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Scintillation event energy measurement via a pulse model based iterative deconvolution
method
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2013 Phys. Med. Biol. 58 7815
(http://iopscience.iop.org/0031-9155/58/21/7815)
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IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 58 (2013) 7815–7827 doi:10.1088/0031-9155/58/21/7815
Scintillation event energy measurement via a pulse
model based iterative deconvolution method
Zhenzhou Deng1,2
, Qingguo Xie1,2
, Zhiwen Duan3
and Peng Xiao1,2
1 Biomedical engineering department, Huazhong University of Science and Technology,
1037 Luoyu Road, Wuhan 430074, People’s Republic of China
2 Wuhan national laboratory for optoelectronics, 1037 Luoyu Road, Wuhan 430074,
People’s Republic of China
3 School of mathematics and statistics, Huazhong University of Science and Technology,
1037 Luoyu Road, Wuhan 430074, People’s Republic of China
E-mail: xiaopeng@hust.edu.cn
Received 26 June 2013, in final form 19 September 2013
Published 21 October 2013
Online at stacks.iop.org/PMB/58/7815
Abstract
This work focuses on event energy measurement, a crucial task of scintillation
detection systems. We modeled the scintillation detector as a linear system and
treated the energy measurement as a deconvolution problem. We proposed a
pulse model based iterative deconvolution (PMID) method, which can process
pileup events without detection and is adaptive for different signal pulse shapes.
The proposed method was compared with digital gated integrator (DGI) and
digital delay-line clipping (DDLC) using real world experimental data. For
singles data, the energy resolution (ER) produced by PMID matched that of
DGI. For pileups, the PMID method outperformed both DGI and DDLC in ER
and counts recovery. The encouraging results suggest that the PMID method
has great potentials in applications like photon-counting systems and pulse
height spectrometers, in which multiple-event pileups are common.
(Some figures may appear in colour only in the online journal)
1. Introduction
Solid scintillation detector is one of the most commonly used devices for high energy photon
and particle detection, due to its high detection efficiency, fast operation speed, low cost,
production capability and radiation hardness (Beer et al 1995, Wernick and Aarsvold 2004).
For scintillation detection systems, event energy measurement is a crucial task. One good
example is the spectrometer, which is widely applied in astronomical observation, geological
exploration, high-energy physics and so on. Its ability of identifying particular elements and
isotopes relies on the accurate energy measurement of the gamma rays emitted by the matter
being detected (Kleinknecht 1998, Knoll 2010). In nuclear medical imaging equipments
like positron emission tomography (PET) and single photon emission computer tomography,
energy information is used to reject Compton scatters and events not generated by the isotopic
0031-9155/13/217815+13$33.00 © 2013 Institute of Physics and Engineering in Medicine Printed in the UK & the USA 7815

7816 Z Deng et al
tracer. As a result, the system’s energy resolution (ER) has a significant influence on its scatter
fraction, sensitivity and noise equivalent counts (Watson 1997, Jaszczak et al 1985, Koral and
Dewaraja 1999). Moreover, energy information is even utilized in Anger logic algorithm to
localize an event’s position in PET systems equipped with position-sensitive photomultiplier
tubes (PMTs) (Anger 1958, Wong et al 1998).
Scintillation detectors are usually considered as a linear system, which will produce a
current signal with amplitude proportional to the energy of the detected photon (Monzo et al
2008, Knoll 2010). All kinds of analogue or digital event energy measurement methods have
been developed. Goulding et al (1994) used a quasi-trapezoidal pulse shaper to change the
original pulse into a flat-top form and then performed peak extraction to measure the energy
as peak’s height. Radeka (1972) proposed the gated integrator (GI) to integrate the gated
section of a signal with a time-invariant prefilter to calculate the energy. The GI filter is
quite complex and is vulnerable to low frequency extraneous noises such as microphony and
power supply ripple (Imperiale and Imperiale 2001). Based on digital signals, moving window
deconvolution algorithm (Stein et al 1996) fits the input pulse to an exponential decreasing
form, whose peak height is measured as the energy. This method requires the pulse tail to
be exponential and does not consider the noise property. Adaptive filters based on least-
mean-squares algorithm are proposed to adopt trapezoidal shaping in the characterization of
PET signals (Monzo et al 2008), with event energy expressed by the pulse height. Generally
speaking, measuring the energy by the pulse height is liable to the influences of signal–noise
rate, count rate and pulse shape (Young et al 2000, Pichler et al 2003, Musrock et al 2003,
Shao et al 2010).
