noise measure-
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⬘ 共u, ␯兲 = SQ共u, ␯兲 + SB共u, ␯兲T2共u, ␯兲 + Sadd
Stot
⬵ k1T2共u, ␯兲 +
FIG. 3. 共a兲 Example CR image of a chest patient 共140 kVp兲. 共b兲 Example
radiograph of the Princess Margaret Hospital 共PMH兲 chest phantom 共see
Ref. 60兲 acquired on the imaging bench 共see Fig. 2兲. The highlighted regions
correspond to realizations from which the NPS analysis was performed. The
realizations were each 100⫻ 100 pixels and 40⫻ 40 pixels for the CR patient images and the phantom radiographs, respectively.
ments and because it provides a useful tool to test the methodology presented in this paper. The variation of the anatomical NPS within each patient was compared to the
variations observed across the patient population as a check
on variation relative to population variations.
The total anatomical NPS measurement was performed
directly on the lung region of the images as illustrated in Fig.
3. The regions of the lungs were manually delineated and
randomly divided into nonoverlapping realizations. The unhighlighted “holes” among the realizations are a result of the
filling algorithm, which places equally sized ROIs throughout the irregular boundaries of the delineated lung. Randomness in placement of the realizations was found to improve
the extent to which the system obeys weak stationarity by
randomizing the location of the realizations with respect to
the structured anatomical noise. The clinical CR data were
⬃2100⫻ 2100 pixel format, and the size of the realizations
were 100⫻ 100 pixels. The size of the realizations was chosen to be approximately the same size or greater than the
structured fluctuations arising from overlying anatomy. This,
in combination with random placement of the ROIs, provided improved stationarity 共mean and standard deviation
equal for all realizations within experimental error兲. While
smaller realizations would have provided a larger number of
ROIs and improved statistical accuracy in the resulting NPS
estimate, smaller ROIs were found to increase the positiondependent mean and standard deviation. The number of realizations varied from approximately 100 to 140 realizations
per patient. The format of the image data acquired using the
chest phantom was smaller 共1024⫻ 1024 pixels兲; therefore,
to obtain realizations corresponding to approximately the
same size with respect to the anatomy as in the CR data, the
realizations were 40⫻ 40 pixels 关see Fig. 3共b兲兴. Fifty radiographs of the chest phantom were acquired with ⬃2500 realizations used to compute the anatomical NPS.
To extract a semiempirical estimate of SB共u , ␯兲 the total
measured relative anatomical NPS was fit to the following
empirical form:
Medical Physics, Vol. 32, No. 5, May 2005
K 2
T 共u, ␯兲 + k2 ,
f␤
共24兲
where k1 , k2 , K, and ␤ are fitting parameters. The k1T2共u , ␯兲
term approximates the quantum noise as proportional to
T2共u , ␯兲. The K / f ␤ term models the anatomical noise, with
the measured MTF of the FPD included outside SB共u , ␯兲 so
that the measurements of K and ␤ are independent of the
imaging system 共i.e., a property of the anatomical structures
alone兲. The k2 term approximates the electronic noise 共taken
as white兲. The four parameters were fit to the ensemble averaged NPS. The terms K and ␤ are thus derived as empirically determined parameters in the generalized DQE,
GDQE共u, ␯兲 =
T2共u, ␯兲
,
K 2
⬘ 共u, ␯兲 + ␤ T 共u, ␯兲 + Sadd
⬘
q̄0 SQ
f
冋
册
共25兲
where SQ
⬘ 共u , ␯兲 is computed using cascaded systems analysis
共Appendix A兲, Sadd
⬘ is modeled as a white NPS from measurements of pixel dark noise and dark-field NPS, q̄0 is determined using Spektr, and the MTF, T共u , ␯兲, is obtained
from a single-parameter Lorentzian fit27 to the measured
MTF.
C. Measurement of the dual-energy image NPS,
anatomical NPS, and generalized DQE
The DE image NPS was analyzed in two ways to verify
Eq. 共10兲. The first was by computing the NPS of the low- and
high-energy images separately and then combining them according to the tissue cancellation parameter as in Eq. 共10兲 to
obtain the DE NPS. The second method was to reconstruct
DE flood images from low- and high-energy flood images
关Eq. 共9兲兴 and compute the NPS directly from the DE floods.
The high-energy kVp technique was 120 kVp with 1.1 mm
Cu added filtration in order to increase the separation between the mean energies. The low-energy technique was 60
kVp with no added filtration. The mAs 共1.0 and 1.6, respectively兲 was chosen to give approximately 50% saturation signal level in the bare beam as measured on the FPD.
The DE anatomical NPS was measured using the imaging
bench 共Fig. 2兲 and the anthropomorphic phantom 共Fig. 3兲
across a variety of kVp, filtration, and exposure conditions.
