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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 62, NO. 3, JUNE 2015 805
Quadratic Programming Time Pickoff Method for
Multivoltage Threshold Digitizer in PET
Zhenzhou Deng, Student Member, IEEE, and Qingguo Xie, Member, IEEE
Abstract—Multi-voltage threshold (MVT) digitization, which
has been implemented in our preclinical scanner–Trans-PET Bio-
CaliBurn, is a low-power sampling solution with a reasonable cost
for fast scintillation pulses. This MVT digitizing scheme employs
a few comparators with programmable reference voltages for
determining the time points when the scintillation pulse crosses
the user-defined voltage thresholds. After obtaining the digital
time samples, use of various sophisticated linear or nonlinear al-
gorithms to improve the accuracy of timing information becomes
possible. In the previous implementation of MVT digitizers, the
arrival time of a scintillation pulse was determined by the linear
fitting (LF) algorithm, which assumes leading edge as a straight
line, and not involves the transition time on the falling edge. It is
not unreasonable to expect achieving a better timing resolution in
MVT-based PET detectors by the combinatorial optimization of
MVT samples both on leading and falling edges. In this paper, a
novel method, referred to as quadratic programming (QP) method
is therefore proposed to mark the time stamps of sampling event
pulses. In this method, the arrival time is directly expressed as
a parametric combination of the MVT samples. The parameters
in the combination are obtained by the quadratic programming
which minimizes variation of timing error. To evaluate the perfor-
mance of QP/MVT method and other time pickoff methods, we
setup two gamma ray detectors using crystal and
Hamamatsu R9800 PMT. The scintillation pulses are directly read
out by Tektronics DPO 71604 digital storage oscilloscope. CTR of
175 ps was obtained by QP/MVT, and 191 ps by LF/MVT, when
four thresholds were employed in each of the two channels. The
experimental results indicate the potential advantage of QP/MVT
in timing determination. Meanwhile, the assumption of linear
leading edge, which is the basis of LF method, was demonstrated
to be improper in the data analysis. For QP method, probably of
even greater significance is the manner to define the parameters
rather than the resulting detector-specified parameters.
Index Terms—Coincidence timing resolution, mean square
error, multi-voltage threshold digitizer, quadratic programming
(QP) time pickoff method.
Manuscript received July 10, 2014; revised November 27, 2014; accepted
February 17, 2015. Date of publication May 08, 2015; date of current ver-
sion June 12, 2015. This work was supported in part by the National Key
Scientific Instrument and Equipment Development Project of China under
Grant 2013YQ030923, by the International S&T Cooperation Program of
China (ISTPC) under Grant 2014DFR10670, by the National Key Technology
R&D Program of China Grant 2012BAI13B06, by the Key Project of Chinese
Ministry of Education 313023, and by the Major International (Regional) Joint
Research Project of the Natural Science Foundation of China under Grant
61210003.
The authors are with the Wuhan National Laboratory for Optoelectronics,
Wuhan 430074, China, and also with the BioMedical Engineering Department,
Huazhong University of Science and Technology, Wuhan 430074, China
(e-mail: dengzhenzhou@gmail.com; qgxie@hust.edu.cn).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNS.2015.2416349
I. INTRODUCTION
TIMING information or time-of-flight (TOF) information
in Positron Emission Tomography (PET) is expected
to improve image quality [1]–[25], which has been widely
accepted by the researchers since the early 1980s. The TOF
information can be used to spatially localize the source of a
gamma pair originating from a positron annihilation and being
detected in coincidence. If a 500 picoseconds (ps) full-width at
half maximum (FWHM) coincidence timing resolution (CTR)
can be achieved in a PET system, many benefits can be realized
for whole-body FDG imaging [1], [5]–[8], [8]–[10], [15].