The above methods are designed to calculate the energy of single events. Their accuracy
is severely affected when two or more pulses are piled up. Such piled-up events are usually
caused by high source activities, long pulse duration and etc. Pileup processing is important in
the energy measurement (Tenney 1984, Germano and Hoffman 1990, Debertin and Schötzig
1977, Gardner and Wielopolski 1977, Cano-Ott et al 1999). Lots of researches have been
focused on it over the past few decades, while the majority of them aimed at detecting the
pileups for rejection (Moszyński et al 1967, Máthé 1963, Tenney 1984, Blatt et al 1968,
Westphal 1979, Barradas and Reis 2006). The pileup rejection sacrifices the counting rate
for the purity of the detected events (Pommé 1999). Some other methods tried to recover
the event information from the detected pileups. The delay-line clipping (DLC) method
shortened the integration duration to measure the energy (Tanaka et al 1979, Woldemichael
2008, Raad and Cheded 2006). However, it will result in a loss of ER due to the clipping
of pulse tail and maybe a further loss of spatial resolution in position-sensitive detectors.
Assuming the pulse tail as an exponential shape, dynamic integrator length methods are
capable of recovering the energy information from pileups (Wong and Li 1998, Tanaka et al
2002).
In this work, we proposed an iterative energy measurement method, which can process
pileup without detection and is independent of signal pulse shape. Our method applies MLEM
algorithm to deconvoluting the digital scintillation pulses and then integrates the voltages of
the resulting spike signals to obtain energy information. The real world experiments showed
that the proposed method provided encouraging performance in ER and count rate recovery.
For singles data, measured energy using our pulse model based iterative deconvolution
(PMID) method is equal to that of the digital gate integrator (DGI). For pileups data, the
ER achieved by PMID at 511 keV is 12.88%, which is better than that of digital DLC
(DDLC). It is even close to the ER produced by DGI for singles data. Meanwhile, the
counts collected by PMID are 5.75 times to those collected by DGI with a 2% energy
window.

Scintillation event energy measurement via PMID 7817
2. Methods
2.1. Characteristics of the scintillation pulse
For a scintillation detector system, an incoming photon can be modeled as an impulse function
fi(t) = Eiδ(t − ti), where δ(t − ti) is a shifted Dirac delta function. For the photon with index
i, its energy and arrival time are denoted as Ei and ti, respectively. The overall inputs of the
system are a series of high energy photons and can be represented as f (t) = fi(t). The
output signals are waveforms denoted as p(t). The scintillation detection procedure is then
expressed as a linear convolution equation
p(t) = f (t) ∗ ϕ(t) + n(t), (1)
where ϕ(t) is the system’s unit impulse response function, n(t) is the noise and ∗ is the
convolution operator. If the arrival times of adjacent events are too close, there will be pileups
in p(t). Otherwise, p(t) contains only single events. To retrieve the energy information Ei, we
need to solve the inverse problem of (1), using the collected signal p(t) to compute f (t).
Solving (1) requires the unit impulse response function ϕ(t) to be formulated. For a fixed
scintillation detector, input signals with different energies will result in output pulses with
different amplitudes but of the same shape (Xie et al 2005). With input signal regarded as an
impulse function, the fixed shape of normalized output pulse is indeed a representation of the
system’s unit impulse response function ϕ(t). More specifically, the ϕ(t) can be defined by
the curve fitting of the normalized mean pulse.
