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April  2020, 14(2): 317-337. doi: 10.3934/ipi.2020014

Simultaneous reconstruction of emission and attenuation in passive gamma emission tomography of spent nuclear fuel

1. 

Department of Mathematics and Statistics and Helsinki Institute of Physics, University of Helsinki, FI-00014 Helsinki, Finland

2. 

Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland

3. 

Helsinki Institute of Physics, University of Helsinki, FI-00014 Helsinki, Finland, and TRIUMF, Vancouver, BC V6T 2A3, Canada

4. 

School of Engineering Science, LUT University, , FI-53850 Lappeenranta, Finland

5. 

Helsinki Institute of Physics, University of Helsinki, , FI-00014 Helsinki, Finland

* Corresponding author: Tatiana A. Bubba

Received  May 2019 Revised  October 2019 Published  February 2020

In the context of international nuclear safeguards, the International Atomic Energy Agency (IAEA) has recently approved passive gamma emission tomography (PGET) as a method for inspecting spent nuclear fuel assemblies (SFAs). The PGET instrument is essentially a single photon emission computed tomography (SPECT) system that allows the reconstruction of axial cross-sections of the emission map of an SFA. The fuel material heavily self-attenuates its gamma-ray emissions, so that correctly accounting for the attenuation is a critical factor in producing accurate images. Due to the nature of the inspections, it is desirable to use as little a priori information as possible about the fuel, including the attenuation map, in the reconstruction process. Current reconstruction methods either do not correct for attenuation, assume a uniform attenuation throughout the fuel assembly, or assume an attenuation map based on an initial filtered back-projection reconstruction. We propose a method to simultaneously reconstruct the emission and attenuation maps by formulating the reconstruction as a constrained minimization problem with a least squares data fidelity term and regularization terms. Using simulated data, we show that our approach produces clear reconstructions which allow for a highly reliable classification of spent, missing, and fresh fuel rods.

Citation: Rasmus Backholm, Tatiana A. Bubba, Camille Bélanger-Champagne, Tapio Helin, Peter Dendooven, Samuli Siltanen. Simultaneous reconstruction of emission and attenuation in passive gamma emission tomography of spent nuclear fuel. Inverse Problems & Imaging, 2020, 14 (2) : 317-337. doi: 10.3934/ipi.2020014
References:
[1]

Treaty on the non-proliferation of nuclear weapons, United Nations Treaty Series, 729, 1995, URL https://www.un.org/disarmament/wmd/nuclear/npt/text. Google Scholar

[2]

C. Bélanger-ChampagneP. PeuraP. EerolaT. HonkamaaT. WhiteM. Mayorov and P. Dendooven, Effect of gamma-ray energy on image quality in passive gamma emission tomography of spent nuclear fuel, IEEE Transactions on Nuclear Science, 66 (2019), 487-496.  doi: 10.1109/TNS.2018.2881138.  Google Scholar

[3]

Y. Berker and Y. Li, Attenuation correction in emission tomography using the emission data - A review, Medical Physics, 43 (2016), 807-832.  doi: 10.1118/1.4938264.  Google Scholar

[4]

S. Bonettini and M. Prato, New convergence results for the scaled gradient projection method, Inverse Problems, 31 (2015), 095008, 20pp. doi: 10.1088/0266-5611/31/9/095008.  Google Scholar

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S. Bonettini, R. Zanella and L. Zanni, A scaled gradient projection method for constrained image deblurring, Inverse Problems, 25 (2009), 015002, 23pp. doi: 10.1088/0266-5611/25/1/015002.  Google Scholar

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A. Bousse, A. Sidlesky, N. Roth, A. Rashidnasab, K. Thielemans and B. F. Hutton, Joint activity/attenuation reconstruction in SPECT using photopeak and scatter sinograms, in Proceedings of 2016 IEEE Nuclear Science Symposium, Medical Imaging Conference and Room-Temperature Semiconductor Detector Workshop, Strasbourg, France, 2016, 1–4. doi: 10.1109/NSSMIC.2016.8069448.  Google Scholar