Furthermore, a target value of 100–200 ps FWHM in CTR
would lead to a spatial precision of 1.5–3 cm along the line of
response (LOR), allowing the localization of the organ under
examination (e.g., pancreas, heart, prostate or lymph nodes)
and an efficient rejection of the events outside the region of
interest. Also, hadron therapy would greatly benefit from a
fast on-line monitoring of the dose delivered during proton
or heavy-ion therapy treatment, which requires very high
resolution, high sensitivity and fast reconstruction imaging of
-emitting isotopes produced by beams or target spallation
processes during the irradiation [26], [27].
The TOF information is extracted from scintillation pulses
with a time pickoff method, which is an important factor for the
CTR of a TOF PET scanner [28]. The most widely used time
pickoff method is the leading edge discriminator (LED) [29],
[30] and the constant-fraction discriminator (CFD) [31], [32].
The development of VLSI and the digital sampling tech-
nology opened a door for the various sophisticated algorithms
of time pickoff to improve accuracy of timing information, such
as statistics-based methods [33]–[39], optimum filter-based
methods [40]–[46], compensating-based method [47], digital
implementations of LED/CFD (DLED/DCFD) [48]–[50] and
curve-fitting methods [50]–[54]. Digital sampling system
provides additional degrees of freedom with respect to more
feasible choices of combination function and parameters. More
degrees of freedom necessarily translate into better optimum
performance, although whether the improvement is sufficient
to motivate the increased complexity should be verified.
Scintillation pulses in a TOF PET are so fast. Hence, a very
high sampling rate digitizer is required to accurately extract
a minimum of three sample points on the rising edge of the
pulse [36]. This practically limits the digital approach in TOF
PET systems due to the high cost and power consumption of
high-speed ADCs.
As an alternative scheme to ADC, the multivoltage threshold
(MVT) digitization [51] has been proposed. This digitizing
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806 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 62, NO. 3, JUNE 2015
Fig. 1. Digitizing scheme of MVT. This digitizing scheme employs a few
discriminators with programmable voltage thresholds for determining the time
point when the scintillation pulse crosses these user-defined thresholds.
scheme employs a few discriminators with programmable
voltage thresholds for determining the time point when the
scintillation pulse crosses these user-defined thresholds as
shown in Fig. 1. Based on MVT method, we have implemented
an LYSO/PSPMT block detector with digital DAQ, which is
then used in a preclinical scanner–Trans-PET BioCaliBurn. In
the initial implementation, we have employed linear fitting (LF)
algorithm to pickoff the arrival time of a scintillation pulse.
CTR of about 300 ps has been obtained, which is comparable
with that of the CFD method [31]. In such implementations,
the fitting process could partially correct the time walk, but
only samples on the leading edge were involved. Considering
falling edges dominate the energy information and time walk is
relevant to the energy information, a better timing resolution is
expected by the combinatorial optimization of MVT samples
both on leading and falling edges.
In this paper, a novel method, referred to as quadratic
programming (QP) method, is proposed to pickoff the event
time of scintillation pulses based on MVT samples. In this
method, the arrival time is directly described as a parameterized
combination of the MVT samples. Quadratic programming is
then used to optimize the parameters of the combination using
the variation minimization criteria. For convenience, LF and
QP method with MVT samples are denoted as LF/MVT and
QP/MVT, respectively. To demonstrate the time performance
improvement of QP/MVT, a comparison experiment based on
a prestored coincidence pulses library was performed. Encour-
aging CTR of 175 ps was obtained by QP/MVT, and 191 ps by
LF/MVT, when four thresholds were employed in each of the
two channels. Meanwhile, QP method produced monotonically
decreasing CTRs as the employed threshold number increased.
II. QUADRATIC PROGRAMMING METHOD FOR MVT SAMPLES
A. Modeling for the MVT Samples
In a scintillation detector system, an incoming photon can be
modeled as an impulse function , where
is a shifted Dirac Delta function, and are random
variables representing the energy and the arrival time of the
photon striking into the detector with index , respectively. The
scintillation pulse without noise is then expressed as
(1)
where is the system response function for the shift-in-
variant condition, which is satisfied for the most of scintilla-
tion detectors, even for nonlinear photoelectric devices, such as
the silicon photomultiplier. In the case of linear shift-invariant
system [55]–[57], the system output can be simplified as the
product . The time of the leading edge for the
threshold voltage , ( ) can be further ex-
pressed using an implicit function
(2)
where is the partial derivatives of with respect to . We
assume all the pulses (1) are reversed to be positive beforehand.
