Deep convolutional neural networks applied to muon tomography images
Lara Lloret & Pablo Martínez Ruiz del Árbol October 2019 1st COMCHA school 1
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Cosmic rays: composition, flux, energy spectrum. C
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Constant flux of high energy particles bombarding the Earth from the space. o
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➔ Composed mainly by protons (98%), α particles (1.8%) and others (< 0.2%)
➔ Cosmic rays are the most energetic particle ever seen so far.
The Oh-My-God Particle 3x108 TeV (1991, several since then) http://www.cosmic-ray.org/reading/flyseye.html#SEC10
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Cosmic muons: flux and energy spectrum. C
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Muons are generated mostly from pion decays in the atmosphere. o
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➔ The flux of muons is mostly proportional to cos2(θ) (angle with the vertical).
➔ Quickly falling spectra → average of 3 GeV when integrating solid angle.
Rule of thumb at the surface:
10000 muons per squared meter and minute.
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Muon interaction with matter: ionization C
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Ionization is one of the most frequent processes for cosmic muons. o
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➔ Energy loss depends on the Z, density, and size of the crossed object.
➔ The Range is the distance for which the particle looses all the energy.
Mean excitation potential
Muon Energy/GeV Material Range/m
1 Water 471 10 Water 4260 1 Concrete 228 10 Concrete 2025 1 Standard Rock 209 10 Standard Rock 1857
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Muons interaction with matter: multiple scattering C
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Coulomb scattering deviates the direction of muons when crossing matter. o
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➔ Scattering angle depends also on the Z, density, and size of the material.
➔ Angular distribution is approximately gaussian (with some tails).
Momentum dependence Muon Material Width/cm Angle/mrad Energy/GeV 1 Iron 1 10 10 Iron 1 1 1 Iron 10 34 10 Iron 10 3 1 Lead 1 19 10 Lead 1 2 1 Lead 10 64 10 Lead 10 6 1 Uranium 1 26 10 Uranium 1 3 1 Uranium 10 88 10 Uranium 10 9
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Two categories on muography applications C
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Absorption Muography Scattering Muography h
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l ➔ Muon flux measured as a function of the direction. ➔ Scattering position and angular shift.
➔ Differential transmittance “T”. ➔ Smaller Masses + shorter exposure times.
➔ Need Large Masses + Long exposure times. ➔ Applications: security, industry, etc.
➔ Applications: vulcanology, geology, archeology… ➔ At least two detectors are needed.
➔ Only one detector needed.
Muon Detector T
θ Δθ Δθ
Muon Detector Muon Detector
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Two categories on muography applications C
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Absorption Muography Scattering Muography h
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l ➔ Muon flux measured as a function of the direction. ➔ Scattering position and angular shift.
➔ Differential transmittance “T”. ➔ Smaller Masses + shorter exposure times.
➔ Need Large Masses + Long exposure times. ➔ Applications: security, industry, etc.
➔ Applications: vulcanology, geology, archeology… ➔ At least two detectors are needed.
➔ Only one detector needed.
Muon Detector T
θ Δθ Δθ
Muon Detector Muon Detector
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Muon detectors: gas proportional chambers C
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Chamber 1 o
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Chamber 2
Surface: 1m x 1m N. wires: 212 Synch Wire: Gold-Tungsten + Wire radius: 25 μm Trigger Chamber 3 Synchronization Gas: Ar-CO2 electronics
Chamber 4
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Muon detectors: gas proportional chambers C
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Image reconstruction in muon tomography C
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➔ h Tomography is a classic “inverse” mathematical problem. o
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l ⃗x Input variables for a given dynamical system Example: Position/direction of a muon at the upper detector.
⃗f Physics laws: evolution of the system.
Notice: Can be deterministic but also probabilistic. ⃗y=⃗f (⃗x ;θ⃗) Example: Moliere’s scattering formula for angular deviations.
θ⃗ Parameters describing the physics environment. Example: Densities and geometry of the crossed material.
⃗y Output variables after the action of ⃗f . Example: Position/direction of a muon at the lower detector.
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Image reconstruction in muon tomography C
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➔ h Tomography is a classic “inverse” mathematical problem. o
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Given l ⃗x Input variables for a given dynamical system Example: Position/direction of a muon at the upper detector. Known ⃗f Physics laws: evolution of the system.
Notice: Can be deterministic but also probabilistic. ⃗y=⃗f (⃗x ;θ⃗) Example: Moliere’s scattering formula for angular deviations. Known θ⃗ Parameters describing the physics environment. Example: Densities and geometry of the crossed material. Unknown ⃗y Output variables after the action of ⃗f . Direct problem Example: Position/direction of a muon at the lower detector.
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Image reconstruction in muon tomography C
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➔ h Tomography is a classic “inverse” mathematical problem. o
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Given l ⃗x Input variables for a given dynamical system Example: Position/direction of a muon at the upper detector. Known ⃗f Physics laws: evolution of the system.