[7]

S. C. Cade, S. Arridge, M. J. Evans and B. F. Hutton, Use of measured scatter data for the attenuation correction of single photon emission tomography without transmission scanning, Medical Physics, 40 (2013), 082506. doi: 10.1118/1.4812686.  Google Scholar

[8]

Y. CensorD. E. GustafsonA. Lent and H. Tuy, A new approach to the emission computerized tomography problem: Simultaneous calculation of attenuation and activity coefficients, IEEE Transactions on Nuclear Science, 26 (1979), 2775-2779.  doi: 10.1109/TNS.1979.4330535.  Google Scholar

[9]

V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography, Inverse Problems, 15 (1999), 931-960.  doi: 10.1088/0266-5611/15/4/307.  Google Scholar

[10]

D. GourionD. NollP. GantetA. Celler and J.-P. Esquerre, Attenuation correction using SPECT emission data only, IEEE Transactions on Nuclear Science, 49 (2002), 2172-2179.  doi: 10.1109/TNS.2002.803862.  Google Scholar

[11]

M. Hanke, A regularizing Levenberg-Marquardt scheme, with applications to inverse groudwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.  Google Scholar

[12]

W. G. Hawkins, C.-H. Tung, D. Gagnon and F. Valentino, Some new sourceless and source-assisted attenuation correction methods for SPECT and PET, in Proceedings of 1999 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, Egmond aan Zee, The Netherlands, 1999, 84–87. Google Scholar

[13]

T. Honkamaa, F. Levai, R. Berndt, P. Schwalbach, S. Vaccaro and A. Turunen, A prototype for passive gamma emission tomography, in Proceedings of Symposium on International Safeguards, Vienna, Austria, 2014. Google Scholar

[14]

S. Jacobsson SvärdS. Holcombe and S. Grape, Applicability of a set of tomographic reconstruction algorithms for quantitative SPECT on irradiated nuclear fuel assemblies, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 783 (2015), 128-141.  doi: 10.1016/j.nima.2015.02.035.  Google Scholar

[15]

S. Jacobsson SvärdL. E. SmithT. A. WhiteV. MozinP. JanssonP. AnderssonA. DavourS. GrapeH. TrellueN. DeshmukhE. A. MillerR. S. WittmanT. HonkamaaS. Vaccaro and J. Ely, Outcomes of the JNT 1955 phase Ⅰ viability study of gamma emission tomography for spent fuel verification, ESARDA Bulletin, 55 (2017), 10-28.   Google Scholar

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A. KrolJ. E. BowsherS. H. ManglosD. H. FeiglinM. P. Tornai and F. D. Thomas, An EM algorithm for estimating SPECT emission and transmission parameters from emission data only, IEEE Transactions on Medical Imaging, 20 (2001), 218-232.  doi: 10.1109/42.918472.  Google Scholar

[18]

A. Krol, I. Echeruo, R. B. Solgado, A. S. Hardikar, J. E. Bowsher, D. H. Feiglin, F. D. Thomas, E. Lipson and I. L. Coman, EM-IntraSPECT algorithm with ordered subsets (OSEMIS) for nonuniform attenuation correction in cardiac imaging, in Proceedings of SPIE, 4684 (2002), 6pp. doi: 10.1117/12.467057.  Google Scholar

[19]

L. A. Kunyansky, A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula, Inverse Problems, 17 (2001), 293-306.  doi: 10.1088/0266-5611/17/2/309.  Google Scholar

[20]

F. Lévai, S. Dési, M. Tarvainen and R. Arlt, Use of High Energy Gamma Emission Tomography for Partial Defect Verification of Spent Fuel Assemblies, Technical Report STUK-YTO-TR 56, STUK, Helsinki, Finland, 1993. Google Scholar

[21]