Similarly, the threshold-crossing time on the falling edge is
(3)
Let’s take the derivative of both sides of (2) and (3), then
(4)
and
(5)
(4) and (5) interpret the linear relation between measurement
expectation of the switching time and entrance time . For
the two-detector condition, . Thus, the time difference
in detection between Detector 1 and 2 for an event at location
from the center of the LOR is
(6)
where is the speed of light in vacuum. and
are the time on both leading and falling edge for the th
threshold and the th detector. If we use the combina-
tion
to estimate
, the following constraint of the combination should be
satisfied:
(7)
This relation can be rewritten as
(8)
(7) and (8) regulate the choice of the combination .
B. Objective Function and Constraints
In order to express the optimization, the event arrival time for
th detector is denoted as
(9)
where , is the denotation for each of the
measured pulses.
is the total set of properties provided by the MVT samples.
For convenience, we rewrite it into a matrix
. Each is a column

DENG AND XIE: QUADRATIC PROGRAMMING TIME PICKOFF METHOD FOR MVT DIGITIZER IN PET 807
vector of elements, which are .
for the simple case of two detectors. Then is the
combined matrix of columns and rows. We consider the
linear expression of (9) as
(10)
where the -element column vector contains the coeffi-
cients which define the time pickoff for the th detector. Then
the time difference is
(11)
We denote the average of as . Then the constructed quadratic
programming equation involved in the CTR is
(12)
Where is the row vector whose elements are all 1. The con-
straint part of (12) defines the scalar of the time shift, which
is numerical expression of (8). Various methods can be used
to figure out the QP problem, including analytical and iterative
methods [58]. And more sophisticated objective function with
more regularization terms can improve robustness and accuracy.
The solution of (12) will be discussed in the Section II-C.
C. Solution of the Quadratic Programming
We define a elementary matrices to apply elementary
column operation
...
...
...
...
...
(13)
which has rows and columns, element to
are , and diagonal elements are all 1, the other elements are
all 0. We denote , , then
. Submitting into
(14)
Here row vector . (14) means that the
constraint of (12) is satisfied, once the first element of
equals to 1. We define a matrix of rows and
columns to extract the lower elements of
...
...
...
...
(15)
The -element , and
and
and then we let vector
, Z is the combination of the centered and
, E is the combination of and . The
constrained QP problem transforms into an unconstrained least
square problem
(16)
(16) has the solution
(17)
and
(18)
The solution expressed as (17) can be applied with greedy
strategy to choose threshold voltages. For more general condi-
tion, (16) can be used to generate nonlinear expression of (9)
by adding high order items in matrix .
D. Relation Between LF Combination and the QP Constraint
QP method chooses an optimized solution from the set of fea-
sible solutions which satisfy the given constraint. In fact, the
LF solution is also an optimal solution which targets on fit-
ting samples to a linear curve. The LF solution employs only
the forward time points as input, so the involved is
an size matrix for LF/MVT. The basic assumption
for LF/MVT is that the pulse voltage is modeled as a linear
function of time, , where is the
slope of the line. After obtaining
, the fitting processing is employed in LF/MVT to define
the parameters of the line, such as slope and intercept. We de-
note the estimation output from fitting processing are slope
and intercept , the mean of the threshold voltages
and the threshold-crossing time are and
, respectively, then
(19)
The LF solution can also be rewritten as the form of
, where is the vector which contains weights of
the th threshold-crossing time as (20).
(20)
where is the threshold index, . We can see (20)
satisfies the constraint of (12), . That
means LF solution is contained in the set of feasible solution of
QP, but the object of LF is not to reduce time error.
