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Nuclear Inst. and Methods in Physics Research, A 906 (2018) 77–82
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research, A
journal homepage: www.elsevier.com/locate/nima
An improved algebraic reconstruction technique for reconstructing
tomographic gamma scanning image
Honglong Zheng a,b , Xianguo Tuo b,c ,∗, Shuming Peng a , Rui Shi b,c , Huailiang Li c , Aijing He c ,
Zhigang Li c , Qiang Han b,c
a
Institute of Nuclear Physics and Chemistry, China Academy of Engineering Physics, Mianyang 621900, China
College of Chemistry and Environmental Engineering, Sichuan University of Science & Engineering, Zigong 643000, China
c
Fundamental Science on Nuclear Wastes and Environmental Safety Laboratory, Southwest University of Science and Technology, Miangyang 621010, China
b
ARTICLE
INFO
Keywords:
Tomographic gamma scanning
Improved algebraic reconstruction technique
Image reconstruction
Total variation minimization
Self-adaptive relaxation factor
ABSTRACT
Tomographic gamma scanning (TGS) is one of the most advanced non-destructive techniques for assaying the
radioactive waste drum. When traditional algorithms are adopted to accurately reconstruct TGS images, the
measurement data must be completely sampled by TGS system, which inevitably leads to a long-term assay.
In order to save time, small number of data is measured by dividing the drum into several large voxels, which
leads to inaccurate TGS images. In this work, an improved algebraic reconstruction technique (IART) is proposed
to reconstruct TGS images. The total variation minimization method and the self-adaptive relaxation factor are
applied to improve the iterative process of traditional algebraic reconstruction technique (ART). Experimental
results show that this IART algorithm can accurately reconstruct TGS images simply based on small amount
of measurement data. Compared with traditional technique, this method can reduce the mean square error
and improve the signal-to-noise ratio of transmission images. And it can improve the positioning accuracy of
radioisotope and the accuracy of reconstructed activity.
1. Introduction
Tomographic gamma scanning (TGS) is one of the most advanced
non-destructive techniques for assaying the radioactive waste drum [1,
2]. Without destructing the drum, TGS technique can obtain the characteristic information of waste materials inside the drum and it has been
successfully applied to measure the heterogeneous waste drum [3]. It
was in 1990s that the Los Alamos National Laboratory (LANL) firstly
put forward the TGS technique and built the prototype [4]. TGS system
consists of two different measurement procedures. One is for transmission tomography and the other is for emission tomography [5]. The
role of transmission tomography is to determine the linear attenuation
coefficient map of voxels and the role of emission tomography is to
locate the radioisotope and determine its activity. The transmission
image is applied to correct emission data so as to improve the accuracy
of activity calculation. Researches of TGS technique have been achieved
in prototype, image reconstruction algorithm and efficiency matrix [6–
9]. When the traditional algorithm is adopted to accurately reconstruct
TGS images, measurement data must be completely sampled by TGS
system, which inevitably leads to a long-term assay. Usually, in order to
save time, small number of data is measured by dividing the drum into
several large voxels, which leads to inaccurate TGS images [10].
The compressed sensing (CS) theory asserts that signal can be
recovered from small number of samples which are far fewer than
the requirement of the nyquist sampling theory [11,12]. The CS
theory is successfully applied to reconstruct medical CT image when
measurement data are sparsely sampled [13–15]. In the TGS image
reconstruction, values of image are scattered and the sparsity is uncertain. However, the image can be considered as piecewise-smooth and its
gradient module is of sparsity [16]. The finite difference transformation
of the image is sparse while the image itself has no sparsity. That is,
the TGS image can be reconstructed under the framework of CS theory.
Therefore, based on the measurement data that sparsely sampled, a
rapid approach to reconstruct accurate TGS images can be achieved
by adopting the total variation minimization to improve the iterative
process of traditional algorithm.
In this work, the total variation minimization method and the selfadaptive relaxation factor are applied to improve the iterative process of
traditional ART algorithm. And the effectiveness of this IART algorithm
is confirmed by the experimental measurement.
2. Method
2.1. Transmission image reconstruction
TGS transmission image reconstruction issue can be expressed by
 = 0 exp(