S. H. Manglos and T. M. Young, Constrained IntraSPECT reconstruction from SPECT projections, in 1993 Nuclear Science Symposium and Medical Imaging Conference Conf. Rec., vol. 3, San Francisco, CA, USA, 1993, 1605–1609. doi: 10.1109/NSSMIC.1993.373561.  Google Scholar

[22]

W. Marshall, B. J. Ade, S. Bowman and J. S. Martinez-Gonzalez, Axial Moderator Density Distributions, Control Blade Usage, and Axial Burnup Distributions for Extended BWR Burnup Credit, Technical Report NUREG/CR-7224, ORNL/TM-2015/544, prepared for the US Nuclear Regulatory Commission by Oak Ridge National Laboratory, Oakridge, TN, USA, 2016. Google Scholar

[23]

M. Mayorov, T. White, A. Lebrun, J. Brutscher, J. Keubler, A. Birnbaum, V. Ivanov, T. Honkamaa, P. Peura and J. Dahlberg, Gamma emission tomography for the inspection of spent nuclear fuel, in Proceedings of 2017 IEEE Nuclear Science Symposium and Medical Imaging Conference, Atlanta, GA, USA, 2017. doi: 10.1109/NSSMIC.2017.8533017.  Google Scholar

[24]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, Philadelphia, PA, USA, 2012. doi: 10.1137/1.9781611972344.  Google Scholar

[25]

J. NuytsP. DupontS. StroobantsR. BenninkL. Mortelmans and P. Suetens, Simultaneous maximum a posteriori reconstruction of attenuation and activity distributions from emission sinograms, IEEE Transactions on Medical Imaging, 18 (1999), 393-403.  doi: 10.1109/42.774167.  Google Scholar

[26]

R. ReisenhoferS. BosseG. Kutyniok and T. Wiegand, A Haar wavelet-based perceptual similarity index for image quality assessment, Signal Processing: Image Communication, 61 (2018), 33-43.  doi: 10.1016/j.image.2017.11.001.  Google Scholar

[27]

A. SalomonA. GoedickeB. SchweizerT. Aach and V. Schulz, Simultaneous reconstruction of activity and attenuation for PET/MR, IEEE Transactions on Medical Imaging, 30 (2011), 804-813.  doi: 10.1109/TMI.2010.2095464.  Google Scholar

[28]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, De Gruyter, Berlin, Boston, 2012. doi: 10.1515/9783110255720.  Google Scholar

[29]

A. O. Seppänen, Correction of Collimator Blurring and Attenuation in Single Photon Emission Computed Tomography, M.Sc. thesis, University of Kuopio, Kuopio, Finland, 2000. Google Scholar

[30]

L. E. Smith, S. Jacobsson-Svärd, V. Mozin, P. Jansson, E. Miller, T. White, N. Deshmukh, H. Trellue, R. Wittman, A. Davour, S. Grape, P. Andersson, S. Vaccaro, S. Holcombe and T. Honkamaa, A Viability Study of Gamma Emission Tomography for Spent Fuel Verification: JNT 1955 Phase I Technical Report, Technical Report PNNL-25995, Pacific Northwest National Laboratory, Richland, WA, USA, 2016. Google Scholar

[31]

R. Thierry, J.-L. Pettier and L. Desbat, Simultaneous compensation for attenuation, scatter and detector response for 2D emission tomography on nuclear waste with reduced data, in 1st World Congress on Industrial Process Tomography, Buxton, United Kingdom, 1999,542–551. Google Scholar

[32]

J. C. Wagner, M. D. DeHart and C. V. Parks, Recommendations for Addressing Axial Burnup in PWR Burnup Credit Analyses, Technical Report NUREG/CR-6801, ORNL/TM-200 1/273, prepared for the US Nuclear Regulatory Commission by Oak Ridge National Laboratory, Oakridge, TN, USA, 2003. doi: 10.2172/885754.  Google Scholar

[33]

Y. Wang and Y. Yuan, Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems, Inverse Problems, 21 (2005), 821-838.  doi: 10.1088/0266-5611/21/3/003.  Google Scholar

