84 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 1, JANUARY 2011
Sufficient Statistics as a Generalization
of Binning in Spectral X-ray Imaging
Adam S. Wang*, Student Member, IEEE, and Norbert J. Pelc
Abstract—It is well known that the energy dependence of X-ray
attenuation can be used to characterize materials. Yet, even with
energy discriminating photon counting X-ray detectors, it is still
unclear how to best form energy dependent measurements for
spectral imaging. Common ideas include binning photon counts
based on their energies and detectors with both photon counting
and energy integrating electronics. These approaches can be
generalized to energy weighted measurements, which we prove
can form a sufficient statistic for spectral X-ray imaging if the
weights used, which we term
functions that can also be used for material decomposition. To
study the performance of these different methods, we evaluate the
Cramér-Rao lower bound (CRLB) of material estimates in the
presence of quantum noise. We found that the choice of binning
and weighting schemes can greatly affect the performance of
material decomposition. Even with optimized thresholds, binning
condenses information but incurs penalties to decomposition
precision and is not robust to changes in the source spectrum
or object size, although this can be mitigated by adding more
bins or removing photons of certain energies from the spectrum.
On the other hand, because
-weighted measurements form a
sufficient statistic for spectral imaging, the CRLB of the material
decomposition estimates is identical to the quantum noise limited
performance of a system with complete energy information of
all photons. Finally, we show that
conspicuity over other methods in a simulated calcium contrast
-weights, are basis attenuation
-weights lead to increased
Index Terms—Cramér-Rao lower bound, dual energy, energy
weighting, photon counting, spectral imaging, sufficient statistic.
dependence can be used for material selective imaging ,
. Therefore, there is information about a measured object
contained not only in the total number of photons transmitted
through the object but also in the energy of each of these
photons. How this information is collected greatly affects the
ability of a system to efficiently utilize the energy dependence
in what is known as spectral X-ray imaging. Early methods for
spectral imaging include the use of dual kVp techniques that
T is well known that a material’s X-ray attenuation is
dependent on the energies of the X-rays and that this
Manuscript received May 23, 2010; revised July 15, 2010; accepted July 17,
2010. Date of publication August 03, 2010; date of current version December
30, 2010. This work was supported in part by GE Healthcare and in part by the
Lucas Foundation. Asterisk indicates corresponding author.
*A. S. Wang is with the Departments of Electrical Engineering and
N. J. Pelc is with the Departments of Radiology, Bioengineering, and
Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail:
Digital Object Identifier 10.1109/TMI.2010.2061862
require two different exposures—one at a lower energy than
the other—or the use of dual layer detectors. Both methods
inherently suffer from spectral overlap, which degrades the
spectral imaging performance –. Energy discriminating
detectors offer the possibility of eliminating spectral overlap
by directly partitioning the transmitted spectrum into sepa-
rate measurements –. Our project began with work to
understand and optimize the partition of the spectrum into
bins and led to the observation that weighted measurements,
a generalization of binning, can achieve universally optimal
performance. Here, we evaluate several methods for forming
energy dependent measurements in the context of ideal photon
counting detectors with energy discriminating capabilities. The
work led us to the discovery of a simple and elegant sufficient
statistic for spectral X-ray imaging.