The above mechanism is general enough to be applied to any scintillation detection system,
but the mean pulse of a particular system has to be obtained individually. Single pulses need
to be sampled and statistically analyzed to determine the shape of mean pulse. To account
for the procedure of formulating ϕ(t), we take the LYSO/PMT detector system used in our
experiments as an example. We collected 5000 pulses from the system with a 50 GSps high
speed oscilloscope. One typical pulse is shown in figure 1(a). Figure 1(b) shows the average
of all pulses with normalized amplitude and aligned arrival time. Based on the resulting mean
pulse, the unit impulse response function ϕ(t) is formulated as follows (Shao 2007):
ϕ(t) = K 1 − exp −
t
τ1
exp −
t
τ
u(t), (2)
where τ1 = 0.7013 ns, τ = 42.55 ns, u(·) is a unit step function and K is the system gain. It
should be noticed that the formula of impulse response function can be different for various
systems and will not affect the validity of the subsequent processing in our method. There is
no prerequisite on the mean pulse being of any particular shape, such as one-sided exponential
and so on.
Using the collected pulses, we also investigated the properties of noise n(t), especially
the relationship between noise level and mean pulse. The noise level is characterized by
the variance of the normalized and aligned pulses. As shown in figure 1(c), the variance of
pulses has the same shape as that of the mean. The decay constants of the exponential tails
are almost equal. Figure 1(d) shows the total variance of pulses with different energies. The
linear correlation coefficient resulted from the linear regression was 0.9993 and the slope was
0.027 93 keV. Based on above observations, we got
ñ(t) = C · ¯p(t), (3)
where the ¯p(t) and ñ(t) denote the mean pulse and the corresponding noise variance,
respectively. C is just a constant. Equation (3) suggests that the main noise component in
scintillation detector could be considered as multiplicative. Furthermore, the scintillation

7818 Z Deng et al
0 50 100 150
0
0.1
0.2
0.3
0.4
Time (ns)
Voltage(V)
(a)
0 50 100 150
0
2
4
x 10
−4
Time (ns)
Voltage(V)
τ = 42.55 ns
(b)
0 50 100 150
0
1
2
3
4
x 10
−9
Time (ns)
Variance(V
2
)
τ = 42.26 ns
(c)
300 400 500 600
6
8
10
12
14
16
18
Energy (keV)
Variation(keV2
)
(d)
Figure 1. (a) A typical scintillation pulse sampled by a high speed oscilloscope. (b) Mean of
the 5000 acquired pulses after normalization and alignment. The exponential fitting result shows
the decay constant is 42.55 ns. (c) Variance of the 5000 acquired pulses after normalization and
alignment. The exponential fitting result shows the decay constant is 42.26 ns. (d) Total variance
of pulses with different energy level.
photon emission can be characterized by an inhomogeneous Poisson process (Hero III 1991,
Choong 2009). Thus, it is safe here to assume the main noise as a filtered Poisson noise (Aykac
et al 2010).
2.2. Inversion of the Toeplitz matrix via MLEM iteration
In practice, signals mentioned above are handled in their discrete forms. We denote f = { fj},
p = {pl} and n = {nl} as the discrete forms of the input, output and noise signal sequences,
respectively. The system impulse response function is also expressed as a vector ϕ = {ϕr},
where ϕr = ϕ(r · t) and t is the sampling interval. r, j and l are element indexes in the
corresponding vectors. For the convenience of computation, the convolution operation in (1)
is rewritten as matrix multiplication. A Toeplitz matrix H = T{ϕ} is generated from ϕ and the
convolution (1) is transferred to
p = H f + n. (4)
The element of H is denoted as
hl j = ϕl−j, (5)
where ϕl−j is the (l − j)th element in vector ϕ. Obtaining f from (4) is an inverse problem
which can be solved by many kinds of well established methods (Stoer et al 1993, Greenbaum
1987), such as singular value decomposition (Mees et al 1987), conjugate gradients (Hestenes
and Stiefel 1952), Newton iteration, MLEM (Shepp and Vardi 1982, Bissantz et al 2008, Zeintl
et al 2010) and etc. When the dominant noise of signals p follows a Poisson-like distribution,

MLEM is especially suitable to realize the inversion of (4) (Shepp and Vardi 1982). The
general form of MLEM algorithm that expresses the updating procedure at the kth iteration is
fk
j =
fk−1
j
l
hl j l
hl j
pl
j
hl j fk−1
j
, (6)
where fk
j denotes jth element of solution f at the kth iteration. With enough rounds of
iterations, the solution will approach to the original impulse function f. Essentially, MLEM
is employed here as an instance of iterative deconvolution methods, which have been
used in other applications like image restorations (Richardson 1972, Schlueter et al 1994),
image enhancement (Kulkarni et al 1997), polymer analysis (Liu and Subhash 2006) and
chromatography (Crilly 1991).