[34]

Z. WangA. C. BovikH. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

[35]

T. White, M. Mayorov, N. Deshmukh, E. Miller, L. E. Smith, J. Dahlberg and T. Honkamaa, SPECT reconstruction and analysis for the inspection of spent nuclear fuel, in Proceedings of 2017 Nuclear Science Symposium and Medical Imaging Conference, Atlanta, GA, USA, 2017. doi: 10.1109/NSSMIC.2017.8532776.  Google Scholar

[36]

T. White, M. Mayorov, A. Lebrun, P. Peura, T. Honkamaa, J. Dahlberg, J. Keubler, V. Ivanov and A. Turunen, Application of passive gamma emission tomography (PGET) for the verification of spent nuclear fuel, in Proceedings of 59th Annual Meeting of the Institute of Nuclear Materials Management, Baltimore, MD, USA, 2018. Google Scholar

show all references

References:
[1]

Treaty on the non-proliferation of nuclear weapons, United Nations Treaty Series, 729, 1995, URL https://www.un.org/disarmament/wmd/nuclear/npt/text. Google Scholar

[2]

C. Bélanger-ChampagneP. PeuraP. EerolaT. HonkamaaT. WhiteM. Mayorov and P. Dendooven, Effect of gamma-ray energy on image quality in passive gamma emission tomography of spent nuclear fuel, IEEE Transactions on Nuclear Science, 66 (2019), 487-496.  doi: 10.1109/TNS.2018.2881138.  Google Scholar

[3]

Y. Berker and Y. Li, Attenuation correction in emission tomography using the emission data - A review, Medical Physics, 43 (2016), 807-832.  doi: 10.1118/1.4938264.  Google Scholar

[4]

S. Bonettini and M. Prato, New convergence results for the scaled gradient projection method, Inverse Problems, 31 (2015), 095008, 20pp. doi: 10.1088/0266-5611/31/9/095008.  Google Scholar

[5]

S. Bonettini, R. Zanella and L. Zanni, A scaled gradient projection method for constrained image deblurring, Inverse Problems, 25 (2009), 015002, 23pp. doi: 10.1088/0266-5611/25/1/015002.  Google Scholar

[6]

A. Bousse, A. Sidlesky, N. Roth, A. Rashidnasab, K. Thielemans and B. F. Hutton, Joint activity/attenuation reconstruction in SPECT using photopeak and scatter sinograms, in Proceedings of 2016 IEEE Nuclear Science Symposium, Medical Imaging Conference and Room-Temperature Semiconductor Detector Workshop, Strasbourg, France, 2016, 1–4. doi: 10.1109/NSSMIC.2016.8069448.  Google Scholar

[7]

S. C. Cade, S. Arridge, M. J. Evans and B. F. Hutton, Use of measured scatter data for the attenuation correction of single photon emission tomography without transmission scanning, Medical Physics, 40 (2013), 082506. doi: 10.1118/1.4812686.  Google Scholar

[8]

Y. CensorD. E. GustafsonA. Lent and H. Tuy, A new approach to the emission computerized tomography problem: Simultaneous calculation of attenuation and activity coefficients, IEEE Transactions on Nuclear Science, 26 (1979), 2775-2779.  doi: 10.1109/TNS.1979.4330535.  Google Scholar

[9]

V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography, Inverse Problems, 15 (1999), 931-960.  doi: 10.1088/0266-5611/15/4/307.  Google Scholar

[10]

D. GourionD. NollP. GantetA. Celler and J.-P. Esquerre, Attenuation correction using SPECT emission data only, IEEE Transactions on Nuclear Science, 49 (2002), 2172-2179.  doi: 10.1109/TNS.2002.803862.  Google Scholar

[11]

M. Hanke, A regularizing Levenberg-Marquardt scheme, with applications to inverse groudwater filtration problems, Inverse Problems, 13 (1997), 79-95.  doi: 10.1088/0266-5611/13/1/007.  Google Scholar