For now, we assume that the materials being measured have
no K-edges within the detected X-ray spectrum; this will be
relaxed later. Spectral imaging in the diagnostic energy range
is often referred to as dual-energy imaging, which is based on
the observation that X-rays in this energy range primarily in-
teract by two physical mechanisms—photoelectric absorption
and Compton scattering—and that in the absence of K-edges
the energy dependence of these mechanisms is independent of
the material . Thus, materials without K-edges within the de-
tected spectrum behave as if they were some linear combination
of material independent basis functions (e.g., photoelectric ab-
sorption as a function of energy and Compton scattering as a
function of energy). In other words, in the energy range of in-
terest, material 1 will have a linear attenuation coefficient at en-
that can be expressed as
Compton interactions, respectively. Moreover, because all ma-
terials without K-edges attenuate X-rays based on the same un-
derlying principles, any material can be expressed as a linear
combination of any other two materials. For materials 1, 2, and
and can be found such that
Thus, any object can be described as an amount
rial 1 and an amount
of material 2. To effectively estimate
0278-0062/$26.00 © 2010 IEEE
WANG AND PELC: SUFFICIENT STATISTICS AS A GENERALIZATION OF BINNING IN SPECTRAL X-RAY IMAGING85
, the object’s attenuation can be measured with
X-rays of different energies. An ideal system based on photon
counting detectors would keep track of the energy of each de-
tected photon. Data handling requirements generally preclude
this, and instead photons are grouped into energy intervals. For
example, a system may measure the number of low and high
energy photons, where a preset threshold energy
whether photons are counted in a low or high energy bin. When
they undergo binning.
For an incident spectrum
the expected number of transmitted photons at energy through
an object of thickness
of material 1 and thickness
terial 2 is
with maximum energy,
where we have simplified the notation of
served spectrum will be a random vector
is the observed number of photons of energy
the detector and is a Poisson random variable with mean
that is independent of the observed number of photons at other
energies . For notational convenience, we will consider all
energies to the nearest integer keV.
Performance bounds on basis material decomposition for any
bound (CRLB) , . The CRLB provides a lower bound
on the covariance of unbiased estimators of deterministic pa-
rameters (e.g., ) and can be computed from the distribution of
the measurements used to estimate the parameter. Due to the
inherent quantum noise of photon statistics, an unbiased esti-
of the amount of each basis material will be noisy and
have some covariance matrix
with corresponding covariance matrix
vides a lower bound so that
sense—understood to mean that the quantity
itive semidefinite. In particular,
. Since the lower bound is depen-
dent on the incident spectrum and detector configuration, the
spectrum and bin thresholds to minimize the estimation noise.
In our work, we consider a fixed incident spectrum and show
that the choice of bin thresholds and, later, weighting functions
greatly affects dual energy performance. Furthermore, we use
the CRLB to compare different measurement schemes to show
that different methods have inherently disparate lower bounds
on their material decomposition precision.
, respectively. However, the ob-
. For any unbiased estimate
, the CRLB
in the matrix inequality
Using Spektr , a Matlab implementation of the X-ray
spectral model TASMIP , we simulated an X-ray tube with
constant tube voltage of 120 kV, 2.5 mm Al total filtration,
and 0.2 mAs exposure illuminating an ideal 1 mm photon
counting energy discriminating detector at 1 m from the source
[Fig. 1(a)]. We take basis materials 1 and 2 to be calcium (den-
sity: 1.55 g/cm ) and water (density: 1.00 g/cm ), respectively
Fig. 1. Simulated120 kV spectrumwith 2.5mmAl filtrationfromSpektr (left)
and calcium and water linear attenuation functions used as basis functions for
dual energy decomposition (right). (a) Spectrum. (b) Basis attenuation.
[Fig. 1(b)] . We chose these as our basis materials because
such a decomposition can have direct applications like bone
densitometry. Furthermore, we can easily convert our calcium
and water decomposition into any other two basis materials via
a linear transformation .
bins with energy thresholds
keV, the measured number of counts
for each bin
is the observed number of photons of energy
is a random vector of the binned counts. In actual
is used to prevent electronic noise from
forming counts, and in this work we assume
above the noise floor. Availability of the full detected spectrum
is a special case of binning in which we have
width 1 keV and is the most information we could ever acquire
from the detector.