E. Numerical Example for 4-Threshold Condition
When the threshold number , there are 4 crossing time
samples of the leading edge and 4 crossing time samples of the
tail edge for each pulse with enough amplitude. For the simple
case of two detectors, each coincidence pulse pair is recorded

as 16 time samples: eight time samples for Detector 1 and eight
for Detector 2. is the total set of time
samples for Detector 1, and for De-
tector 2. Suppose there are 100 000 pulse pairs, then both matrix
and matrix have eight columns and 100 000 rows. Time
pickoff method applied on the two detectors outputs 100 000
pairs of time value, which are the functions of the eight input
time samples.
When only the first order items are considered
(21)
and
(22)
the parameters and are the
required values, which are defined by the quadratic program-
ming. The parameters satisfied the constraint
(23)
and
(24)
and the minimizing objective function is the square error of the
time difference
(25)
After the elementary operation (13) and (15), the constrained
optimization problems transforms into an unconstrained least
square problem (16).
(26)
and is an 100 000 14 size Matrix
(27)
where
Then we can obtain the 14-element vector
according (17). Both and
have seven elements. The solution of and
are
(28)
and
(29)
Submit the parameters (28) and (29) to (21) and (22), then the
arrival times for Ch1 and Ch2 are
(30)
and
(31)
III. EXPERIMENT SETTINGS AND RESULTS
A. Experiment Settings and Pulses Dataset Description
To evaluate the performance of the time pickoff methods, we
carried out experiments to obtain event data with two gamma
ray detectors as shown in Fig. 2. The : crystals
of the size as shown in Fig. 3 were
optically coupled to Hamamatsu R9800 photomultiplier tubes
(PMT) via the round glass of the bottom facets, while the
other facets were wrapped in Aluminium housings. The supply
voltages of the both PMTs were set to 1300 V and the PMT’s
outputs were directly connected to a Tektronics DPO 71604
digital storage oscilloscope with a 50- input impedance. The
oscilloscope was operated with bandwidth of 16 GHz and
sampling rate of 50 GSps per channel. One or three tubules
of 1.2-mm inner diameter filled with solution
were used as the radioactive source. A pair of detectors worked
in the coincidence mode, which ensured that the collected
data were generated by the solution. The coarse
coincidence timing window width was set to 4 ns. The oscil-
loscope was triggered by an AND-logic event generated by
the two : /PMT detectors, therefore ensuring that the
majority of the resulting event pairs were coincidences. The
trigger voltage was set to 180 mV to reduce false triggering.
Each of the two pulses in a coincidence was sampled by the
oscilloscope for 100 ns, resulting in 5000 data points. An
energy window of keV was applied in the energy
discrimination. The measured gain ratio of the two detectors
was 0.600. Although the chosen experimental setup is far away
from the state-of-the-art data acquisition systems, it has the
fundamental advantages of being easily reproducible, as no
custom digital electronics is required, while still allowing a full
and direct comparison between the various digital time pickoff
under test. Experimental system like this has been used in [43],
[45], [46], [50], and [51].
B. Timing Comparison Between DLED, LF/MVT, QP/MVT
Related With Leading-Edge, Double-Edge Samples, and DCFD
Four threshold-based methods: DLED, LF/MVT, QP/MVT
with leading-edge samples (LESs) and double-edge samples
(DESs) were investigated in this comparison. Four voltage
thresholds were employed in LF/MVT and QP/MVT. Consid-
ering the two channels worked with different gain, 7.5, 223.5,
439.5, and 655.5 mV were employed for Channel 1, and 12.5,
372.5, 732.5, and 1092.5 mV for Channel 2. The employed

Fig. 2. Experimental system with coincidence of 4 ns time window. All the
pulse pairs are saved to be analyzed in the off-line mode. The FDG is contained
in a test tube, which is fixed between the crystals.