[12]

W. G. Hawkins, C.-H. Tung, D. Gagnon and F. Valentino, Some new sourceless and source-assisted attenuation correction methods for SPECT and PET, in Proceedings of 1999 International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, Egmond aan Zee, The Netherlands, 1999, 84–87. Google Scholar

[13]

T. Honkamaa, F. Levai, R. Berndt, P. Schwalbach, S. Vaccaro and A. Turunen, A prototype for passive gamma emission tomography, in Proceedings of Symposium on International Safeguards, Vienna, Austria, 2014. Google Scholar

[14]

S. Jacobsson SvärdS. Holcombe and S. Grape, Applicability of a set of tomographic reconstruction algorithms for quantitative SPECT on irradiated nuclear fuel assemblies, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 783 (2015), 128-141.  doi: 10.1016/j.nima.2015.02.035.  Google Scholar

[15]

S. Jacobsson SvärdL. E. SmithT. A. WhiteV. MozinP. JanssonP. AnderssonA. DavourS. GrapeH. TrellueN. DeshmukhE. A. MillerR. S. WittmanT. HonkamaaS. Vaccaro and J. Ely, Outcomes of the JNT 1955 phase Ⅰ viability study of gamma emission tomography for spent fuel verification, ESARDA Bulletin, 55 (2017), 10-28.   Google Scholar

[16]

C. T. Kelley, Iterative Methods for Optimization, chapter 3.2.3 and 3.3.5, SIAM, Philadelphia, PA, USA, 1999. doi: 10.1137/1.9781611970920.  Google Scholar

[17]

A. KrolJ. E. BowsherS. H. ManglosD. H. FeiglinM. P. Tornai and F. D. Thomas, An EM algorithm for estimating SPECT emission and transmission parameters from emission data only, IEEE Transactions on Medical Imaging, 20 (2001), 218-232.  doi: 10.1109/42.918472.  Google Scholar

[18]

A. Krol, I. Echeruo, R. B. Solgado, A. S. Hardikar, J. E. Bowsher, D. H. Feiglin, F. D. Thomas, E. Lipson and I. L. Coman, EM-IntraSPECT algorithm with ordered subsets (OSEMIS) for nonuniform attenuation correction in cardiac imaging, in Proceedings of SPIE, 4684 (2002), 6pp. doi: 10.1117/12.467057.  Google Scholar

[19]

L. A. Kunyansky, A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula, Inverse Problems, 17 (2001), 293-306.  doi: 10.1088/0266-5611/17/2/309.  Google Scholar

[20]

F. Lévai, S. Dési, M. Tarvainen and R. Arlt, Use of High Energy Gamma Emission Tomography for Partial Defect Verification of Spent Fuel Assemblies, Technical Report STUK-YTO-TR 56, STUK, Helsinki, Finland, 1993. Google Scholar

[21]

S. H. Manglos and T. M. Young, Constrained IntraSPECT reconstruction from SPECT projections, in 1993 Nuclear Science Symposium and Medical Imaging Conference Conf. Rec., vol. 3, San Francisco, CA, USA, 1993, 1605–1609. doi: 10.1109/NSSMIC.1993.373561.  Google Scholar

[22]

W. Marshall, B. J. Ade, S. Bowman and J. S. Martinez-Gonzalez, Axial Moderator Density Distributions, Control Blade Usage, and Axial Burnup Distributions for Extended BWR Burnup Credit, Technical Report NUREG/CR-7224, ORNL/TM-2015/544, prepared for the US Nuclear Regulatory Commission by Oak Ridge National Laboratory, Oakridge, TN, USA, 2016. Google Scholar

[23]

M. Mayorov, T. White, A. Lebrun, J. Brutscher, J. Keubler, A. Birnbaum, V. Ivanov, T. Honkamaa, P. Peura and J. Dahlberg, Gamma emission tomography for the inspection of spent nuclear fuel, in Proceedings of 2017 IEEE Nuclear Science Symposium and Medical Imaging Conference, Atlanta, GA, USA, 2017. doi: 10.1109/NSSMIC.2017.8533017.  Google Scholar

