Bayesian Activity Estimation and Uncertainty Quantification of Spent Nuclear Fuel Using Passive Gamma Emission Tomography
Abstract
:1. Introduction
 We formulate the pin activity estimation problem within a Bayesian framework and assign a Bernoulli truncatedGaussian (BtG) prior model to the intensity field to be estimated. To the best of our knowledge, no work attempted to sample from such a highlymultimodel joint posterior distribution using an SPA sampler. This allows for estimating the activity of spentfuel, including the assessment of fuel rod presence/absence.
 We compare the performance of two different noise models for pin activity estimation from PGET sinograms simulated while accounting for pin selfattenuation (i.e., not simulated using a linear model).
 In addition to estimating the activity profile, the proposed algorithms allow the automated estimation of the crucial model hyperparameters, like regularization parameters, which might affect the resulting estimated activity.
2. Problem Formulation
3. Hierarchical Bayesian Model
3.1. Likelihood
3.2. Prior Distributions
3.3. Joint Posterior Distribution
4. Bayesian Inference
Algorithm 1 Split and augmented—partially collapsed Gibbs sampling algorithm for activity estimation in PGET—version I. 

5. Simulations Using Synthetic Datasets
5.1. Data Creation
5.2. Quantitative Analysis
5.3. Qualitative Analysis
5.3.1. The Proposed Method
5.3.2. Comparison with Existing Methods
6. Simulations Using Realistic Datasets
6.1. The IDEAL Response Matrix
6.1.1. Results Using the Proposed Approach
6.1.2. Comparison with Existing Methods
6.2. The FULL Response Matrix
6.2.1. Results Using the Proposed Approach
6.2.2. Comparison with Existing Methods
6.3. The EMPIRICAL Response Matrix
7. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Algorithm A1 Split and augmented—partially collapsed Gibbs sampling algorithm for activity estimation in PGET—version II. 

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SPAPBtG  SPAGBtG  SPAP${\mathbf{\ell}}_{1}$  SPAG${\mathbf{\ell}}_{1}$  PIDAL${\mathbf{\ell}}_{1}$  FISTA${\mathbf{\ell}}_{1}$  Ground Truth  

Case 1  2.00  1.49  2.244  1.65  1.512  1.524  1.51 
Case 2  2.44  1.54  2.89  1.72  1.47  1.513  1.70 
Case 3  2.10  1.58  2.37  1.77  1.49  1.624  1.70 
Case 4  2.25  1.61  1.68  1.83  1.57  1.68  1.70 
Case 5  2.16  1.65  2.47  1.80  1.69  1.69  1.70 
Method  SPAPBtG  SPAGBtG  SPAP${\mathbf{\ell}}_{1}$  SPAG${\mathbf{\ell}}_{1}$  PIDAL${\mathbf{\ell}}_{1}$  FISTA${\mathbf{\ell}}_{1}$ 

Computation time (min)  50  60  45  55  75  3 
Uncertainty maps  Yes  Yes  Yes  Yes  No  No 
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Eldaly, A.K.; Fang, M.; Di Fulvio, A.; McLaughlin, S.; Davies, M.E.; Altmann, Y.; Wiaux, Y. Bayesian Activity Estimation and Uncertainty Quantification of Spent Nuclear Fuel Using Passive Gamma Emission Tomography. J. Imaging 2021, 7, 212. https://doi.org/10.3390/jimaging7100212
Eldaly AK, Fang M, Di Fulvio A, McLaughlin S, Davies ME, Altmann Y, Wiaux Y. Bayesian Activity Estimation and Uncertainty Quantification of Spent Nuclear Fuel Using Passive Gamma Emission Tomography. Journal of Imaging. 2021; 7(10):212. https://doi.org/10.3390/jimaging7100212
Chicago/Turabian StyleEldaly, Ahmed Karam, Ming Fang, Angela Di Fulvio, Stephen McLaughlin, Mike E. Davies, Yoann Altmann, and Yves Wiaux. 2021. "Bayesian Activity Estimation and Uncertainty Quantification of Spent Nuclear Fuel Using Passive Gamma Emission Tomography" Journal of Imaging 7, no. 10: 212. https://doi.org/10.3390/jimaging7100212