In addition, for each solution produced during the iteration, the energy sum of the solution
fk
is always equal to that of the system output p. Without changing the sum, each iteration
just produces a redistribution of energy among the elements of solution. This is the so-called
energy conservation property of the MLEM algorithm (Pruksch and Fleischmann 1998), which
can be denoted as
j
fk
j =
l
pl. (7)
Another feature of MLEM is the nonnegativity of the solution fk
j , since it is updated by
multiplication with a positive value. The nonnegativity conforms to the real world situation. In
contrast, some methods, such as the DLC method, may yield a negative solution. It is also worth
mentioning that the choice of initial solution may affect the final result. It is not unreasonable
to set the initial solution of MLEM iteration here as the output of the deconvolution filtering,
which is usually close to the convergent solution (Fessler 1995).
Ideally, the solution after deconvolution should be a series of impulses with zero duration.
However, due to the noise and inaccuracy in signal measurement, the results are in fact spike-
like signals. Figures 2(b) depicts the outcome of MLEM iteration as the deconvolution result
for the scintillation signal shown in figure 2(a). Although not ideal impulses, most resulting
spikes are narrow enough to be well separated from each other. We can conveniently retrieve the
event arrival time using the peak extraction technique (Raad and Cheded 2006) and calculate
the event energy by integrating each individual spike’s voltage.
The above processing would be sufficient for all the singles and most pileups. However,
there may still be a very small fraction of resulting spikes which remain inseparable from
each other, like those in the dashed boxes in figure 2(b). Such inseparable spikes indicate that
the corresponding events are piled up too closely. If the handling of these seriously piled-up
events is necessary, we can conduct an additional two-step processing which is similar to the
method proposed in Wong and Li (1998), Tanaka et al (2002). The first step is estimating the
pulse amplitude using the shape information given by the mean pulse, the truncated integral of
the frontier scintillation pulse and the events’ arrival time given by peak extraction on spikes.
The second step is subtracting the estimated pulse from the original piled-up pulses. The two
steps are executed recursively for N-1 times for N-event pileups.
3. Experiment
3.1. Experimental setup
To evaluate the performance of the PMID method using MLEM algorithm on energy
measurement, we carried out experiments to obtain real world event data with a gamma

7820 Z Deng et al
0 100 200 300 400 500 600 700 800
0
0.1
0.2
0.3
Time (ns)
Voltage(V)
(a)
0 100 200 300 400 500 600 700 800
0
1
2
3
4
Time (ns)
Voltage(V)
(b)
Figure 2. (a) The original scintillation pulses, containing pileups and singles. (b) The solution after
the MLEM iteration in the PMID method.
(a ) (b )
Dark box
Digital oscilloscope
Radioactive source
Scintillation crystal
PMT
(c )
Figure 3. (a) The LYSO scintillation crystal is in the size of 16.5×16.5×10 mm3. (b) The detector
consists of an LYSO crystal coupled to a Hamamatsu R9800 photomultiplier tube. (c) The pileups
data experiment setup. Only one channel of oscilloscope was used.
ray detector shown in figure 3. An LYSO crystal was optically coupled to a Hamamatsu
R9800 PMT via one of its six facets, while the other five were wrapped in aluminum foil.