To compare different binning schemes, we turn to the CRLB,
whose formulation for binning is shown in Appendix A. The
CRLB for the full spectrum
of the material decomposition for any information collection
precision is the inherent quantum noise. We compare the CRLB
(A.3) of any binning scheme to
suboptimality in estimating
when binning the detected spec-
trum. If we normalizeby
for binning. For instance, if calcium is material 1,
represents how much higher the best case
variance of the estimated amount of calcium,
as a result of binning the full detected spectrum since
corresponds to the minimum variance of estimate
ning. Clearly, the performance of binned measurements can be
no better than that of the full detected spectrum, which encom-
passes all information available at the detector, so
itself is a function of the binning threshold
energies, so naturally we would like to find the threshold(s) that
minimize some objective function of the CRLB for any given
object and incident spectrum. For this discussion, we only con-
sider threshold energies at integer keV values to keep the set of
possible threshold energies finite, and we take as our objective
to minimize the variance of the calcium estimate
keV to be
bins each of
(A.4) represents the abso-
to illustrate the
, we get a penalty factor
, will be
86IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 1, JANUARY 2011
Fig. 2. Calcium estimate penalty factor ?
energy between two bins for three different objects.
as a function of the threshold
resulting optimal threshold(s) will depend on the object’s mate-
rial decomposition .
A. Two Bins
Fig. 2 highlights the shortcomings of binning with a single
fixed threshold. It shows the penalty factor for calcium estima-
tion as a function of binning threshold for measurements with
the incident spectrum of Fig. 1 and three objects characterized
by different amounts of calcium and water. The penalty factor
varies drastically as a function of threshold energy and
across objects. If the threshold energy is chosen to be 55 keV to
minimize the penalty for
rial decomposition of 0.5 cm Ca and 20 cm water, the penalty
factor is higher than the minimum for object
suggests that using two bins is far from ideal (because even the
is at least 50% for these objects) and
far from robust (because using the wrong threshold incurs an
even higher penalty).
, an object with a mate-
B. Improving Performance
The binning penalty and robustness can be improved by in-
creasing the number of bins, but another interesting possibility
is removing photons from the spectrum—either at the source
or the detector—to increase spectral separation of the two bins.
Here we present an idealized scenario where photons in an en-
ergy interval are removed without impacting the remainder of
the spectrum. We examine the effect of removing photons from
the spectrum on the CRLB and the noise penalty. Removing
photons from the source spectrum also reduces radiation dose
but that effect is not treated here.
If we allow for idealized notch filters that remove all photons
within a certain range of energies, these gaps in the spectrum
ning thresholds since the goal is to increase spectral separation
of bins. Then each bin
will effectively have lower and higher
energy thresholds (
keV. The measured counts of bin
. Note that this scenario is a generalization of
the simple binning case, which can be recovered by removing
all notch filters so that
and the resulting penalties for object ? ? ????????. The unoptimized three bin
case is included for comparison with the two bins with a notch filter.
To optimize our thresholds in regular binning, we perform an
exhaustive search over the set of integer
keV, finding the set of
, thus enabling the most precise estimate of
bins and incident spectrum . If notch filters are allowed, the
search is over the space of all effective thresholds
search is too taxing, e.g., when the number of bins is large, then
, comparison was made to the CRLB with the full spectrum
(i.e., without photons removed) in all cases.
Fig. 3 shows the results for four binning strategies:
and 3, with and without all energies, for a fixed object
. The effective energy ranges of the bins are shown as
bars and notch filters are shown as gaps. Thecolumn to theright
lists the calcium estimate penalty
photons in the middle of the spectrum from either of the two
case. Of course we would expect that retaining these photons
in a separate bin would be even better. While this indeed pro-
vides a further improvement, the additional benefit is marginal
because this is an unoptimized three bin design. The optimized
threebindesignperforms evenbetterandhasthresholds thatare
the number of bins is increased, the penalty factor goes to unity.
object may have different optimal thresholds.
In some hardware implementations (e.g., using comparators)
inserting a gap between two bins is essentially as complex as
adding an additional bin. In those cases, retaining the photons
in an (optimized) additional bin is preferred. However, an im-
portant point in the results presented above is that much of the
benefit of additional bins can come from providing separation
in the spectra contained in each bin. If the notch filter is imple-
mented at the source, the benefit to material decomposition also
comes with a benefit in dose reduction without additional com-
plexity at the detector.