The total anatomical NPS in DE images of the phantom was
measured in the same manner as described above for plain
radiography 共patient and/or phantom images; Fig. 3兲, where
realizations were formed within a ROI in the lung, and the
anatomical parameters, K and ␤, were measured as fitting
parameters to yield estimates of anatomical noise. These results were taken as semiempirical input to CSA of the generalized DQE for a DE image according to Eq. 共21兲. The
effective fluence 关Eq. 共16兲兴 was computed using the measured exposure and fluence-per-unit exposure for the lowand high-energy images. The MTF for the low- and highenergy images were assumed equivalent, supported by mea-
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FIG. 4. The solid line shows the scintillator MTF with K fluorescence
共T3TKtot兲, and the dashed line shows the scintillator MTF without K fluorescence 共T3,兲. Curves shown are Lorentzian fits based on measurements of the
system MTF and modeling of TKtot as in Que et al. 共see Ref. 62兲.
surements in our laboratory that show the MTF to vary only
slightly 共within experimental error兲 for the two energies
used.
IV. RESULTS
A. Cascaded systems analysis for a CsI FPD
Since DE imaging uses images acquired over a broad
range of energies, it was necessary to validate CSA calculations of NPS and DQE in comparison to measurements for a
broad range of techniques. Also, the parallel cascaded systems model described in Appendix A extends the modeling
of K fluorescence to include variation in the optical gain as
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an energy-dependent phenomenon, also requiring validation
of theory in comparison to measurement. To consider the
impact of K fluorescence on the DQE, the MTF was calculated with and without K fluorescence, as shown in Fig. 4.
Measurements on the experimental bench gave the total system MTF 关T3共u , ␯兲T5共u , ␯兲TKtot共u , ␯兲兴, with the pixel aperture
MTF 关T5共u , ␯兲兴 described well by a sinc function.61 The measured MTF divided by T5共u , ␯兲 gave the scintillator MTF
关T3共u , ␯兲TKtot共u , ␯兲兴, with TKtot共u , ␯兲 computed using the
model of Que et al.62 共see Appendix A兲. The curves in Fig. 6
are Lorentzian fits to T3共u , ␯兲 and T3共u , ␯兲TKtot共u , ␯兲. In calculation of the DQE 共below兲, cases with and without K fluorescence were modeled with the MTF as T共u , ␯兲
= T3共u , ␯兲T5共u , ␯兲TKtot共u , ␯兲 and T共u , ␯兲 = T3共u , ␯兲T5共u , ␯兲, respectively.
Figure 5 shows the typical agreement between the NPS
and DQE measurements and calculations using cascaded systems analysis. Table I provides a summary and glossary of
CSA parameters. Results without and with K fluorescence
can be appreciated by considering the quantum gain in
Branch A, ḡ2A, and the effective quantum gain, ḡ2, respectively. The NPS and DQE measurements shown in Fig. 5
were extracted from 2D measurements along a diagonal in
the Fourier domain up to the Nyquist frequency 共uNyq , ␯Nyq兲,
since the imaging system is sampled. Theoretical calculations demonstrated that the influence of K fluorescence on
the NPS is subtle but non-negligible, increasing the NPS
particularly at low frequencies. The DQE is degraded at all
frequencies because of the increase in the NPS at low frequencies and the degradation of the MTF at higher frequencies. Each column shows the NPS and DQE at a given kVp.
FIG. 5. NPS 共top row兲 and DQE 共bottom row兲 at various kVp. Measurements were performed on the imaging bench using a FPD with a 250 mg/ cm2 CsI:TI
converter at 400 ␮m pixel pitch. The solid and dashed lines show theoretical calculations with and without K fluorescence, respectively. Inclusion of K
fluorescence improves the agreement of the theoretical model, especially for the DQE. NPS measurements at various kVp had slightly different exposure
levels due to limitations in technique selection, but show reasonable agreement with CSA calculations. For comparison at various kVp, calculations are shown
for the case of 1.0 mR exposure, where the NPS is seen to increase with kVp.
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TABLE I. Glossary and summary of parameters used in the cascaded systems analysis model. Values were computed for a flat-panel detector incorporating a
250 mg/ cm2 CsI:TI x-ray converter.
