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 78 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. 79 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. 80 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. 81 H. Zheng et al. Nuclear Inst. and Methods in Physics Research, A 906 (2018) 77–82 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. 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