This strategy of applying notch filters is beneficial when es-
(and also ) but may not apply for other tasks
where the total number of photons is more important than spec-
tral separation of the bins, such as measuring the total attenua-
tion or forming monoenergetic images. In those cases, retaining
the gap photons in a separate bin is a very significant benefit.
. If the exhaustive
. We find that removing
WANG AND PELC: SUFFICIENT STATISTICS AS A GENERALIZATION OF BINNING IN SPECTRAL X-RAY IMAGING87
Fig. 4. Energy weighting functions for three special cases: (a) CIX - PC/EI, (b) binning with all energies, and (c) binning with notch filters.
V. WEIGHTED MEASUREMENTS
A further generalization of binning counts is forming
weighted measurements. The general concept is that photon
counting energy discriminating detectors can be used to form
weighted measurements of photon counts, where the weighting
are a function of photon energy. The
result is a set of measurements
The concept of energy weighting or weighting functions is
not new , –, and when at least two measurements
composition can be performed. A special case is a counting and
and energy integration, providing measurements with weights
special case is binning—the weighting functions are binary and
nonoverlapping [Fig. 4(b)]. When notch filters are considered,
the weights are zero for energies filtered out, creating separa-
tion between the bins [Fig. 4(c)]. In the most general case, the
weighting functions are arbitrary and can take on any real value
to no more than one bin measurement, here photons can con-
tribute to all weighted measurements. As with binning, forming
weighted measurements reduces the amount of data that has
to be stored and processed. Since binning, with or without ex-
cluding some energies, are special cases, there must be sets of
weighting functions that perform at least as well as binning. We
seek to characterize the performance as a function of the choice
of weights and ultimately to find the optimal weighting func-
For weighted measurements, the random vector
posed of weighted sums of independent, Poisson distributed
values. Every photon of energy
. Except for the special case of binary, nonoverlap-
ping weights, since the weighted measurements
from the same realization of
no longer independent nor Poisson distributed. In such cases,
multivariate Gaussian models are commonly used because of
their convenient analytical form, tractability, and ability to ac-
curately capture the first- and second-moments of
a sufficiently high transmitted number of photons, this model
– [Fig. 4(a)]. Another
(4) is com-
contributes valueto mea-
, the elements of vectorare
the object, the weighted sum of discrete Poisson distributions
may not be well-approximated by a smooth, continuous distri-
bution. Nonetheless, assuming a jointly Gaussian distribution,
the CRLB approach can be used to evaluate estimator perfor-
mance as a function of incident spectrum and energy weights
that should be valid at least at high signal-to-noise ratio (SNR)
(Appendix B). For comparison, we also find the exact CRLB
sum of Poisson random variables is the convolution of the indi-
vidual scaled Poisson distributions, but has no known tractable
form. However, the exact distribution and CRLB
computed numerically using empirical characteristic functions
A. Optimal Weights
In principle, we can find the optimal weights by brute force,
treating the weights as variables over which we minimize some
objective function , . A more elegant approach, if pos-
sible, is to find weights that form a sufficient statistic for the
that are sufficient statistics for estimating , then the weighted
measurements would contain as much information about mate-
as the full spectrum .
More formally, let
be a random vector whose distribution
is parameterized by . Then, a statistic
if the conditional distribution of
independent of . Intuitively, simple examples of sufficient
statistics include counting the number of heads in a sequence
of coin flips rather than keeping track of the individual flips to
estimate the probability of heads or using the sample mean of
a sequence of independent, identically distributed random vari-
ables to estimate the mean of a normal or Poisson distribution.