[24]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, Philadelphia, PA, USA, 2012. doi: 10.1137/1.9781611972344.  Google Scholar

[25]

J. NuytsP. DupontS. StroobantsR. BenninkL. Mortelmans and P. Suetens, Simultaneous maximum a posteriori reconstruction of attenuation and activity distributions from emission sinograms, IEEE Transactions on Medical Imaging, 18 (1999), 393-403.  doi: 10.1109/42.774167.  Google Scholar

[26]

R. ReisenhoferS. BosseG. Kutyniok and T. Wiegand, A Haar wavelet-based perceptual similarity index for image quality assessment, Signal Processing: Image Communication, 61 (2018), 33-43.  doi: 10.1016/j.image.2017.11.001.  Google Scholar

[27]

A. SalomonA. GoedickeB. SchweizerT. Aach and V. Schulz, Simultaneous reconstruction of activity and attenuation for PET/MR, IEEE Transactions on Medical Imaging, 30 (2011), 804-813.  doi: 10.1109/TMI.2010.2095464.  Google Scholar

[28]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, De Gruyter, Berlin, Boston, 2012. doi: 10.1515/9783110255720.  Google Scholar

[29]

A. O. Seppänen, Correction of Collimator Blurring and Attenuation in Single Photon Emission Computed Tomography, M.Sc. thesis, University of Kuopio, Kuopio, Finland, 2000. Google Scholar

[30]

L. E. Smith, S. Jacobsson-Svärd, V. Mozin, P. Jansson, E. Miller, T. White, N. Deshmukh, H. Trellue, R. Wittman, A. Davour, S. Grape, P. Andersson, S. Vaccaro, S. Holcombe and T. Honkamaa, A Viability Study of Gamma Emission Tomography for Spent Fuel Verification: JNT 1955 Phase I Technical Report, Technical Report PNNL-25995, Pacific Northwest National Laboratory, Richland, WA, USA, 2016. Google Scholar

[31]

R. Thierry, J.-L. Pettier and L. Desbat, Simultaneous compensation for attenuation, scatter and detector response for 2D emission tomography on nuclear waste with reduced data, in 1st World Congress on Industrial Process Tomography, Buxton, United Kingdom, 1999,542–551. Google Scholar

[32]

J. C. Wagner, M. D. DeHart and C. V. Parks, Recommendations for Addressing Axial Burnup in PWR Burnup Credit Analyses, Technical Report NUREG/CR-6801, ORNL/TM-200 1/273, prepared for the US Nuclear Regulatory Commission by Oak Ridge National Laboratory, Oakridge, TN, USA, 2003. doi: 10.2172/885754.  Google Scholar

[33]

Y. Wang and Y. Yuan, Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems, Inverse Problems, 21 (2005), 821-838.  doi: 10.1088/0266-5611/21/3/003.  Google Scholar

[34]

Z. WangA. C. BovikH. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

[35]

T. White, M. Mayorov, N. Deshmukh, E. Miller, L. E. Smith, J. Dahlberg and T. Honkamaa, SPECT reconstruction and analysis for the inspection of spent nuclear fuel, in Proceedings of 2017 Nuclear Science Symposium and Medical Imaging Conference, Atlanta, GA, USA, 2017. doi: 10.1109/NSSMIC.2017.8532776.  Google Scholar

[36]

T. White, M. Mayorov, A. Lebrun, P. Peura, T. Honkamaa, J. Dahlberg, J. Keubler, V. Ivanov and A. Turunen, Application of passive gamma emission tomography (PGET) for the verification of spent nuclear fuel, in Proceedings of 59th Annual Meeting of the Institute of Nuclear Materials Management, Baltimore, MD, USA, 2018. Google Scholar