Fig. 3. : scintillation crystal pair contained by aluminium boxes of
glass bottom windows.
voltages in DLED were 37.5 and 62.5 mV for Channels 1 and
2, respectively. The CTR (FWHM) was calculated from the
Gaussian fitting to the photopeak of the histogram of the time
difference. Besides LED method, CFD is also a commonly
used standard timing method. Thus, we additionally evaluated
the digital version of CFD, referred to as DCFD method, using
the same data. In the DCFD method, the attenuation and delay
time are respectively 0.1538 and 1.6 ns. In Table I CTRs and
their corresponding confidence bounds are listed. The table
shows a better timing resolution is obtained by use of DES
QP/MVT (Fig. 4).
The time spectrum using DES QP/MVT for radioactive
source of three points is also shown in Fig. 5. The fact that the
mean time differences agree with space differences justifies the
scalar of the QP time pickoff.
C. Linear Range of Leading Edge After Aligned
The basic assumption of LF/MVT can be verified by a simple
numerical testing. If leading edges of pulses are linear, the mean
value of aligned samples on leading edges will also be linear. In
other words, if the mean value of aligned samples on leading
edges is not linear, leading edges of pulses cannot be linear.
Fig. 4. TS produced by QP/MVT with samples on the double edges. This his-
togram contains 100 000 counts.
TABLE I
CTR OF DIFFERENT TIME PICKOFF METHODS
Fig. 5. Timing spectra recorded with DES QP/MVT method, using a
source of three points located at mm mm (Left Peak),
mm mm (middle peak), and mm mm
(right peak). The red curves indicate Gaussian fits to the data. Each histogram
contains 210 000 counts.
Fig. 6(a) and (b) show aligned leading edge and double edge
samples for Channels 1 and 2. Only a part of the leading edge
acts as a line. The lower part of the leading edge is nonlinear, and
contains important information about timing. This result reveals
that the leading edge shape is too complicated to be regarded as
a line. Figs. 7 and 8 show the coincidence event pulse pair output
by the two detectors.
D. Threshold-Specified Evaluation Between LF/MVT and
QP/MVT
We evaluated the time performance for LF/MVT, LES
QP/MVT, and DES QP/MVT when different thresholds were
employed. In this evaluation, we added the employed voltage
threshold one by one. Every time a new employed threshold
was added into the calculation, the old employed thresholds
were unchanged. The th additional threshold voltage is
(32)

Fig. 6. Averaged scintillation pulses output by the detector pairs triggered by
coincidence events. (a) Leading edge part. (b) Leading edge and tail edge part.
where is
(33)
.
The th increasing voltage was set as far as possible from the
nearest old threshold between and . and
was set according to the gain of the two channels.
mV and mV.
In Fig. 9, the performance comparison shows QP/MVT out-
strips LF/MVT for each number of employed thresholds, no
matter when only LES or both DES are involved. And QP/MVT
with DES is better than that with only LES. This result reveals
another property of QP method: the CTRs monotonically de-
creased as the number of the chosen threshold increased. The ad-
ditional degree of freedom does not deteriorate the CTR, when
the QP optimization is employed. In the worst condition that the
newly added degree of freedom is independent of the existing
time information, the QP solution will deﬁne the weight of the
newly added degree of freedom to be zero, and the existing time
information is kept. All data points have error bars representing
Fig. 7. Scintillation pulses output by the detector pairs triggered by a coinci-
dence event. (a) Leading edge part. (b) Leading edge and tail edge part.
Fig. 8. Overlays of several scintillation pulses output by the detector pairs trig-
gered by coincidence events.
the errors in the measurement. CTRs and the corresponding Er-
rors with some typical threshold numbers are shown in Table II.
In Fig. 9, the timing results using the LF/MVT method (the
blue curve) produce a repeating pattern with adding threshold

Fig. 9. CTR as function of involved thresholds number for LF/MVT, LES QP/MVT and DES QP/MVT. CTR of the QP methods show monotone decreasing
feature while LF/MVT not. LF/MVT performs worse when the newly added voltage threshold is already too high.