To find a sufficient statistic, we use the Factorization Theorem
, which states that if
function (pmf) of
given , then statistic
if and only if there exist functions
such that, for alland
can be used to estimate as well as the original
Thus, for our case, the pmf of the detected spectrum
the material thicknesses
needs to be decomposed into the
product of one function of the statistic
of the detected spectrum that does not depend on . We begin by
writing the joint pmf of the detected spectrum
of the pmf at each energy since they are independent given
. We use our knowledge that the photon counts are Poisson
is a sufficient
is the joint probability mass
and , and a function
as the product
88IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 1, JANUARY 2011
distributed and assume that the incident spectrum
and can be treated as a constant
Therefore, we have proven that two weighted measurements
using the attenuation basis functions
weights form a sufficient statistic for dual energy imaging. By
definition, this sufficient statistic (i.e., these weighted measure-
ments) should achieve optimal performance for estimating any
with any incident spectrum. Our only assumption is that the
call these weights
and as their
To illustrate that the -weighted measurements form a suffi-
cient statistic,we cancomputetheCRLB.Weexpecttheperfor-
mance of the
-weighted measurements to be identical to that
of the full detected spectrum
addition, because the
-weighted measurements only require
two values derived from the detector signals, for comparison
we compute the CRLB for two other schemes that report two
measurements: optimal binning with two bins and using a CIX
detector that does both photon counting and energy integration.
for all objects. In
A. Performance Curves
Consider a range of relevant
cm). For the 120 kVp spectrum [Fig. 1(a)], the min-
tion of the amount of water and calcium is shown in Fig. 5(a) as
log contour lines. The uncertainty increases with increasing ob-
increase in the variance of
) when two bins are used with a
fixedthreshold of57keV, thethreshold thatminimizesthemax-
imum (over ) penalty factor, and Fig. 5(c) shows similar data
for the CIX detector. Binning with a threshold of 57 keV incurs
a penalty by a factor of 3.4 at the lower left corner
and by at least a factor of 1.7 throughout. A CIX detector incurs
lower penalties, especially at larger but still incurs a penalty of
at least 22% and up to a factor of two in Var
The ratio of
Gaussian model is shown in Fig. 5(d) for
thicker objects where the number of photons is low. However,
we know that any unbiased estimator cannot truly have better
performance than an estimator having knowledge of the full
detected spectrum. Therefore, the multivariate Gaussian model
may not be accurate for weighted measurements when the
number of photons detected is low. For example, the expected
number of photons detected for the thickest object
is only 39. Poisson distributions with low mean counts can only
take ona set of discrete values and are positivelyskewed, unlike
Gaussian distributions. This conclusion is supported by the fact
that the exact CRLB of the -weighted measurements from the
empirical characteristic function (Appendix C) is well within
0.1% of the CRLB of the full detected spectrum [Fig. 5(e)]. The
accuracy of this method simply depends on the discretization
of the Fourier space variable
and becomes a numerical issue.
These results show that
-weighted measurements achieve
the same performance in estimating
full spectrum. Although not shown here, the
andfor -weighted measurements also match those
of the full spectrum, while those of binning and the CIX exhibit
similar penalty ratios to their
different measurement schemes.
It is possible to have objects that decompose into a negative
amount of calcium or water (but not both). While only quadrant
of the calcium/water decomposition space
is shown in Fig. 5, our results extend smoothly into the valid
negative amounts of either basis material.
-weights when using the
. This ratio is
as knowledge of the
counterparts under the
B. Wedge Phantom
We also simulated the dual-energy performance of the three
different binning or weighting schemes on a phantom designed
to compare the calcium detectability of the methods. Consider
a projection image with 0.2 mAs exposure of a water wedge
phantom varying from 0 to 40 cm thickness in the horizontal
direction with square calcium contrast elements ranging in
area from 1 mm to 1 cm overlaid on top [Fig. 6(a)]. These el-
ements have a thickness such that the predicted (full area) SNR
of each calcium element from the full detected spectrum is 4 (at
the threshold of detectability). Because the necessary amount
of calcium to maintain a constant SNR increases as the water
gets thicker and as the elements becomes smaller, the calcium
images are displayed on a spatially varying grayscale window
ified as a function of horizontal position
[Fig. 6(b) and (c)].
