{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T18:22:10Z","timestamp":1776882130267,"version":"3.51.2"},"reference-count":42,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T00:00:00Z","timestamp":1634169600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S000631\/1"],"award-info":[{"award-number":["EP\/S000631\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Royal Academy of Engineering under the Research Fellowship scheme","award":["RF201617\/16\/31"],"award-info":[{"award-number":["RF201617\/16\/31"]}]},{"name":"EPSRC under Grant number","award":["EP\/T028270\/1"],"award-info":[{"award-number":["EP\/T028270\/1"]}]},{"name":"Nuclear Regulatory Commission (NRC) Faculty Development Grant","award":["31310019M0011"],"award-info":[{"award-number":["31310019M0011"]}]},{"name":"Department of Energy STTR","award":["DE-SC0020733"],"award-info":[{"award-number":["DE-SC0020733"]}]},{"name":"the UK MOD University Defence Research Collaboration","award":["UK MOD UDRC"],"award-info":[{"award-number":["UK MOD UDRC"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>In this paper, we address the problem of activity estimation in passive gamma emission tomography (PGET) of spent nuclear fuel. Two different noise models are considered and compared, namely, the isotropic Gaussian and the Poisson noise models. The problem is formulated within a Bayesian framework as a linear inverse problem and prior distributions are assigned to the unknown model parameters. In particular, a Bernoulli-truncated Gaussian prior model is considered to promote sparse pin configurations. A Markov chain Monte Carlo (MCMC) method, based on a split and augmented Gibbs sampler, is then used to sample the posterior distribution of the unknown parameters. The proposed algorithm is first validated by simulations conducted using synthetic data, generated using the nominal models. We then consider more realistic data simulated using a bespoke simulator, whose forward model is non-linear and not available analytically. In that case, the linear models used are mis-specified and we analyse their robustness for activity estimation. The results demonstrate superior performance of the proposed approach in estimating the pin activities in different assembly patterns, in addition to being able to quantify their uncertainty measures, in comparison with existing methods.<\/jats:p>","DOI":"10.3390\/jimaging7100212","type":"journal-article","created":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T09:10:49Z","timestamp":1634202649000},"page":"212","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Bayesian Activity Estimation and Uncertainty Quantification of Spent Nuclear Fuel Using Passive Gamma Emission Tomography"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2266-6992","authenticated-orcid":false,"given":"Ahmed Karam","family":"Eldaly","sequence":"first","affiliation":[{"name":"Institute of Sensors, Signals and Systems, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ming","family":"Fang","sequence":"additional","affiliation":[{"name":"Department of Nuclear, Plasma, and Radiological Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Angela","family":"Di Fulvio","sequence":"additional","affiliation":[{"name":"Department of Nuclear, Plasma, and Radiological Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stephen","family":"McLaughlin","sequence":"additional","affiliation":[{"name":"Institute of Sensors, Signals and Systems, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2327-236X","authenticated-orcid":false,"given":"Mike E.","family":"Davies","sequence":"additional","affiliation":[{"name":"Institute for Digital Communications & The Joint Research Institute for Signal and Image Processing, The University of Edinburgh, Edinburgh, EH9 3JL, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yoann","family":"Altmann","sequence":"additional","affiliation":[{"name":"Institute of Sensors, Signals and Systems, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yves","family":"Wiaux","sequence":"additional","affiliation":[{"name":"Institute of Sensors, Signals and Systems, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"711","DOI":"10.2307\/2199482","article-title":"The treaty on the non-proliferation of nuclear weapons","volume":"63","author":"Firmage","year":"1969","journal-title":"Am. J. Int. Law"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"White, T., Mayorov, M., Deshmukh, N., Miller, E., Smith, L.E., Dahlberg, J., and Honkamaa, T. (2017, January 21\u201328). SPECT reconstruction and analysis for the inspection of spent nuclear fuel. Proceedings of the 2017 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS\/MIC), Atlanta, GA, USA.","DOI":"10.1109\/NSSMIC.2017.8532776"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"White, T., Mayorov, M., Lebrun, A., Peura, P., Honkamaa, T., Dahlberg, J., Keubler, J., Ivanov, V., and Turunen, A. (2018, January 22\u201326). Application of passive gamma emission tomography (PGET) for the verification of spent nuclear fuel. Proceedings of the INMM 59th Annual Meeting, Baltimore, MD, USA.","DOI":"10.1109\/NSSMIC.2017.8533017"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mayorov, M., White, T., Lebrun, A., Brutscher, J., Keubler, J., Birnbaum, A., Ivanov, V., Honkamaa, T., Peura, P., and Dahlberg, J. (2017, January 21\u201328). Gamma emission tomography for the inspection of spent nuclear fuel. Proceedings of the 2017 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS\/MIC), Atlanta, GA, USA.","DOI":"10.1109\/NSSMIC.2017.8533017"},{"key":"ref_5","first-page":"487","article-title":"Effect of gamma-ray energy on image quality in passive gamma emission tomography of spent nuclear fuel","volume":"66","author":"Peura","year":"2018","journal-title":"IEEE Trans. Nucl. Sci."