60 kVp
Parameters
q̄0 / X
ḡ1
␰␻
ḡ2A
ḡ2B
ḡ2C
ḡ2
␧g2A
␧g2B
␧g2C
fK
ḡ4
80 kVp
100 kVp
120 kVp
共Filtration: 4 mm Al+ 0.6 mm Cu兲
Glossary
Fluence per exposure 关x rays/ 共mm2 mR兲兴
QDE
Probability of producing a K x ray
Conversion gain to optical photons
Branch A
Branch B
Branch C
Weighted sum of the three branches
Poisson excess in ḡ2A
Poisson excess in ḡ2B
Poisson excess in ḡ2C
Probability that the K x ray is reabsorbed
Optical coupling gain to photodiode
140 kVp
2.59⫻ 105
0.94
0.73
2.77⫻ 105
0.82
0.73
2.83⫻ 105
0.72
0.73
2.74⫻ 105
0.64
0.73
2.64⫻ 105
0.59
0.73
1528
521
1117
1371
134
86
81
0.78
0.65
1823
819
1117
1670
129
109
65
0.78
0.65
2019
1011
1117
1860
142
169
43
0.78
0.65
2159
1151
1117
2000
178
234
40
0.78
0.65
2259
1251
1117
2099
216
299
37
0.78
0.65
For the NPS 共top row兲, the results appear nonmonotonic with
kVp due primarily to differences in the exposure, X, required
to give ⬃50% detector signal saturation; the NPS computed
at X = 1.0 mR is included for comparison, showing a monotonic increase in the NPS with kVp. Correspondingly, the
DQE 共bottom row兲 shows a monotonic reduction with kVp.
The agreement between measured and calculated NPS and
DQE is good across all frequencies, giving confidence that
the extended CSA model provides a useful theoretical tool
for analysis of imaging performance across the broad range
of conditions considered below.
B. Generalization of the DQE
1. The anatomical NPS
Figure 6共a兲 shows the total anatomical NPS measurements along a diagonal in the Fourier domain up to the Nyquist frequency obtained from 90 CR patient chest images.
The measurements were made along the u = ␯ diagonal because of artifactual power in the measured NPS observed
along the u and ␯ axes. The anatomical NPS differs markedly
from the quantum NPS in that the former has a very strong
low-frequency characteristic. Also, a large amount of variability was observed across the population data, as reported
by Samei et al.6 Figure 8共b兲 shows the variability 共i.e., the
variability of the anatomical NPS兲 within one patient in comparison to the population variation. The error bars represent
two standard deviations in NPS estimates across ⬃100 realizations from a single patient image. This variation is important to examine because it quantifies the nonstationarity
within an image, which is of the same order as the nonstationarity across the population. Therefore, the validity of the
measured NPS estimate with respect to nonstationarity is
about the same as the validity of the empirical model across
the population average. Finally, Fig. 6共c兲 compares the patient population anatomical noise with the anatomical noise
measured in the chest phantom 共60–140 kVp兲. Although the
overall magnitude of the NPS is less in the chest phantom,
due in part to differences in performance between the CR
and FPD systems and in part due to the imperfect simulation
FIG. 6. 共a兲 Measurements of total anatomical NPS on 90 CR chest patients acquired at 140–150 kVp. 共b兲 The shaded area is the region spanned by the 90 CR
chest patients. The error bars represent the variation within one patient showing that NPS variations across one radiograph are of the same order as that of the
population variation. 共c兲 Total anatomical NPS for the patient population 共shaded area兲 in comparison to that measured using the chest phantom. While the
phantom provides a reasonable experimental tool for anatomical noise measurements, the anatomical noise is found to be less than that in human images;
therefore, the effects demonstrated in the generalized DQE are fairly conservative estimates.
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FIG. 7. 共a兲 Anatomical noise parameters, K and ␤, measured as a function of kVp 共60–140 kVp兲. The solid
line 共left axis; K兲 quantifies the magnitude of anatomical noise. The
dashed line 共right axis; ␤兲 quantifies
the frequency content in the anatomical noise. 共b兲 Radiographs of lung regions at various kVp illustrate the reduction in magnitude of anatomical
noise 共i.e., reduction in contrast of
overlying bone兲 as kVp increases.
of real anatomy, the frequency dependence is similar, and the
chest phantom provides a useful tool to test the methodology
for measuring thoracic anatomical noise. Since the anatomical noise measured in patients was greater than that measured in the phantom, the impact of generalization demonstrated in this paper is conservative in its effect on DQE.
Measurements of anatomical noise were performed using
the chest phantom, and empirical estimates of K and ␤ were
derived as a function of kVp 关see Fig. 7共a兲兴. K, which quantifies the magnitude of the anatomical noise, was observed to
decrease considerably as the kVp was increased. On the
other hand, ␤, which quantifies the frequency content of the
anatomical noise, did not appear to vary much across the
measured range of kVp. This can be observed qualitatively in
the images of Fig. 7共b兲: As the kVp increases, the rib contrast
共related to K兲 is reduced, while the “clumpiness,” ␤, of the
anatomical noise does not change.
2. The generalized DQE and detectability index
Figure 8共a兲 shows the calculated DQE for a 250 mg/ cm2
CsI:TI-based FPD with a 400 ␮m pixel pitch and 80% fill
factor at an exposure of 0.1 mR and various kVp 共4 mm Al
and 0.6 mm Cu added filtration兲. Theoretical calculations
were performed at 60, 80 100, 120, and 140 kVp, showing a
monotonic reduction in DQE with kVp. However, if anatomical noise, SB, is included in the total NPS by taking the
empirically determined anatomical parameters K and ␤ from
Fig. 6共a兲, the landscape changes entirely as shown in Fig.