WANG AND PELC: SUFFICIENT STATISTICS AS A GENERALIZATION OF BINNING IN SPECTRAL X-RAY IMAGING 89
Fig. 5. CRLB performance over a wide range of material thicknesses for (a) the full detected spectrum (cm ) and the multiplicative penalty factor ? for (b) two
bins with a 57 keV threshold (no notch), (c) counting and integrating, (d) optimal ?-weights with a Gaussian model, and (e) optimal ?-weights with the exact
numerically computed CRLB. Note the different contour levels in the subfigures. (a) ??????
? penalty: CIX. (d) ????? ?
? penalty: ?-weights, Gaussian. (e) ????? ?
?: Full spectrum. (b) ????? ?
? penalty: ?-weights, Exact.
? penalty: 2 bins, ? ? ?? keV. (c)
The numerical phantom was designed so that all contrast ele-
ments project onto an integer number of detectors and are per-
fectly aligned with the detector grid so as to avoid any partial
volume effects. Furthermore, while it is a practical concern, we
leave aside the issue of spatial resolution, count rates, and en-
ergy resolution by assuming an idealized photon counting de-
tector with no cross-talk, no count rate losses, and perfect en-
ergy resolution. Transmission of the X-ray spectrum and detec-
tion with energy resolution of 1 keV and perfect detection effi-
1 keV energy. The simulated noisy detected spectra were then
and weighted measurements were obtained from the same real-
ization of detected spectra.
Maximum-likelihood estimation (MLE) is a commonly used
method for estimating
given noisy measurements
 and was a convenient choice given the extensive use of the
log-likelihood functions (A.2) and (B.1) in deriving the CRLB.
We used Matlab’s Optimization Toolbox (v3.1.1) to find the
Although more sophisticated methods may exist for estimating
, these are beyond the focus of this paper. For the CIX and
pare theMLdecompositionofmeasurementstaken with:1)two
bins with a 57 keV threshold (no notch); 2) a CIX detector (i.e.,
); 3) -weights.
are shown in Fig. 6. Detection of the targets is challenging in
all the images since the phantom was designed to make this
so, but the -weights image is noticeably superior. The images
for increased conspicuity of all contrast elements as compared
with a two bin or CIX approach. The estimates are shown with
the spatially varying grayscale window
expected from the CRLB comparisons (Fig. 5), the noise in the
calcium image is highest when using two bins and lowest when
using -weights. The empirical SNR as measured with 10 000
realizations of this experiment is consistent with the theoretical
SNR using the CRLB (Fig. 5), although a bias in the estimates
becomes evident at the thick end of the wedge. As predicted by
Fig. 5(c), the advantage of -weights over CIX is largest in the
lower left corner of the wedge and drops off to the upper right.
VII. DISCUSSION AND CONCLUSION
A simple explanation for the improvement of binning per-
formance from including a notch between bins is that the sen-
sitivity of the ML estimate
to detected photons depends on
the photon energy. Fig. 7 plots the expected transmitted spec-
and the change in the material decomposition estimates
using the full detected spectrum when one additional photon of
is detected beyond the expected number of counts
for the object . Photons near 64 keV provide very
little information to the calcium estimate, whereas additional
photons at lower energies would induce the MLE to predict less
90IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 1, JANUARY 2011
Fig. 6. Calcium thickness estimation task for overhead projection with different weighting schemes, displayed on a spatially varying grayscale range ?????? ?
?????, where (a) is an illustration of the water wedge with calcium contrast elements of different sizes and thicknesses overlaid, (b) is the true calcium thickness,
(c) shows the grayscale window as a function of horizontal direction ?, and the estimated calcium thickness is from (d) two bins with a 57 keV threshold (no
notch), (e) counting and integrating, and (f) optimal ?-weights. (a) Wedge phantom. (b) True ?
. (c) Spatially varying ?. (d)? ?
: 2 bins, ? ? ?? keV. (e)? ?