},{"key":"ref_6","unstructured":"Virta, R., Backholm, R., Bubba, T.A., Helin, T., Moring, M., Siltanen, S., Dendooven, P., and Honkamaa, T. (2020). Fuel rod classification from Passive Gamma Emission Tomography (PGET) of spent nuclear fuel assemblies. arXiv."},{"key":"ref_7","unstructured":"Backholm, R., Bubba, T.A., B\u00e9langer-Champagne, C., Helin, T., Dendooven, P., and Siltanen, S. (2019). Simultaneous reconstruction of emission and attenuation in passive gamma emission tomography of spent nuclear fuel. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3133","DOI":"10.1109\/TIP.2010.2053941","article-title":"Restoration of Poissonian images using alternating direction optimization","volume":"19","author":"Figueiredo","year":"2010","journal-title":"IEEE Trans. Image Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2345","DOI":"10.1109\/TIP.2010.2047910","article-title":"Fast image recovery using variable splitting and constrained optimization","volume":"19","author":"Afonso","year":"2010","journal-title":"IEEE Trans. Image Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1109\/TIP.2010.2076294","article-title":"An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems","volume":"20","author":"Afonso","year":"2010","journal-title":"IEEE Trans. Image Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1111\/j.1365-2966.2012.21605.x","article-title":"Sparsity averaging reweighted analysis (SARA): A novel algorithm for radio-interferometric imaging","volume":"426","author":"Carrillo","year":"2012","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1109\/LSP.2013.2259813","article-title":"Sparsity averaging for compressive imaging","volume":"20","author":"Carrillo","year":"2013","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Eldaly, A.K., Altmann, Y., Akram, A., Perperidis, A., Dhaliwal, K., and McLaughlin, S. (2019, January 8\u201311). Patch-based sparse representation for bacterial detection. Proceedings of the 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019), Venice, Italy.","DOI":"10.1109\/ISBI.2019.8759297"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1109\/TCI.2018.2811939","article-title":"Deconvolution and restoration of optical endomicroscopy images","volume":"4","author":"Eldaly","year":"2018","journal-title":"IEEE Trans. Comput. Imaging"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Eldaly, A.K., Altmann, Y., Perperidis, A., and McLaughlin, S. (2018, January 10\u201313). Deconvolution of irregularly subsampled images. Proceedings of the 2018 IEEE Statistical Signal Processing Workshop (SSP), Freiburg im Breisgau, Germany.","DOI":"10.1109\/SSP.2018.8450801"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Tachella, J., Altmann, Y., Pereyra, M., McLaughlin, S., and Tourneret, J.Y. (2018, January 3\u20137). Bayesian restoration of high-dimensional photon-starved images. Proceedings of the 2018 IEEE 26th European Signal Processing Conference (EUSIPCO), Rome, Italy.","DOI":"10.23919\/EUSIPCO.2018.8553175"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Vono, M., Dobigeon, N., and Chainais, P. (2019, January 12\u201317). Bayesian image restoration under Poisson noise and log-concave prior. Proceedings of the ICASSP 2019\u20142019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK.","DOI":"10.1109\/ICASSP.2019.8683031"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1648","DOI":"10.1109\/TSP.2019.2894825","article-title":"Split-and-augmented Gibbs sampler\u2014Application to large-scale inference problems","volume":"67","author":"Vono","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1007\/s11222-015-9567-4","article-title":"Proximal markov chain monte carlo algorithms","volume":"26","author":"Pereyra","year":"2016","journal-title":"Stat. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1137\/16M1108340","article-title":"Efficient bayesian computation by proximal markov chain monte carlo: When langevin meets moreau","volume":"11","author":"Durmus","year":"2018","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_21","unstructured":"Iyengar, A.S., Hausladen, P., Yang, J., Fabris, L., Hu, J., Lacy, J., and Athanasiades, A. (2017). Detection of Fuel Pin Diversion via Fast Neutron Emission Tomography, Oak Ridge National Lab.(ORNL). Technical Report."