The PMT’s operation voltage was set to 1300 V and its output was directly connected to a
Tektronics DPO 71604 digital storage oscilloscope with a 50 termination. The oscilloscope
was operated with a 16 GHz bandwidth and a 50 GSps sampling rate per channel. With an inner
diameter of 5.0 mm, a test tube filled with F18-FDG solution was placed near the crystal’s top
facet as the radiative source. The whole system setup is illustrated in figure 3(c).

0 100 200 300 400 500 600
0
1
2
3
x 10
4
Energy (keV)
Counts
DGI :ER=12.62%
DDLC:ER=13.74%
PMID :ER=12.62%
Figure 4. The energy histogram obtained from 900 000 single events. Results based on events
acquired at channel 1 of the digital scope is displayed here.
The PMID method was compared with the digital versions of GI (Weng 1995) and
DLC (Hatch 1977) methods, referred to as DGI and DDLC respectively, in two experiments
performed with different source activities. The first experiment aimed to acquire singles data
with a 0.05 mCi source. In this experiment, totally 900 000 singles were acquired. Each single
event was sampled by the oscilloscope for 200 ns, resulting in 10 000 data points. The second
experiment was to collect pileups data. It used a 1.35 mCi source and thus the count rate was
high enough to generate pileup events. The pulse data were collected as frames, each of which
contained 5 × 107
data points continuously sampled in a duration of 1 ms. Totally 150 frames
were obtained and used for evaluating the pileup processing performance of DGI, DDLC and
PMID methods. Both the DGI and DDLC methods calculated the energy by digital integration
of a duration of the signal sequence, which was triggered by the signal’s surpassing of 60 mV.
The length of integration duration was 200 ns for DGI and 42 ns for DDLC. However, DDLC
will clip the pulse tails before the integration, by subtracting a delayed copy of the signal from
a amplified copy. The delay time was set to 42 ns, approximately equal to the time constant of
the pulse tail.
3.2. Experimental results
At first, energy measurement performances of DGI, DDLC and PMID methods were evaluated
with singles data and the results are compared in figure 4. The ER was calculated from the
Gaussian fitting to the photopeak of energy histogram. Both DGI and PMID obtained an ER
of 12.62% at 511 KeV, while DDLC got 13.74%. For DDLC method, its short integration
duration limited the collection of scintillator photons and thus deteriorated the ER (Tanaka
et al 2002).
The energy measurement results for the pileups data are shown in figure 5. There was
a peak around 1000 keV in the energy spectrum of DGI method, which has no pileup
discrimination. This peak represented the misinterpretation of pileups as singles. There was
no such pileup peak in the energy spectra of DDLC and PMID, both of which were capable of
pileup processing. As expected, ERs of all the methods were worse than those for singles data
due to pulse aliasing. The corresponding ERs were 22.48%, 14.88% and 12.88% at 511 keV
for DGI, DDLC and PMID, respectively. DGI method produced the worst ER, because its
long integral duration was very vulnerable to the disturbance of the piled-up event. For the
two methods capable of pileup discrimination and processing, PMID method outperformed
DDLC and its ER was even close to that of singles data.

7822 Z Deng et al
0 200 400 600 800 1000 1200 1400 1600
0
0.5
1
1.5
2
x 10
4
Energy (keV)
Counts
DGI :ER=22.46%
DDLC:ER=14.88%
PMID :ER=12.88%
Figure 5. The energy histogram obtained from pileups data.
50G 25G 10G 1G 500M 200M 100M
14
16
18
20 No Shaper
3ns Leading Edge
5ns Leading Edge
7ns Leading Edge
Sampling Rate (Sps)
EnergyResolution(%)
Figure 6. The ER evaluation of PMID on pileups data with different sampling rates and pulse
shapes.