8共b兲. The generalized DQE is severely degraded by the presence of low-frequency anatomical noise, and GDQE peaks at
midfrequencies. Even more interesting, the GDQE increases
significantly with kVp, corresponding to the reduction in
anatomical noise at higher kVp.
Figure 9 summarizes these results in terms of the detectability computed as a function of kVp using conventional
and generalized formulations of DQE 共without and with anatomical noise, respectively兲. According to the conventional
description, detectability index decreases with kVp. However, generalization of the DQE tells the opposite story: The
detectability increases with kVp when anatomical noise is
included. This is observed qualitatively in Fig. 7, showing
reduced rib contrast and increased nodule conspicuity as the
kVp is increased. This analysis is compelling because it
brings image theory in line with techniques used clinically in
chest radiography, where the techniques are typically in the
higher kVp range.63
C. Extension to DE imaging
1. The DE anatomical NPS
The NPS for DE flood-field images was computed in two
ways as described in Sec. III C. Both results were observed
to produce the same results for the DE NPS, thus verifying
the expression in Eq. 共10兲 共derived in Appendix B兲. Figure
FIG. 8. 共a兲 Theoretical calculations of
DQE at various kVp showing monotonic reduction in DQE as the kVp increases. 共b兲 Theoretical calculations of
the generalized DQE, using K and ␤
values measured at various kVp,
showing improvement in GDQE with
kVp. The distinction highlights the importance of the generalized approach.
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Figure 11共a兲 shows the strong dependence observed between the anatomical noise parameters, K and ␤, and the
tissue cancellation parameter, wt, with minimization of K,
corresponding to a reduction in bone contrast 关Fig 11共b兲兴.
However, a close inspection of the images in Fig. 11共b兲 reveals an increase in quantum noise at the same level of
wt共⬃0.4兲. This suggests a tradeoff between the reduction of
anatomical noise and the amplification of quantum noise.
Hence, an analysis, such as the generalized CSA—which
takes into account the quantum noise and anatomical
noise—is essential in the analysis of DE imaging performance.
2. The DE generalized DQE and detectability index
FIG. 9. The conventional and generalized detectability index for radiographic chest images as a function of kVp. While detectability based on the
conventional description of DQE degrades monotonically with kVp, the
generalized detectability index 共which includes anatomical nose in GDQE兲
increases with kVp, bringing imaging theory in line with typical chest imaging techniques 共see Ref. 63兲.
10共a兲 shows the measured and calculated NPS for a DE image, indicating the amplification of quantum noise due to DE
image processing. The measured exposure at the detector
was 1.8 and 0.8 mR in the low- and high-energy images,
respectively. The solid line with squares corresponds to the
NPS for the DE image computed using CSA, where reasonable agreement is observed between the theory and measurements. The discrepancies at higher frequencies arise from
differences between the measured MTF and the Lorentzian
model incorporated in theoretical calculations. Figure 10共b兲
shows the measured total anatomical noise and shows the
strong reduction in anatomical noise achieved in DE image
reconstruction. Taken together, these results clearly show
that DE imaging slightly increases quantum noise, while significantly reducing anatomical noise.
Figure 12共a兲 shows a grayscale image plot of the generalized DQE across a range of tissue cancellation parameters.
Figure 12共b兲 shows the generalized DQE for selected values
of wt. The generalized DE DQE was computed using CSA
with measured values of K and ␤ 关Fig. 11共a兲兴. The generalized DE DQE exhibits a strong dependence on the tissue
cancellation parameter and is maximized when the anatomical noise is minimized. While a slight spatial-frequency dependence is noted in the optimal value of wt in Fig. 12共a兲, as
shown in Fig. 12共b兲, a value of wt ⬃ 0.4 essentially maximizes GDQE at all spatial frequencies.
The generalized DE detectability index computed for a
uniform-function detection task is shown in Fig. 13, calculated using the generalized DQE for DE images at various
tissue cancellation parameters. The results demonstrate a distinct optimum at wt = 0.4. This result agrees well with the
value that was chosen qualitatively by a single observer
viewing a series of DE images reconstructed using various
tissue cancellation parameter values. Also, the detectability
index for this idealized detection task is found to increase
considerably for DE imaging compared to plain radiography.
Specifically, in the DE images, detectability is ⬃0.35 at optimal tissue cancellation, whereas in plain radiography de-
FIG. 10. 共a兲 Plots of the low- and high-energy NPS with the DE NPS measured using two different techniques to verify Eq. 共10兲 共see Appendix B兲. The line
is the NPS computed on the floods processed according to Eq. 共9兲 with wt = 0.4. The x symbols represent the NPS calculated using the high- and low-energy
NPS and Eq. 共10兲 with wt = 0.4. The line with square symbols shows the theoretical prediction for the DE image. 共b兲 Total anatomical NPS measurements for
low-energy, high-energy, and DE images. Measurements were performed on the lung region of the PMH chest phantom 共see Fig. 3兲. While DE processing
increases the quantum NPS 共a兲 it reduces the total NPS 共b兲 due to reduction in anatomical noise.