Notice: Can be deterministic but also probabilistic. ⃗y=⃗f (⃗x ;θ⃗) Example: Moliere’s scattering formula for angular deviations. Unknown θ⃗ Parameters describing the physics environment. Example: Densities and geometry of the crossed material. Known ⃗y Output variables after the action of ⃗f . Inverse problem Example: Position/direction of a muon at the lower detector.
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The simple geometric approach: POCA C
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➔ h This complex problem can be approximated doing some physical assumptions o
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➔ The scattering occurs only at one point during the whole trajectory.
➔ Works very well for scenarios that are essentially empty with some high density chunks.
Point of Closest Approach (POCA)
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The simple geometric approach: POCA C
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The simple geometric approach: POCA C
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o Front view Side view l
Top view
wear
wear wear
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Measurement of the width of an insulated pipe C
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20 cm
Steel 20 cm 24 cm Rock wool 50 cm R Aluminium 24.06 cm
20 cm
DOI:10.1098/rsta.2018.0054
Deep convolutional neural networks applied to muon tomography images 16 1
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Measurement of the width of an insulated pipe C
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20 cm
Steel 20 cm 24 cm Rock wool 50 cm R Aluminium 24.06 cm
20 cm
DOI:10.1098/rsta.2018.0054
Deep convolutional neural networks applied to muon tomography images 17 1
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Measurement of the width of an insulated pipe C
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20 cm
Steel 20 cm 24 cm Rock wool 50 cm R Aluminium 24.06 cm
20 cm
DOI:10.1098/rsta.2018.0054
Deep convolutional neural networks applied to muon tomography images 18 1
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Measurement of the width of an insulated pipe C
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20 cm
Steel 20 cm 24 cm Rock wool 50 cm R Aluminium 24.06 cm
20 cm
DOI:10.1098/rsta.2018.0054
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Measurement of the width of an insulated pipe C
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Simulation Details:
CRY (*) generator.
Geant4 + local chamber response.
Assuming perfect resolution.
About 1 hour of data taking.
https://nuclear.llnl.gov/simulation/doc_cry_v1.7/cry.pdf
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Reconstructed images for different widths C
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Thickness = 1.2 cm Thickness = 1.6 cm Thickness = 1.8 cm h
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Thickness = 2.0 cm Thickness = 2.4 cm Thickness = 2.6 cm
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Simulations with realistic detector C
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Thickness = 1.2 cm Thickness = 1.6 cm Thickness = 1.8 cm c
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Thickness = 2.0 cm Thickness = 2.4 cm Thickness = 2.6 cm
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Simulations with realistic detector C
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Thickness = 1.2 cm Thickness = 1.6 cm Thickness = 1.8 cm c
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QUESTION:
Thickness = 2.0 cm Thickness = 2.4 cm Thickness = 2.6 cm How to quantify and discern the thickness of the pipes?
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Convolutional neural networks (brief reminder) C
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➔ o CNN are Machine Learning algorithms designed to operate with images. o
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➔ Images are 2x1 dimension tensors: horizontal-vertical pixel x color/depth.
➔ A CNN is a network of filters operating over sub-spaces of the input.
Deep convolutional neural networks applied to muon tomography images 24 1
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Convolutional neural networks (brief reminder) C
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➔ o Filters operating on a particular sub-space produce a single number. o
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Multiplying and adding = (50*30) + (50*30) + (50*30) + (20*30)+(50*30) = 6600
Multiplying and adding = 0
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Convolutional neural networks (brief reminder) C
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➔ o As the filter spans the whole initial tensor it produces a new “image”. o
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Filter 5x5 Features map
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Convolutional neural networks (brief reminder) C
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➔ o Layers with filters are combined with Pooling and Dense Layers. o
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➔ Pooling layers are just reducing the dimension of the feature maps.
➔ Dense layers are normal NN layers useful specially in the last stages.
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Idea: Classify Muon Tomography images with CNN C
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➔ o Train a CNN with Muon Tomography images of different thicknesses. o
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➔ Quantify the thickness of an unknown image as the one with better compatibility.
4 mm
6 mm
8 mm 10 mm 12 mm
14 mm 16 mm
18 mm Thickness = 2.6 cm
9 categories are defined. Hundreds of simulations for each category.
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A simple keras model for Muon Tomography C
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A simple keras model for Muon Tomography C
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ImageDataGenerator (from Keras)
Automatically reads train and test images from a train and test directories.
Each sub-directory within train and test will be taken as a category.
Image colors are automatically rescaled.
The size of the images have to be provided as an argument.
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A simple keras model for Muon Tomography C
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Model definition c
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o Combination of Conv2D, MaxPooling2D and Dense layers. l
But also using Dropout and Flatten.
Defining also the type of loss function, optimizers and metrics to use.
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A simple keras model for Muon Tomography C
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Model fitting
Definition of number of epochs, batch, shuffling, etc.