The above results were obtained from data sampled at 50 GSps. The pre-stored pulses data
with a high sampling rate allow electronic designer to evaluate their DSP algorithms with good
operability and flexibility. However, 50 GSps is not necessary for the PMID method to process
the pileups. We investigated the sampling rate’s effect on the PMID method. Data points were
picked up at different intervals from the original sequence to simulate different ADC sampling
rates, ranging from 100 MSps to 50 GSps. Thereafter, the re-sampled waveform was screened
and processed by the PMID method. Figure 6 shows that the ER of PMID method remained
almost unchanged until the sampling rate was decreased to 1 GSps. With a 500 MSps sampling
rate, the PMID method still maintained an ER of 13.26% at 511 keV. Since different pulse
shapes of single-photon responses will affect the required sampling rate, we evaluated another
three pulses with the rise time of 3, 5 and 7 ns. These pulses are obtained by applying different
digital Gaussian shaper on the original pulses and the relative ER versus sampling rate curves
are shown in figure 6. The results indicate that the requirement on sampling rate decreases
with the increasing of the rise time.
We also evaluated the counting rate performance with pileups and depicted the results
in figure 7. Since DGI cannot discriminate and process pileups, its counting rates were the
worst with every energy window. DDLC and PMID recovered valid counts from the piled-up
events and provided much better counting rates. Furthermore, the counting rates of PMID
were higher than those of DDLC, mainly because of PMID’s better ER. Those results verified
the superior pileup processing ability of PMID.
The computing burden of PMID was analyzed to investigate its feasibility. At first,
we estimated its computational complexity. One iteration of PMID involves mainly
two convolution operations, which can be further decomposed as an FFT operation, a

2 5 8 10 15 20
0
2
4
x 10
5
DGI
DDLC
PMID
Energy Window (%)
Counts
Figure 7. The restored counting evaluation. The chosen energy window is 2%, 5%, 8%, 10%, 15%
and 20% of 511 keV energy peak.
0 10 20 30 40 50
0
0.5
1
1.5
2
2.5
3
3.5
x 10
4
Total Number of Iterations
RequiredComputingCapability(Gflops)
Sample Rate = 1 GSps
Figure 8. Computing capacity requirements of PMID in real-time mode with different data
sampling rates and total iteration numbers.
multiplication, a division and an IFFT operation. Considering the symmetry of FFT and
IFFT, we can approximate one PMID iteration as four FFT operations. If the number of
samples is assumed as N, the computational complexity of PMID should be O(N log(N)), on
the same level as that of FFT. On the other hand, DGI algorithm requires an over-threshold
operation and an integration operation, while DDLC requires one more multiplication and one
more subtraction. Thus, both of their computational complexities can be estimated as O(N).
Although PMID’s computational complexity is O(log(N)) times larger than those of DGI and
DDLC, it will become a problem only in applications with real-time processing requirement.
We further assessed the computing capability required to realize PMID in real-time mode. If
the signal sequence’s length is 1 us and the sampling rate is x GSps, there is N = 1000x. Since
the computing burden of FFT is (34/9) log2(N)N floating-point operations when N > 64
(Johnson and Frigo 2007), that of one PMID iteration is (136/9) log2(N)N floating-point
operations. The computing capability required to execute the PMID with 20 iterations in
real-time mode is (136/9) log2(N)N ×20/1000 ≈ 302.2 log2(1000x)x Gflops (Giga floating-
point operations per second). With a sampling rate of 1 GSps, the corresponding computing
capability requirement is about 3000 Gflops as shown in figure 8). Such requirement can be
satisfied by nowadays high-end GPU, such as the Tesla K20X, whose peak floating point
performance is 3950 Gflops (NVIDIA 2012).
To further understand the performance of the PMID method, we monitored the changes of
the pulse shape, event energy, ER and corresponding elapsed time during the first 50 iteration.