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FIG. 11. 共a兲 Measurement of anatomical noise parameters, K and ␤, as a
function of the DE tissue cancellation
parameter, 共wt = 0.2− 0.6兲, showing a
strong dependence on wt. 共b兲 DE images at various wt, demonstrating removal of the ribs from the image
where K is minimized and ␤ is maximized. Note: The darker spheres evident in the wt = 0.6 共black—bones兲 image are acrylic. The lighter spheres in
the wt = 0.4 image represents the simulated nodules.
tectability varied from ⬃0.10- 0.18, depending on beam energy. Therefore, DE imaging increased detectability by more
than a factor of 2.
V. DISCUSSION AND CONCLUSIONS
Anatomical noise was included in a generalized DQE
analysis of flat-panel radiography and extended to DE imaging, yielding descriptions of spatial-frequency-dependent imaging performance that are completely distinct from conventional approaches. In the case of radiography, the analysis
reveals a quantitative and intuitive description of increasing
detectability with increasing kVp for chest radiography when
the “generalized detectability” is considered, thus bringing
imaging theory in line with clinical techniques.
For DE imaging, the extension of cascaded systems
analysis provides a powerful theoretical framework for the
evaluation of the DE DQE, generalized DQE, and detectability index. Just as such an analysis has proven invaluable to
the development and optimization of flat-panel detector systems for radiography, application in DE imaging allows identification of the factors limiting DE imaging performance
and offers a guide to the development of high-performance
systems. The generalized DE detectability index gives a
quantitative tool to optimize the tissue cancellation parameter, with a quantitative account of the tradeoffs between
quantum and anatomical noise. A conventional description
共ignoring anatomical noise兲 would be insufficient in describing DE imaging performance, because the tradeoffs between
quantum noise and anatomical noise are central to optimal
DE reconstruction. The ability of DE imaging to improve
diagnostic performance is apparent in the analysis above,
where the detectability index for a simple idealized detection
task was a factor of 2 greater for DE imaging compared to
radiography, in agreement with the improved conspicuity offered by DE techniques.12,23
These results are promising because they point to other
directions of investigation, such as optimal selection of dual
kVp settings in DE imaging and optimal allocation of dose
between high- and low-energy images. Also, the Fourierbased analysis of anatomical noise in DE images provides a
fast robust technique for optimal selection of tissue cancellation parameter through minimization of the anatomical
NPS, corresponding to minimization of bony contrast. By
operating on a simply defined ROI 共the lung fields兲 without
complex segmentation, a fast Fourier-based tissue cancellation technique is under development.
In the same manner as CSA was extended above to the
case of DE imaging to yield a theoretical framework for
generalized DQE and system optimization, the approach may
be similarly extended to other advanced FPD applications,
such as tomosynthesis and cone-beam CT.64 For example,
such a generalized analysis would provide a useful guide to
understanding tradeoffs between quantum noise, anatomical
noise, number of projection views, slice thickness, and selection of optimal reconstruction filters. Such an analysis offers
to quantify improvements in detectability afforded by these
FIG. 12. 共a兲 Grayscale image plot of
the generalized DQE as function of
DE tissue cancellation parameter.
Maximum GDQEDE occurs at the optimal tissue cancellation parameter.
共b兲 Generalized DQE plots for selected values of wt. GDQEDE is
maximized near wt = 0.4, corresponding to minimization of the anatomical NPS and cancellation of overlying ribs 关Fig. 11共b兲兴.
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FIG. 13. Generalized DE detectability index computed for a simple constant
frequency-weighting task function versus tissue cancellation parameter. Detectability was computed using the DE generalized DQE of Fig. 12, demonstrating a distinct optimum at wt = 0.4.
advanced applications in terms of progressively reduced anatomical noise, from radiography to DE imaging to tomosynthesis and cone-beam CT.
Although a constant task function was used as the simplest case to illustrate the methodology, other more complex
detection tasks can be used in computing detectability. Future work involves generalized DQE analysis combined with
more sophisticated model observer performance descriptors
for evaluation of the detectability index for higher-order
frequency-dependent tasks, such as discrimination and localization tasks for the advanced applications mentioned above.
If this approach shows to be accurate through observer studies that validate the detectability index as described in this
manuscript, it will provide a valuable means to configure
novel systems and investigate imaging performance across a
broad range of detector configurations, imaging conditions,
and imaging tasks. Incorporation of a generalized DQE
analysis in the evaluation of detectability provides a quantitative tool that begins to bridge the gap between detector
performance and observer performance in such advanced
modalities.