Fig. 7. The expected transmitted spectrum ? for object ? ? ???????? and the
sensitivity of estimates? ?
to the number of photons ? at energy ?
when using the full detected spectrum. The shaded region is the range of photon
energies that should be removed when estimating ?
with two bins.
calcium and more water and vice-versa for additional photons
at higher energies because the ratio of calcium to water atten-
uation is much higher at lower energies. While photons in the
different energies within bin lose their individual energy iden-
in the same bin that have much more information. They there-
fore weaken the overall material discrimination ability.
Material decomposition is not only insensitive to photons in
the shaded energy range (Fig. 7), but these photons also move
the average energy of each bin closer to each other and reduce
the spectral separation between the bins. It has been shown for
dual kVp and dual layer methods that spectral separation is de-
sired –. Introducing a notch increases the spectral separa-
tion. While this is at the expense of fewer photons per bin, it
nonetheless improves the precision of the calcium (and water)
estimate. In analogous prior work on dual energy imaging with
dual layer detectors, it was found that adding a metal filter be-
tween the front and back layers improves material separation
performance. In that case and as in our work, discarding pho-
tons of intermediate energy improves the precision of material
decomposition even though it decreases the number of photons
that contribute to the measurements –.
Ideally, X-rays of the discarded energies would be removed
from the incident spectrum to reduce radiation dose, and the
net effect is equivalent to binning with a notch between bins.
If a notch filter is realizable (either at the source or detector),
material decomposition precision can be improved with the
same data storage, transmission, and estimation requirements
as without one. In practice such selectivity may not be possible
to achieve, but the large fraction of omitted photons (Fig. 7)
highlights the important role that source filtration and opti-
mization can play in dose efficiency. It is important to note that
the implementation of two bins with a notch at the detector
using comparators requires the same number of comparators
as using three bins. Without a doubt, forming a third bin from
the discarded photons instead and utilizing it enables better
material decomposition estimation.
Although adding more bins and notch filter(s) improves the
ning scheme is universally optimal. Therefore, binning cannot
WANG AND PELC: SUFFICIENT STATISTICS AS A GENERALIZATION OF BINNING IN SPECTRAL X-RAY IMAGING91
Fig. 8. Two sets of equivalent weights where the weights on the left are simply
the linear attenuation functions of calcium, water and iodine and the weights on
the right are linearly independent combinations of the attenuation functions. (a)
, and ? (b) Linear combinations.
for a large number of bins, the difference in performance may
be negligible. Using -weights, the detector performance is op-
timized for all spectra and objects, and thus we have decoupled
the joint optimization of the spectrum and detector weights into
simply an optimization of the spectrum.
-weighted measurements could perhaps be formed at
the detector using analog processing with two parallel circuits
that map pulse energy to
and, respectively, or digitally by determining
the energyof a detected photon from its pulse heightand adding
andto two different accumulators that represent
and, respectively. The values of
be embedded in hardware or stored in a look-up table. Incor-
-weights directly into the detectors may require
significant hardware redesign. Alternatively, the -weights can
reduce the dimensionality of the measurements to only two, al-
stant approximation to the attenuation curves and would have to
be further investigated. Because any linearly independent com-
and , the attenuation due to the photoelectric
effect and Compton scattering, respectively, can be used as
and , there are infinitely many selections for
Should the object possibly contain an additional material
that deviates from the dual basis material
assumption, such as a material with a K-edge within the de-
tected spectrum, the attenuation of this material can be used
as an additional basis function. The sufficient statistic proof
(5) can be extended to show that only three weighted measure-
ments are needed, with weights
previous two points, the selected weights could be
[Fig. 8(a), where material 3 is iodine, density: 4.93 g/cm ],
or they could be a more complicated linear combination, such
as normalized and flattened weights, where the third set of
weights has values near 0 for energies greater than 60 keV
[Fig. 8(b)]. The performance would be identical since any set
of three weighted measurements is entirely recoverable from
any other set of three weighted measurements through a simple
linear transformation, even in the presence of quantum noise.