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-021-82031-8","article-title":"Quantitative imaging and automated fuel pin identification for passive gamma emission tomography","volume":"11","author":"Fang","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Newstadt, G.E., Hero, A.O., and Simmons, J. (July, January 29). Robust spectral unmixing for anomaly detection. Proceedings of the 2014 IEEE Workshop on Statistical Signal Processing (SSP), Gold Coast, QLD, Australia.","DOI":"10.1109\/SSP.2014.6884587"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1214\/009053604000001147","article-title":"Spike and slab variable selection: Frequentist and Bayesian strategies","volume":"33","author":"Ishwaran","year":"2005","journal-title":"Ann. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1080\/01621459.1993.10476353","article-title":"Variable selection via Gibbs sampling","volume":"88","author":"George","year":"1993","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"17","DOI":"10.2307\/3315687","article-title":"Bayesian variable selection with related predictors","volume":"24","author":"Chipman","year":"1996","journal-title":"Can. J. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1438","DOI":"10.1198\/016214508000000869","article-title":"High-dimensional sparse factor modeling: Applications in gene expression genomics","volume":"103","author":"Carvalho","year":"2008","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2727","DOI":"10.1109\/TSP.2012.2190066","article-title":"Blind deconvolution of sparse pulse sequences under a minimum distance constraint: A partially collapsed Gibbs sampler method","volume":"60","author":"Kail","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2840","DOI":"10.1109\/TBME.2010.2076809","article-title":"P- and T-wave delineation in ECG signals using a Bayesian approach and a partially collapsed Gibbs sampler","volume":"57","author":"Lin","year":"2010","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Kail, G., Tourneret, J.Y., Hlawatsch, F., and Dobigeon, N. (2010, January 14\u201319). A partially collapsed Gibbs sampler for parameters with local constraints. Proceedings of the 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, Dallas, TX, USA.","DOI":"10.1109\/ICASSP.2010.5495806"},{"key":"ref_31","unstructured":"Ge, D., Idier, J., and Le Carpentier, E. (2008, January 25\u201329). A new MCMC algorithm for blind Bernoulli-Gaussian deconvolution. Proceedings of the 2008 IEEE 16th European Signal Processing Conference, Lausanne, Switzerland."},{"key":"ref_32","unstructured":"Robert, C., and Casella, G. (2013). Monte Carlo Statistical Methods, Springer Science & Business Media."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.media.2019.06.009","article-title":"Bayesian Bacterial Detection Using Irregularly Sampled Optical Endomicroscopy Images","volume":"57","author":"Eldaly","year":"2019","journal-title":"Med. Image Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1080\/01621459.1994.10476829","article-title":"The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem","volume":"89","author":"Liu","year":"1994","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1198\/016214508000000409","article-title":"Partially collapsed Gibbs samplers: Theory and methods","volume":"103","author":"Park","year":"2008","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1023\/A:1013172322619","article-title":"Marginal maximum a posteriori estimation using Markov chain Monte Carlo","volume":"12","author":"Doucet","year":"2002","journal-title":"Stat. Comput."},{"key":"ref_37","unstructured":"Robert, C. (2007). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, Springer Science & Business Media."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Goorley, J.T., James, M.R., Booth, T.E., Brown, F.B., Bull, J.S., Cox, L.J., Durkee, J.W., Elson, J.S., Fensin, M.L., and Forster, R.A. (2013). Initial MCNP6 Release Overview-MCNP6 Version 1.0, Los Alamos National Lab. (LANL). Technical Report.","DOI":"10.2172\/1086758"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Gelman, A., Stern, H.S., Carlin, J.B., Dunson, D.B., Vehtari, A., and Rubin, D.B. (2013). Bayesian Data Analysis, Chapman and Hall\/CRC.","DOI":"10.1201\/b16018"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1137\/19M1283719","article-title":"Accelerating proximal Markov chain Monte Carlo by using an explicit stabilized method","volume":"13","author":"Pereyra","year":"2020","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1945","DOI":"10.1137\/20M1339829","article-title":"Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach Part I: Methodology and Experiments","volume":"13","author":"Vidal","year":"2020","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_42","first-page":"1858","article-title":"Gaussian sampling by local perturbations","volume":"23","author":"Papandreou","year":"2010","journal-title":"Adv. Neural Inf. Process. Syst."}],"container-title":["Journal of Imaging"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2313-433X\/7\/10\/212\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:12:57Z","timestamp":1760166777000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2313-433X\/7\/10\/212"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,14]]},"references-count":42,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["jimaging7100212"],"URL":"https:\/\/doi.org\/10.3390\/jimaging7100212","relation":{},"ISSN":["2313-433X"],"issn-type":[{"value":"2313-433X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,14]]}}}