ACKNOWLEDGMENTS
The authors would like to thank N. Paul 共University
Health Network, Toronto, Canada兲 for providing the CR
chest patient data, K. Brown 共Elekta Oncology Systems,
Inc.兲 for supplying the flat-panel detector, and M. K. Gauer
共PerkinElmer兲 for technical information concerning the flatpanel detector. This research was funded in part by the Ontario Student Opportunity Trust Fund.
APPENDIX A: PARALLEL CASCADED SYSTEMS
ANALYSIS
Several authors have studied parallel-cascaded systems
analysis,29,30,43 allowing a description of signal and noise
transfer processes occurring simultaneously in the imaging
chain, for example, K fluorescence. A brief overview of
Stage 2 for parallel CSA of FPDs is provided for completeMedical Physics, Vol. 32, No. 5, May 2005
FIG. 14. Flow-chart representation of cascaded systems analysis for a FPD.
Each stage represents a physical process in the imaging chain. At Stage 2, a
parallel cascade models K fluorescence: Branch A represents the case in
which all the energy of the incident x ray is deposited locally in the scintillator, Branch B is the case in which only a fraction of the energy is deposited locally after K x ray production, and Branch C corresponds to remote
deposition of the K x ray.
ness, with extension 共not covered in previous work兲 to give a
general account of variance in the conversion gain by inclusion of the Poisson excess 共Swank factor兲.
1. Stage 2- conversion gain
Stage 2 describes conversion of x rays into optical photons in the scintillator. A secondary parallel process may happen when an x ray interacts with an electron in the K shell of
an atom in the scintillator and produces a K x ray that may be
reabsorbed in the scintillator and produce light at a remote
location. Figure 14 illustrates the three possible scenarios
whereby an x ray produces optical photons. Branch A corresponds to all of the energy of the x ray being converted into
optical photons. Branch B represents the case in which a
fraction of the energy is deposited locally and produces optical photon when a K shell interaction occurs. Branch C
corresponds to the resultant energy in the K x ray being
deposited remotely.
2. K-fluorescence parameters
␰ denotes the probability that, when an incident photon
interacts in the screen, it undergoes a K-shell interaction. ␻ is
the fluorescent yield of K-shell photoelectric interactions;
hence, ␰␻, quantifies the probability that a K x ray will be
produced. For the detector considered 共CsI:TI兲, Cs and I
have similar physical properties, such as density, ␰ , ␻, and
K-edge energies 共EK兲; therefore, an effective value was computed from their respective fractional weights65 in CsI 关␰
= 0.834, ␻ = .870, EK = 35 keV兴. Since no K x rays are produced below the K-edge energy, an energy dependent probability of producing K x rays is defined:
Richard et al.: Generalized DQE analysis
1410
␰␻共E兲 = 0 for E ⬍ EK or ␰␻ for E ⬎ EK .
1410
共A1兲
f K denotes the probability of reabsorption in the screen for
each photoelectric interaction producing a K x ray and was
computed analytically using a multilayer model initially developed by Vyborny66 共extended by Shuping and Judy,67 and
Chan and Doi兲.68 f̄ K was computed as:
f̄ K =
冕
E
共A2兲
.
q1共E兲␰␻共E兲dE
E
TK is the MTF corresponding to the spread of K x rays in
the phosphor. The point-spread function of the K x ray was
computed using a multilayer model as described by Que.62
Therefore, an effective gain at Stage 2 can be defined,
ḡ2 = 共1 − ␰␻兲ḡ2A + ␰␻ḡ2B + ␰␻ f̄ Kḡ2C .
The fluence in each branch is q̄0 multiplied by the gains,
From the transfer relations from Rabbani et al.
= gi共1 + ␧gi兲 for i = 2A, 2B, 2C,
E
· ḡesc ,
q̄1共E兲共1 − ␰␻共E兲兲dE
S2C共u, ␯兲 = q̄0ḡ1␰␻ f̄ Kḡ2C共ḡ2C + 1 + ␧g2c兲,
The Swank factor I2i was computed using the moments of
the absorbed energy distribution 共AED兲:
ḡ2B =
冕
E
冕
EK
E
I2i =
冕
冕
q1共E兲dE
E
· ḡesc ,
共A4兲
q̄1共E兲␰␻共E兲dE
M 2i共E兲q1共E兲dE
where,
M 2i共E兲 =
2
g2 共E兲
M 1i
共E兲
= 2i
,
M 0i共E兲I共E兲
I共E兲
共A9兲
冉 冊
1
− 1 − 1.
I2i
共A10兲
S2BC共u, ␯兲 = q̄0ḡ1 f̄ Kḡ2Bḡ2CTK共u, ␯兲.