Figure 9(a) suggests that the resulting signals were narrowing down as the iteration went on,

7824 Z Deng et al
0
0.1
0.2 Original Sequence
0
0.5 Iteration 2
0
1
2
Iteration 10Voltage(V) A B C D E
0
5 Iteration 20 F
200 400 600 800 1000 1200 1400
0
5
Time (ns)
Iteration 40
(a)
12 5 10 20 30 40 50
350
400
450
500
Energy(keV)
Iterations
2−Event Pileup: A B
Single Event: C
3−Event Pileup: D E F
A B C D E F
(b)
0 10 20 30 40 50
350
360
370
380
390
400
ElapsedTime(s)
Iterations
0 10 20 30 40 50
0.12
0.13
0.14
0.15
0.16
0.17
EnergyResolutionElapsed Time
Energy Resolution
(c)
Figure 9. (a) The original pileup scintillation sequence and the deconvoluted spikes at the 2nd,
10th, 20th, 40th iteration. (b) The energy convergence in the first 50 iterations. (c) The energy
resolution and corresponding elapsed time in the first 50 iterations.
until they were spike-like. In this way, most piled-up events became distinguishable from each
other. The energy values of single events such as C in figure 9(b) remained constant, equal
to the integration of the original scintillation pulses. That conformed to the PMID’s energy
conversation property mentioned in subsection 2.2. For the two-event pileup (like events A and
B in figure 9(b)) or three-event pileup (events D, E and F in figure 9(b)), the total energy also
remained constant while its distribution among the piled-up events would change during the
iteration until a convergence was reached. Both figures 9(b) and (c) indicate that convergence

happens at around 20 iterations, when the variation of the mean energy was less than 0.1%.
Figure 9(c) also shows that the elapsed time of PMID is around 370 s per iteration. That is
obtained in the Matlab environment running in a PC with an Intel Core i3-3220@3.30 GHz
CPU and 8GB 1600 MHz Kingston DDR3 memory.
4. Discussion and conclusion
Based on the mean pulse of scintillation signals, we constructed a linear model for scintillation
detection systems and thus transferred the measurement of event energy to a deconvolution
problem. Thereafter, we applied the MLEM algorithm to solve this problem. Experiments
with real world signals showed that this method provided good ERs with both the singles and
pileups. The counting rate performance of this method was also better than the widely used
DGI and DDLC methods.
The proposed method does not require the mean pulse of scintillation signals to be in a
particular shape, as long as it is ﬁxed. In contrast to it, dynamic integrator methods proposed
in Wong and Li (1998), Tanaka et al (2002) are based on the one-sided exponential model.
Since different front-end electronics in detection systems may produce different pulse shapes,
the adaptability of the proposed method would be appreciated in applications.
Another issue worth mentioning is that the majority of pileups and singles are processed
uniformly in the proposed method, without any speciﬁc detection or processing for pileups.
Only a small number of seriously piled-up events, i.e. less than 3% of the total events under
a 10 Mcps counting rate, need further processing. Thus, the proposed method is unlikely to
encounter the false pileup detection caused by noise. This feature may have made a contribution
to its good performance.
In the future works, we will apply the PMID method to data obtained by multi-voltage
threshold digitizer instead of the high speed ADC. Since both nulti-voltage threshold method
and the PMID method require the prior knowledge of pulse shape, they could be well-
matched. Other iterative algorithms, such as maximum a posterior or ordered subset expectation
maximization, can also be considered in the PMID method.
Acknowledgments
The authors would like to thank Mr Yanzhao Li and Mr Xiaoqing Cao for the inspiring discus-
sions, Ms Yongqian Chen and Mr Mohammed Abdullah Abdusalam Ahmed for proofreading
the paper, Ms Yawen Zheng, Mr Ming Niu and Mr Jun Zhu for assistance in the experiment
setup. This work was supported in part by the Natural Science Foundation of China (NSFC)
grant #U1201256 and #61027006, the Ministry of Science and Technology of China (MOST)
grant #2012DFG31970, the Research and Development Programme of Hubei Province grant
#2011BFA005, the Wuhan Programs for Science and Technology Development grant
#201231234461 and the Jiangsu Province Natural Science Foundation grant #BK2011329.
Some related computations were performed on the High Performance Computing Center exper-
imental testbed in SCTS/CGCL (http://grid.hust.edu.cn/hpcc) and the High Performance Com-
puting platform provided by Computer Technology Application Key Lab of Yunnan Province.
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