∑
−  )
=1
∗ Corresponding author at: College of Chemistry and Environmental Engineering, Sichuan University of Science & Engineering, Zigong 643000, China.
E-mail address: myconnectionmail@126.com (X. Tuo).
https://doi.org/10.1016/j.nima.2018.07.095
Received 28 June 2018; Received in revised form 12 July 2018; Accepted 31 July 2018
Available online xxxx
0168-9002/© 2018 Elsevier B.V. All rights reserved.
(1)
H. Zheng et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 77–82
them. Nevertheless, the constant relaxation factor cannot accurately
describe this relationship [19]. Therefore, we use a binomial function
to describe this relationship as
where 0 is the photon counting rate for the transmission source without
attenuation and it is depend on the activity of the source.  is the
photon counting rate attenuated by the matrix in the drum in the ith
transmission measurement.  is the linear attenuation length of the
gamma ray when it is emitted from the transmission source, enters
the detector and passes through the jth voxel in the ith transmission
measurement.  is the linear attenuation coefficient of the jth voxel. J
is the total number of voxels. The transmission measurement data can
be expressed in a vector form as I=(1 , 2 , . . . ,  , . . . ,  )T . I is the total
number of times of transmission measurement.
Define  = −ln( ∕0 ), then Eq. (1) is converted as

∑
  = 
 =  ⋅ 2 +  ⋅  + 
(10)
where a, b, c are the parameters of the function. The value range
of  is from 0.2 to 0.8. The TGS image can be seemed as a twodimensional function and , is the value of mth row and nth column,
1≤m≤M, 1≤n≤N. The finite difference transformation is used as a sparse
transformation and its L1 norm (Total Variation, TV) is used as an
objective function for optimization problem [20]. Image values can be
obtained by solving the objective function as following
(2)
min ‖‖  , .. =  ⋅ 
=1
(11)

⋅=
(3)
The gradient formula of total variation is
where X is a × sized matrix consisting of each element  , U=(1 ,
2 , . . . ,  , . . . ,  )T , P=(1 , 2 , . . . ,  , . . . ,  )T . In the jth voxel, the
functional relationship between linear attenuation coefficient and the
energy can be represent as
 () = 0 + 1 exp(−∕2 ) + 3 exp(−∕4 )
2, − −1, − ,−1
 ‖‖ 
≈ √
,
 + (, − −1, )2 + (, − ,−1 )2
(4)
−√
where  ( = 0, 1, … , 4) is parameters of the function.
,+1 − ,
 + (,+1 − , )2 + (+1, − +1,−1 )2
2.2. Emission image reconstruction
−√
TGS emission image reconstruction issue can be expressed by
ℎ =

∑
   
(5)
 =
exp(−  )
(6)
=1
where ℎ is the counting rate for the drum in the ith emission measurement.  is the detection efficiency for the jth voxel in the ith emission
measurement.  is the activity of the jth voxel. f is the branching
ratio of interested gamma ray.  is the linear attenuation length of
the gamma ray in the kth absorbing voxel when it is emitted from the
jth voxel in the ith emission measurement.  is the linear attenuation
coefficient of the kth absorbing voxel.
Define  =   f, then Eq. (5) can be converted as
ℎ =

∑
 
⃖⃖⃗(2 −1) () =

 ‖‖ 
| ( −1)
, = 2  ()
(13)
∧ (2 −1)
⃖⃖⃗(2 −1) ()

(14)
| ⃖⃖⃗(2 −1) |
()|
|
|
|
The iterative format of total variation minimization along the gradient direction is like

() =
( −1)
∧ 2
( −1)
( )
 2  () =  2  () −  ()
()
(15)
‖ (2 )
‖
(1)
‖
 () = ‖
(16)
‖   ( − 1) −    ()‖
‖
‖
where  is the relaxation index. 2 is the iterative number of total
variation minimization and 2 = 1, 2, . . . , 2 . k is the iterative number
of IART,  = 1, 2, . . . , K.
The steps of the IART algorithm are as following:
(1) Define (1)
( = 1) = , c is initial value.