共A11兲
Hence, by substitution:
S2共u, ␯兲 = q̄0ḡ1关共1 − ␰␻兲ḡ2A共ḡ2A + 1 + ␧g2A兲 + ␰␻ḡ2B共ḡ2B
+ 1 + ␧g2B兲… + ␰␻ f̄ Kḡ2C共ḡ2C + 1 + ␧g2C兲
ḡ2C = W · EK · ḡesc ,
where W is the mean number of optical photons produced
per keV absorbed, taken to be 56 photons/ keV.29 ḡesc quantifies the fraction of optical photons that escape the scintillator which was assumed to be 0.55, a similar value to that of
Hillen et al.69 Similarly for the total fluence at Stage 2:
+ 2f̄ K␰␻ḡ2Bḡ2CTK共u, ␯兲兴.
ḡ2
共A12兲
Finally, S2共u , ␯兲 can be rewritten in terms of PK共u , ␯兲:
S2共u, ␯兲 = q̄0ḡ1ḡ2关PK共u, ␯兲 + 1兴,
共A13兲
where,
关共1 − ␰␻兲ḡ2A共ḡ2A + ␧g2A兲 + ␰␻ḡ2B共ḡ2B + ␧g2B兲 + ␰␻ f̄ Kḡ2C共ḡ2C + ␧g2C兲 + 2f̄ K␰␻ḡ2Bḡ2CTK共u, ␯兲兴
Medical Physics, Vol. 32, No. 5, May 2005
共A8兲
,
Also, the cross NPS term as described by Cunningham,
兲 · q̄1共E兲␰␻共E兲dE
E
PK共u, ␯兲 =
2
g2i
␧gi = ḡ2i
E
W · E · 共1 −
共A7兲
where M 0i , M 1i, and M 2i are the zeroth, first, and second
moments of the AED. Finally, the Poisson excess for each
branch of the system was computed using the Swank factor:
W · E · q̄1共E兲共1 − ␰␻共E兲兲dE
冕
i
E
q̄2C = q̄0ḡ1␰␻ f̄ Kḡ2C,where each gain is integrated with the
normalized spectrum at that stage,
ḡ2A =
with ␴g2
S2B共u, ␯兲 = q̄0ḡ1␰␻ḡ2B共ḡ2B + 1 + ␧g2B兲,
q̄2B = q̄0ḡ1␰␻ḡ2B ,
共A3兲
冕
共A6兲
34
3. The fluence and NPS
q̄2A = q̄0ḡ1共1 − ␰␻兲ḡ2A,
共A5兲
S2A共u, ␯兲 = q̄0ḡ1共1 − ␰␻兲ḡ2A共ḡ2A + 1 + ␧g2A兲,
f K共E兲q1共E兲␰␻共E兲dE
冕
q̄2 = q̄0ḡ1关共1 − ␰␻兲ḡ2A + ␰␻ḡ2B + ␰␻ f̄ Kḡ2C兴.
.
共A14兲
1411
Richard et al.: Generalized DQE analysis
1411
S7共u, ␯兲 = q̄0a4pdḡ1ḡ2ḡ4关1 + ḡ4 PK共u, ␯兲T32共u, ␯兲兴
It is the spatial-frequency-dependent cross term in PK共u , ␯兲
that is mainly responsible for the increase of the NPS and
goes as TK共u , ␯兲; therefore, the effect of K fluorescence is
greater at lower frequencies as can be seen in Fig. 4. Note the
special case when ␰␻ = 0, i.e., when no K x rays are produced: PK共u , ␯兲兩␰␻=0 = ḡ2 + ␧g2.
⫻T52共u, ␯兲* *III共u, ␯兲 + Sadd .
APPENDIX B: NPS FOR A DE IMAGE
The DE NPS, SDE
⬘ 共u , ␯兲, may be derived in terms of the
NPS of the low- and high-energy images and the tissue cancellation parameter, wt, as shown in Eq. 共10兲. We first derive
the autocovariance for a general random variable, c共x兲, arbitrarily comprised of random variables a共x兲 and b共x兲. By Fourier transform, we show that the NPS of c共x兲 is related to the
NPS and first derivative of a共x兲 and b共x兲. From this general
analytic approach, we treat the specific case in which c共x兲
represents the DE image, IDE, and a共x兲 and b共x兲 are the lowand high-energy images as in Eq. 共10兲.
Given c共x兲 = f关a共x兲 , b共x兲兴, where a共x兲 , b共x兲 and c共x兲 are
random variables, and a共x兲 and b共x兲 have absolute errors, ⳵a
and ⳵b, respectively. From a Taylor expansion,70
4. Stage 3-spreading of optical quanta
This stochastic blurring stage quantifies the spreading of
the optical photons in the scintillator
S3共u, ␯兲 = q̄0ḡ1ḡ2关1 + PK共u, ␯兲T32共u, ␯兲兴,
共A15兲
where T3共u , ␯兲 was determined from measurements of
T3共u , ␯兲T5共u , ␯兲TKtot共u , ␯兲, where T5共u , ␯兲 is a sinc function,61
and TKtot共u , ␯兲, is the normalized weighted sum of the MTFs
at the three branches:
TKtot共u, ␯兲 =
冋
共1 − ␰␻兲ḡ2A + ␰␻ḡ2B + ␰␻ f̄ Kḡ2CTK共u, ␯兲
ḡ2
册
⌬c共x兲 = ⌬a共x兲
.