TABLE II
CTRS AND THE CORRESPONDING ERRORS IN FIG. 9 (PS)
one by one. We can derive the time difference using LF/MVT
from (20), (10), and (11)
(34)
and when , . We denote the
centered as , then the coincidence time resolu-
tion of LF/MVT is
(35)
Since the newly added threshold voltage periodically increase
from to with doubling period, the calculated
with increasing threshold number make the cyclically changed
contribution of the thresholds from to . In the mean-
time, the has exhibited the two properties: 1) significant
correlations were found between the crossing time of adjacent
thresholds [Fig. 10(a)]; 2) the time jitters initially decreased
and then increased with increasing voltage of the threshold
[Fig. 10(b)]. Properties of and (Fig. 10) result in that
the in (35) approximately periodically changes in Fig. 9.
This effect can not occur in QP/MVT. Since the weight
of QP method is derived not by threshold voltage setting but
by the optimization process, none periodicity exists in of
QP method in the threshold setting (33). QP/MVT optimizes the
weight coefficient in the combination of the given time points.
Then QP/MVT involving more thresholds produces CTR not
worse than that involving less thresholds. From Fig. 9, we also
can see that the employed samples on the falling edge result
in better performance. In practice, no extra electronic hardware
is required for extracting DES than extracting LES, when the
speed of data access is enough. So it is feasible to employ DES
in most instances, even though the improvement is small.
IV. CONCLUSION AND DISCUSSION
In this work, we have proposed a PET event timing pickoff
method using QP to optimize the combination of MVT sam-
ples on both leading and falling edges. QP method aims at min-
imizing the variation of time differences with the constraint of
the fixed time shift scale. We derived the solution of this QP
problem. Experimental results showed the potential advantages
of using QP/MVT to retrieve timing information.
In addition, we have analyzed the LF solution. The LF solu-
tion has a similar structure to QP solution. Then the computa-
tional complexity is nearly the same. Furthermore, we found
the LF solution is belonging to the set of the feasible solu-
tion in QP/MVT. However, the goal of LF is to approach a
linear leading edge. The assumption of linear leading edge was
demonstrated to be improper in the data analysis.
At the same time, one distinct feature of the QP method is that
the QP method with more degrees of freedom does better than
with less. This feature is due to the programming: the weight
of the newly added degree of freedom will be set as zero, if the
newly added degree of freedom is irrelevant to the existing time
difference.
We also found that samples on the falling edge have contri-
bution to time resolution. The falling edge dominates the energy
information, which can be used to calibrate time walk in the time
pickoff. The QP time pickoff involving DES not only extends
the degrees of freedom, but also makes full use of the available
energy information in falling edge. In practice, no extra elec-
tronic hardware is required for extracting DES than extracting

Fig. 10. (a) Correlation coefficient matrix of in (34). (b) Time jitters mea-
sured from for different threshold voltages.
LES, when the speed of data access is enough. So it is feasible
to employ DES in most instances, even though the improvement
is small.
Another issue worth mentioning is that additional thresholds
will be rewardless for CTR improvement, when more than 10
thresholds have been applied in QP/MVT method. This reveals
the truth that regular ADC, which works with many quantiza-
tion levels, has much redundancy to measure the arrival time
of the scintillation pulses. In contrast, MVT digitizing scheme
avoids the data redundancy by employing the tunable reference
voltages. Since comparators and TDCs are relatively easy to im-
plement, MVT digitizing scheme is expected to be used exten-
sively in the future due to its low cost.
Last but not least, the proposed QP method can also be ap-
plied to other systems of multiple pulse samples, such as mul-
tiple CFD systems, PSPMT readout systems, and light-sharing
systems. It offers a new way of combining the obtained samples
with optimal parameters by the programming.
ACKNOWLEDGMENT
The authors would like to thank S. Patrick, Y. Li, and M.
Ahmed for carefully proof reading the manuscript, and the two
anonymous reviewers for their helpful comments on an earlier
draft of this paper.
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