(1 )
(2) 
() is obtained by Eqs. (9) and (10) of 1 times iteration.
(1 )
(3) Then define (1)
() = 
().
 
(2 )
(4)    () is obtained from Eq. (12) to Eq. (16) of 2 times
iteration.
( )
( )
(5) If ‖ 2 () −  2 ( − 1)‖ < , or  ≥ , the iteration ends.
If not, then the algorithm continues from step (2) and (1)
( + 1) =

( )
 2 ().
(7)
=1
⋅=
(12)
 + (+1, − , )2 + (+1, − +1,−1 )2
where  is a minimal positive value and its value is 10−8 .  is introduced
to avoid the denominator to be zero. The gradient descent method is
applied to solve the total variation minimization (TVM). The gradient
and the gradient direction of the image total variation can be calculated
as Eq. (13) and Eq. (14).
=1

∏
+1, − ,
(8)
where E is a I ×J sized matrix consisting of each element  , A=(1 , 2 ,
. . . ,  , . . . ,  )T , H=(ℎ1 , ℎ2 ,. . . , ℎ , . . . , ℎ )T .
2.3. Image reconstruction algorithm
Among various algorithms of image reconstruction, the algebraic
reconstruction technique (ART) is a common choice [17,18]. In the
ith TGS measurement, the iterative format of ART algorithm can be
expressed as
∑
( −1)
 − =1   1
( −1)
( )
 1 =  1 + 