The propagation of the NPS in remaining stages follows
directly from the transfer equations yielding the NPS of the
flat-panel imaging system:
再
冉 冊冉 冊
冉 冊 冉 冊
冉 冊冉 冊
⳵c
⳵a
=
⳵c
⳵a
⳵c
⳵a
冉 冊
⳵c
⳵a
冉 冊
⳵c
⳵a
2
Ka共x兲 +
冉 冊
⳵c
⳵b
2
Ka共x兲 +
⳵c
⳵b
+ ⌬b共x⬘兲⌬b * 共x + x⬘兲
冉 冊
⳵c
⳵b
2
+
冎
⳵c
共E兵⌬a共x⬘兲⌬b * 共x + x⬘兲其 + E兵⌬a * 共x + x⬘兲⌬b共x⬘兲其兲 + …,
⳵b
共B3兲
共B4兲
where F兵其 denotes the Fourier transform. Taking the Fourier
transform of Eq. 共B3兲, we obtain:
Medical Physics, Vol. 32, No. 5, May 2005
2
Kb共x兲 +
The NPS, Sa共u兲, for the random variable a共x兲 is given by the
Fourier transform of the autocovariance:71
Sa共u兲 = F兵Ka共x兲其,
共B1兲
2
2
Kb共x兲.
冉 冊
⳵c
⳵c
+ ⌬b共x兲
+ …,
⳵a
⳵b
⳵c
共⌬a共x⬘兲⌬b * 共x + x⬘兲 + ⌬a * 共x + x⬘兲⌬b共x⬘兲兲 + …
⳵b
where * indicates the complex conjugate. Under the assumption that a and b are uncorrelated, then the cross terms are
zero and the expression reduces to:
Kc共x兲 =
冉 冊
where ⌬a共x兲 = a共x兲 − E兵a共x兲其 and similarly for ⌬b共x兲 and
⌬c共x兲, where E兵a共x兲其 is the expectation value of a共x兲. From
the definition of the autocovariance, which describes the correlation of ⌬a共x⬘兲 with itself at a location displaced by x, we
have:71
共A16兲
Kc共x兲 = E兵⌬c共x⬘兲⌬c * 共x + x⬘兲其 = E ⌬a共x⬘兲⌬a * 共x + x⬘兲
共A17兲
Sc共u兲 =
冉 冊
⳵c
⳵a
2
Sa共u兲 +
冉 冊
⳵c
⳵b
共B2兲
2
Sb共u兲.
共B5兲
Taking a共x兲 , b共x兲, and c共x兲 as the low-energy, high-energy,
and DE images, respectively: and extending to two dimensions:
SDE共u, ␯兲 =
冉 冊
冉 冊
⳵ IDE 2
⳵ IDE 2
SL共u, ␯兲 +
SH共u, ␯兲,
⳵ IL
⳵ IH
共B6兲
where the 2D DE soft tissue image is defined as previously
关Eq. 共9兲兴:
1412
Richard et al.: Generalized DQE analysis
IDE共x,y兲 =
ILwt共x,y兲
IH共x,y兲
1412
共B7兲
.
The assumption of statistical independence between a共x兲 and
b共x兲—that is, IL共x , y兲 and IH共x , y兲—is reasonable, since they
represent independently acquired projection images. Correlation between the two projections could be introduced by image lag,35 representing a possible limitation to this assumption to be considered in future work.
If we rewrite Eq. 共B6兲 in terms of the relative NPS, S⬘,
which is defined as the NPS divided by the signal squared,
and substitute the derivatives:
⬘ 共u, ␯兲 =
SDE
=
SDE共u, ␯兲
I2DE
冉
wtILwt−1
IH
,
冊
冉 冊
2
I Lwt
SL共u, ␯兲 +
冉 冊
I Lwt
I H2
2
2
SH共u, ␯兲
,
IH
=w2t
SL共u, ␯兲
IL
2
+
SH共u, ␯兲
I H2
,
⬘ 共u, ␯兲.
=w2t SL⬘ 共u, ␯兲 + SH
共B8兲
Therefore, the NPS for a DE 共e.g., soft-tissue-only兲 image is
given by the NPS of the high-energy image plus the NPS of
the low-energy image weighted by square of the tissue cancellation parameter, as verified in Fig. 10共a兲.
a兲
Author to whom correspondence should be addressed: Ontario Cancer
Institute, Suite 7-502, Princess Margaret Hospital, 610 University Ave.,
Toronto,
ON,
Canada
M5G
2M9;
electronic
mail:
[email protected]
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