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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 and Imaging, 2020, 14 (2) : 317-337. doi: 10.3934/ipi.2020014
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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
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
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)
(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 $i$
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
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
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
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
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)
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
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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