(9)
∑
2
=1 
3. Experiment
where 1 is the iterative number of ART (1 = 1, 2, . . . , 1 ). i is the serial
number of TGS measurement (1≤i≤I ). j is the serial number of voxel
( )
( −1)
(1≤j≤J ).  1 and  1 are the new and current estimates respectively.
 is the relaxation factor.  is the ith measurement data.  is the
element of system matrix.
In the TGS image reconstruction, the relaxation factor  is related to
the measurement data  . When  increases, the corrected offset value
will be raised in the iteration, and there exists a direct ratio between
The tomographic gamma scanner includes an external collimated
source 152 Eu, a platform for the radioactive waste drum, a machine
control system and a gamma ray spectrometer system which consists of a collimated HPGe detector and a multichannel analyzer. The
tomographic gamma scanner is shown in Fig. 1. By controlling the
platform, the radioactive waste drum can be rotated coaxially and
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H. Zheng et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 77–82
4. Results and discussion
4.1. Transmission image
Taking the center of the drum as the origin point in polar coordinates,
one layer is divided into 96 (4 circles × 24 angles) or 864 (12 circles × 72
angles) voxels and three methods, including 96-ART, 864-ART and 864IART, are used to reconstruct transmission images. The three methods
above are adopted to reconstruct the transmission images of six gamma
rays as 0.122, 0.344, 0.779, 0.964, 1.112, 1.408 MeV and the results are
shown in Fig. 3. Fig. 3(a-1) to (a-6) are reference images from of these
six gamma rays. In reference images, the linear attenuation coefficients
of the seven materials are obtained from the individual attenuation
experiments. Fig. 3(b-1) to (b-6) are transmission images of 0.122,
0.344, 0.779, 0.964, 1.112, 1.408 MeV reconstructed by 96-ART. Since
the images consisting of 96 voxels, the spatial resolution is low and the
images cannot accurately describe the distribution of different materials
in Fig. 2. Fig. 3(c-1) to (c-6) are transmission images reconstructed
by 864-ART. This is an ill-posed problem since 864 unknown values
are supposed to be solved with only 96 measured data. Results show
that the transmission images cannot be accurately reconstructed with
ART algorithm. Although the spatial resolution is improved, images are
unreliable from Fig. 3(c-1) to (c-6). Fig. 3(d-1) to (d-6) are transmission
images reconstructed by 864-IART. The spatial resolution of images is
improved and the shape of images is consisted with the distribution of
different materials in Fig. 2. It is notes that the IART algorithm works
much better than traditional ART algorithm in solving this ill-posed
problem.
Fig. 1. Tomographic gamma scanner.
moved horizontally. And the gamma spectrum can be acquired with the
spectrometer system.
In the transmission measurement, gamma rays of 0.122, 0.344,
0.779, 0.964, 1.112, 1.408 MeV are emitted by 152 Eu (2.892×108 Bq)
and collimated to pass through the drum. Seven materials have been
used for constructing the heterogeneous matrix of the drum including:
fiber of 0.21 g cm−3 (1 ), water of 1.00 g cm−3 (2 ), polyethylene of
1.04 g cm−3 (3 ), plastic of 1.41 g cm−3 (4 ), glass of 1.44 g cm−3 (5 ),
concrete of 2.02 g cm−3 (6 ) and aluminum of 2.70 g cm−3 (7 ) in Fig. 2.
A gamma point source 137 Cs (3.273 × 105 Bq) is placed in the drum as
point a in Fig. 2.
The TGS experimental measurement includes two patterns of movements, namely coaxial rotation and horizontal movement shown in
Fig. 2. And the measurement steps can be described as following:
Step 1, put the drum into Position 1(245 mm away from the center).
Step 2, make the drum rotate coaxially in Position 1 with 15◦ for one
rotation leading to 24 sectors in all.
Step 3, move the drum horizontally to the Position 2(175 mm away
from the center)
Step 4, make the drum rotate coaxially in Position 2 with 15◦ for one
rotation with 15◦ for one rotation leading to 24 sectors in all.
...
The following steps are the same as above.
Only one layer measured in the experiment. 96 measurement data
of transmission and 96 of emission are measured.
The mean square error (MSE) and signal-to-noise ratio (SNR) of
transmission images are calculated by
 