Using the fact that
-weighted measurements form a suf-
ficient statistic, we have shown that a multivariate Gaussian
[i.e., an energy dependent
] followed by integrators that
and. To illustrate the
model for the distribution of weighted measurements may not
be accurate at low photon counts (the CRLB is lower than pos-
sible). This suggests that using a Gaussian model for the esti-
mation of the thicknesses can yield biased estimates. The loss
of accuracy in a Gaussian model is not limited to just -weights
and extends to CIX detectors and binning because the under-
lying problem is that the measurements are a weighted sum
of Poisson random variables instead of the commonly assumed
ficient estimators are beyond the scope of this paper, we expect
them to be important issues in accurate and SNR-efficient dual
energy decomposition. One solution may be to select a more
appropriate likelihood function, such as a Gaussian approxima-
tion when the number of photons is large and switching to a
discrete distribution at lower counts using the empirical charac-
teristic function. Future work will also focus on other practical
detectors, such as incorporating spectral response functions ,
In conclusion, the choice of binning and weighting schemes
can greatly affect the performance of dual-energy material
decomposition. Even with optimal thresholds, binning incurs
penalties to decomposition precision and is not robust to
changes in the source spectrum or object size, although this
can be mitigated by adding more bins. On the other hand,
-weighted measurements in theory form a sufficient
statistic for recording the detected photons and in our simulated
experiment do well when implemented with a multivariate
Gaussian approximation to the true probability mass function.
-weighted measurements outperform binned or CIX
measurements in our contrast wedge phantom simulation.
Estimating two unknown parameters is at the core of dual-en-
ergy theory, and we have proven that only two
measurements are sufficient to accomplish this without any loss
whose elements are given by
is over , a random variable. Then, if is an unbiased estimator
of , theCRLBprovidesalowerboundonthecovariancematrix
by stating that, where
Poisson random variables. Therefore, each bin count
a Poisson random variable, parameterized by
is the log-likelihood function of acquiring mea-
for object and the expected value of the argument
92IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 1, JANUARY 2011
notch filters are used. For measurements of an object character-
ized by thickness vector , the log-likelihood function (ignoring
the constant term) is
As shown in , the resulting CRLB for binning,
. One interpretation
special case of binning when we use the full detected spectrum
, the CRLB for the full spectrum is
is that each expected measurement
is weighted by
ance matrix of measurements
Ignoring the constant term, the log-likelihood function for a
Gaussian model of weighted measurements is
taken with weighting
, the expected value of mea-
. In addition,
will be , the covari-
The elements of the Fisher information matrix are derived in
,  and given by
eled by a Gaussian distribution is
EMPIRICAL CHARACTERISTIC FUNCTIONS
The characteristic function of a distribution is the Fourier
transform of its probability density function. For a random vari-
having a Poisson distribution with mean
teristic function is
, the charac-
where in this section
, then random variable
function of a sum of independent random variables is equiva-
lent to the product of their characteristic functions . That is,
. The characteristic function has
will have a characteristic
. Furthermore, the characteristic
has a characteristic function
the characteristic function of the multivariate random variable
able is the Fourier transform of its distribution
This can be numerically computedby discretizing and eval-
at these points, giving what is known as the empir-
ical characteristic function , . Then, taking the 2-D in-
verse discrete Fourier transform of the empirical characteristic
function gives us the empirical distribution of
depends on the sampled range of , while the range
depends on the spacing of the samples of
Expanding this method, we can also compute the Fisher ma-
. The reso-
is the support of . Then the exact CRLB comes from
. We have shown that
(C.5), and the other terms in the integral can be numerically
computed as well. These terms all contain partial derivative(s),
can be found using
WANG AND PELC: SUFFICIENT STATISTICS AS A GENERALIZATION OF BINNING IN SPECTRAL X-RAY IMAGING93
of one another, the partial derivative with respect to can move
inside the Fourier transform. For example,
The other partial derivatives can be similarly computed nu-
merically for any
. Then, because a discretized form of each
term in the integrand of (C.6) can be found, the Riemann inte-
gral can be found by summing the expression over all samples
to arrive at
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