Figure 1.  Simplified schematics of the PGET instrument. (a) Two detector banks on opposite sides of an SFA being measured. (b) Collimator slit profile and the location of the detectors with respect to the fuel rods
Figure 2.  Scaled-down example of a discrete emission map $ \lambda $ (left), a discrete attenuation map $ \mu $ (right), and the pixel indexing. Next to the maps is the detector array at the position corresponding to measurement angle zero. The collimators are in blue, and the detectors, shown with their indexing, are in red
Figure 3.  Line from the center of pixel $ p $ to the center of detector $ i $ (left), and $ d_{i, p} $ which tells for every pixel in the grid the length that the aforementioned line travels inside that pixel (right)
Figure 4.  (a) The volume that pixel $ p $ represents is divided into voxels. (b) The cone spanned by the visible part of detector $ i $ from voxel $ s $ defines a solid angle. (c) Spatial responses $ r_{i, p} $ of detector $ i $ for all the pixels $ p $ in the grid. (d) The angle $ \alpha_s $ between the line from pixel $ p $ to detector $ i $ and the line from voxel $ s $ to detector $ i $
Figure 5.  Examples of the basis images used by the geometry aware prior in the scaled-down setting with four rod positions. On the left there is $ r_{1} $, an image containing only one of the rods, and on the right there is the water image $ w $. Due to the low resolution and the round shape of the rods, all the rod pixels are partly water, which is why the water image has non-zero values in the rod pixels. This is also the case for the edge pixels of the fuel rods in the full-scale setting
Figure 6.  The linear bounds illustrated in the emission-attenuation-plane along with points that correspond to spent fuel (high emission, high attenuation), fresh fuel (no emission, high attenuation) and water (no emission, low attenuation). The values inside the triangle are allowed by the bounds. The triangle is slightly larger than necessary to allow the three materials mentioned, which is to simulate error from estimating the bounds. The attenuation values of the three points shown are the linear attenuation coefficients (mm-1) of water and UO2 for 662 keV gamma-rays from 137Cs. The emission values are arbitrary
Figure 7.  The ground truth and the reconstruction images cropped to include only the $ 69\times 69 $ pixel area that includes the fuel assembly. In the top row there are the emission images and in the bottom row the attenuation images. In columns from left to right: ground truth, the FBP reconstruction, the iterative reconstruction using the smoothness prior, and the same using the geometry aware prior
Figure 8.  The difference of the emission value of a rod position from the average value of its neighboring positions plotted against the distance of the position from the assembly center. From left to right: FBP reconstruction, iterative reconstruction using the smoothness prior, and iterative reconstruction using the geometry aware prior
Figure 9.  The emission and attenuation values of each rod position plotted in the emission-attenuation-plane for the iterative reconstructions using the smoothness prior (left) and the geometry aware prior (right)
Figure 10.  Reconstructions done using box bounds, i.e., lower and upper bounds for emission and attenuation that form a square in the emission-attenuation-plane. In the top row are the emission images and in the bottom row the attenuation images. In columns from left to right: ground truth, the iterative reconstruction using the smoothness prior, and the same using the geometry aware prior
Table 1.  Metrics comparing the reconstruction to the ground truth: relative error (RE), structural similarity index (SSIM) and Haar wavelet-based perceptual similarity index (HaarPSI)
Emission $ \lambda $ Attenuation $ \mu $
RE (%) SSIM HaarPSI RE (%) SSIM HaarPSI
Filtered back-projection 58.7 0.470 0.256 - - -
Smoothness prior 21.8 0.907 0.694 21.9 0.908 0.694
Geometry aware prior 10.5 0.970 0.866 9.44 0.972 0.850
Emission $ \lambda $ Attenuation $ \mu $
RE (%) SSIM HaarPSI RE (%) SSIM HaarPSI
Filtered back-projection 58.7 0.470 0.256 - - -
Smoothness prior 21.8 0.907 0.694 21.9 0.908 0.694
Geometry aware prior 10.5 0.970 0.866 9.44 0.972 0.850
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