1 ∑∑
( (, ) −  (, ))2
 =1 =1 
∑ ∑
2
=1
=1 ( (, ) −  (, ))
]
 = −10 log[
∑ ∑
2
=1 ( (, ))
=1
 =
(17)
(18)
where  (m, n) and  (m, n) are the reconstruction value and
reference value of mth row and nth column of image respectively. The
MSE and SNR of transmission images obtained by 96-ART, 864-ART and
864-IART are in Table 1. From Fig. 3(b-1) to (b-6), images are divided
into 864 voxels to calculate their MSE and SNR. Overall, the MSE of 864IART is lower than both 96-ART and 864-ART. And the SNR of 864-IART
is higher than both 96-ART and 864-ART. It indicates that 864-IART
method works much better in reconstructing transmission images when
the small number of data is measured.
Fig. 2. Experimental measurement of TGS.
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H. Zheng et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 77–82
Fig. 3. Transmission images of 0.122, 0.344, 0.779, 0.964, 1.112, 1.408 MeV with 96 transmission data. (a-1)–(a-6) Reference images, 12×72 voxels. (b-1)–(b-6) Reconstructed with
ART, 4×24 voxels. (c-1)–(c-6) Reconstructed with ART, 12×72 voxels. (d-1)–(d-6) Reconstructed with IART, 12×72 voxels.
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H. Zheng et al.
Nuclear Inst. and Methods in Physics Research, A 906 (2018) 77–82
Fig. 4. Efficiency maps of four detection positions. (a) Position 1. (b) Position 2. (c) Position 3. (d) Position 4.
Fig. 5. Emission images of 0.662 MeV with 96 emission data. (a) Reconstructed with ART, 4×24 voxels. (b) Reconstructed with ART, 12×72 voxels. (c) Reconstructed with IART, 12×72
voxels.
Table 1
The MSE and SNR of reconstructed images.
Table 2
The emission reconstruction results of 137 Cs activity.
Energy (MeV) MSE
0.122
0.344
0.779
0.964
1.112
1.408
SNR (dB)
Method
Reconstructed (Bq)
Actual (Bq)
Error (%)
96-ART
864-ART
864-IART
96-ART 864-ART 864-IART
3.22 × 10−3
1.91 × 10−3
8.60 × 10−4
7.39 × 10−4
5.89 × 10−4
5.06 × 10−4
5.45 × 10−3
2.56 × 10−3
1.28 × 10−3
1.12 × 10−3
8.57 × 10−4
7.76 × 10−4
3.58 × 10−3
1.64 × 10−3
8.18 × 10−4
7.20 × 10−4
5.46 × 10−4
4.92 × 10−4
12.09
9.76
10.90
11.14
10.64
11.22
96-ART
864-ART
864-IART
3.572 × 105
2.771 × 105
3.378 × 105
3.273 × 105
3.273 × 105
3.273 × 105
9.14
−15.34
3.21
6.82
6.86
6.93
6.95
6.89
6.95
11.02
11.27
11.40
11.40
11.41
11.50
The emission reconstruction results of 137 Cs activity are shown in
Table 2. From their relative errors, the IART algorithm greatly improves
the accuracy of the reconstructed activity.
4.2. Efficiency map
5. Conclusion
A gamma point source 137 Cs of 0.662 MeV is placed in the drum
and its branching ratio is 85%. Detection efficiency of 0.662 MeV in
different voxels can be measured by experimental method. Efficiency
maps of four detection positions (Position 1, 2, 3, 4 in Fig. 2) are shown
in Fig. 4. For one detection position, when the drum is coaxially rotated
with equal angles, the distribution of efficiency in current measurement
is same with that of the next measurement.
An improved algebraic reconstruction technique is proposed to
reconstruct TGS images. The measurement data of transmission and
emission are sampled by TGS system. The total variation minimization
method and the self-adaptive relaxation factor are applied to improve
the iterative process of traditional ART algorithm. Compared with traditional algorithm, this method could not only reduce the mean square
error and improve the signal-to-noise ratio of transmission images, but
also improve the positioning accuracy of radioisotope and the accuracy
of activity calculation. In conclusion, experimental results show that this
method is effective.
4.3. Emission image
Emission images represent the distribution of radioisotope activity in
the drum. According to the transmission images of 0.122, 0.344, 0.779,
0.964, 1.112, and 1.408 MeV, the linear attenuation coefficient map of
0.662 MeV in voxels are calculated by Eq. (4). Based 96 emission data
and combined with the efficiency maps as well as the linear attenuation
coefficient map of 0.662 MeV, emission images are reconstructed by
three methods as 96-ART, 864-ART and 864-IART in Fig. 5. Fig. 5(a) is
an emission image with 96 voxels reconstructed by ART algorithm. An
image with only 4×24 voxels cannot accurately locate the radioisotope.
Fig. 5(b)is an image with 864 voxels reconstructed by ART algorithm
which shows the same result as Fig. 5(a). Fig. 5(c) with 864 voxels is
reconstructed by IART algorithm. The positioning accuracy of radioisotope in Fig. 5(c) is much better than those of Fig. 5(a) and Fig. 5(b).
Acknowledgments
This work is supported by the Young Scientists Fund of the National
Natural Science Foundation of China (No. 41604154), the Sichuan Major
Science and Technology Achievement Transformation Program of China
(No. 2017CC0076), and the Sichuan Science and Technology Program
of China (No. 2017GZ0362).
Appendix A. Supplementary data
Supplementary material related to this article can be found online at
https://doi.org/10.1016/j.nima.2018.07.095.
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References
[10] L. Zhe, Z. Li, Review of -ray CT for radioactive waste assay, CT Theory Appl. 23
(2014) 1025–1040.
[11] D.L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006) 1289–
1306.
[12] E.J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform.
Theory 52 (2006) 489–509.
[13] A. Boudjelal, Z. Messali, A. Elmoataz, et al., Improved simultaneous algebraic
reconstruction technique algorithm for positron-emission tomography image reconstruction via minimizing the fast total variation, J. Med. Imaging Radiat. Sci.
48 (2017) 385–393.
[14] E.Y. Sidky, X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol. 53 (2008) 4777–
4807.
[15] C. Zhang, The Research of Compressed Sensing Algorithm for CT Reconstruction,
University of Chinese Academy of Sciences, 2016.
[16] K. Kim, H. Cho, U. Je, et al., Improvement of image characteristics in high-voltage
computed tomography (CT) by applying a compressed-sensing (CS)-based image
deblurring scheme, Ndt & E Int. 84 (2016) 11–19.
[17] R.J. Estep, T.H. Prettyman, G.A. Sheppard, Tomographic gamma scanning to assay
heterogeneous radioactive waste, Nucl. Sci. Eng. J. Amer. Nucl. Soc. 118 (1994)
145–152.
[18] P.P. Bruyant, Analytic and iterative reconstruction algorithms in spect, J. Nucl.
Med. Off. Publ. Soc. Nucl. Med. 43 (2001) 1343–1358.
[19] R. Jia, Research on CT Image Reconstruction Technology Based on Compressed
Sensing, Northeastern University, 2014.
[20] J. Ma, C. Chen, C. Li, et al., Infrared and visible image fusion via gradient transfer
and total variation minimization, Inf. Fusion 31 (2016) 100–109.
[1] W. Gu, C. Liu, N. Qian, et al., Study on detection simplification of tomographic
gamma scanning using dynamic grids applied in the emission reconstruction, Ann.
Nucl. Energy 58 (2013) 113–123.
[2] K. Wang, Z. Li, W. Feng, Reconstruction of finer voxel grid transmission images in
Tomographic Gamma Scanning, Nucl. Instrum. Methods Phys. Res. A 755 (2014)
28–31.
[3] T.H. Anh, T.Q. Dung, Evaluation of performance of gamma tomographic technique
for correcting lump effect in radioactive waste assay, Ann. Nucl. Energy 28 (2001)
265–273.
[4] R.J. Estep, TGS [underscore] FIT: Image reconstruction software for quantitative,
low- resolution tomographic assays, Los Alamos National Laboratory, 1993.
[5] J.C. Palacios, L.C. Longoria, J. Santos, et al., A PC-based discrete tomography
imaging software system for assaying radioactive waste containers, Nucl. Instrum.
Methods Phys. Res. 508 (2003) 500–511.
[6] M. Han, Z. Guo, H. Liu, et al., Influence of different path length computation
models and iterative reconstruction algorithms on the quality of transmission
reconstruction in Tomographic Gamma Scanning, Nucl. Instrum. Methods Phys.
Res. A 861 (2017) 16–22.
[7] R. Venkataraman, M. Villani, S. Croft, et al., An integrated tomographic gamma
scanning system for non-destructive assay of radioactive waste, Nucl. Instrum.
Methods Phys. Res. A 579 (2007) 375–379.
[8] T. Roy, M.R. More, J. Ratheesh, et al., Active and passive CT for waste assay using
LaBr3(Ce) detector, Radiat. Phys. Chem. 130 (2017) 29–34.
[9] Y. Cheng, Study on the scale model of efficiency matrix of Tomographic Gamma
Scanning, Shanghai Jiaotong University, 2007.
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