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Training of the network C
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➔ o It is important to monitor the training process to make sure there are no: o
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➔ Overfiting problems: The model is too complex. “Learning random noise”.
➔ Bias problems: The model is too simple. “Not catching all correlations”.
In this example 70% of images for training and 30% of images for testing.
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Training of the network C
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➔ o It is important to monitor the training process to make sure there are no: o
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➔ Overfiting problems: The model is too complex. “Learning random noise”.
➔ Bias problems: The model is too simple. “Not catching all correlations”.
A little bit of overfitting
In this example 70% of images for training and 30% of images for testing.
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Final results: having a look at the confusion matrix C
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➔ o Use the test sample to evaluate the performance of your classifier. o
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➔ In about 82% of the cases the answer provided by the CNN is correct.
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Final results: having a look at the confusion matrix C
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➔ o Use the test sample to evaluate the performance of your classifier. o
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➔ In about 82% of the cases the answer provided by the CNN is correct.
A resolution of about 2 mm (granularity of the training) can be claimed !!!!
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Going beyond classification C
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➔ o Can we solve the “Inverse Problem” and the classification in just one go? o
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➔ Need to think about the nature of a probabilistic Inverse Problem.
Deterministic Inverse Problem Probabilistic Inverse Problem
⃗y=⃗f (⃗x;θ⃗) ⃗y∼ pdf (⃗x ;θ⃗)
A pair of x and θ does not determine y.
A pair of x and θ completely determines y. Only distributions of many (x, y) can do it.
Loss function can use differences between y and f. Loss functions involving these distributions or
estimators on these distributions.
⃗ ⃗ Loss= g(h ( y , x ;θ⃗),h ( y , x ;⃗θ),...) Loss=∑ g(‖⃗yi−f (⃗x ;θ)‖) ∑ 1 ⃗ ⃗ 2 ⃗ ⃗ i i
Index “i” running on events. Index “i” running on measurements of many events.
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Going beyond classification: quantile distributions. C
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➔ Consider the quartiles of distributions of the following parameters as input to DNN o
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➔ Entrance position and direction of the muon. (x, y)
➔ Deviation in position and direction as measured by second muon. (x,y)
➔ Products of these variables.
➔ Consider the density of every voxel as the output of the DNN.
Paper in preparation
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Conclusions C
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o ➔ Convolutional Neural Networks are a very powerful tool to classify images. l
➔ They can be easily applied to many problems in science and the industry.
➔ I have shown an application in the context of muon tomography.
➔ All using standard tools and not many more than 100 lines of code.
➔ However some advices when dealing with these technologies:
➔ They are not always a/the best solution to your problem. Don’t get obsessed.
➔ These solutions are easy to code but often very hard to understand.
➔ No general rules about what is the best structure → testing, testing, testing
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Thanks
Muon Tomography Systems SL www.muon.systems Av. Altos Hornos de Vizcaya 33 48901 Barakaldo (Bizkaia)
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BACKUP
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Application to vulcanology C
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Monitoring of Vesuvious vulcano. c
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https://doi.org/10.1098/rsta.2018.0050 l
Only 1 scintillator detector.
Exposure time ~ year.
Dinamic studies on magma convenction.
Monitoring of Asama & Iwujima vulcanos. https://doi.org/10.1098/rsta.2018.0142 Deep convolutional neural networks applied to muon tomography images 42 1
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Application to mineral mines. https://doi.org/10.1098/rsta.2018.0061 C
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Search for uranium/lead clusters in McArthur River (Canada) and Pend Oreille (USA) c
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o Canadian Private company: l
CRM Geotomography Technologies, Inc.,
Grid of scintillator detectors in the galleries.
Underground mapping for pipe location.
https://doi.org/10.1098/rsta.2018.0133 Israel Private company:
Lingacom S.L,
Cylindrical-scintillator detectors.
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Application to archeology/civil engineering. C
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Search for hidden galleries in Mount Echia (Bourbon Tunnel) h
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Only 1 scintillator detector.
Exposure time ~ 26 days.
Monitoring of the dome of Santa Maria del Fiore. https://doi.org/10.1098/rsta.2018.0136
One of the only scattering-based applications in civil engineering.
Two drift tube chambers located outside and inside the dome.
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Application to border security. C
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Truck inspection using cosmic muons. https://doi.org/10.1098/rsta.2018.0051 S
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o Using CMS-like chambers at Torino. l
For this application: 4 chambers.
Detection of a 7cmx7cmx7cm lead box.
Decision Science
First commercial system.
First contract in Singapur to install portal.
Multi-technology detection portal.
https://decisionsciences.com/
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Application to the nuclear industry. C
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Nuclear waste characterization. S
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Lynkeos Technology Ltd (University of Glasgow) l
UK Nuclear Decommissioning Authority.
Commercial Scintillator detector. https://doi.org/10.1098/rsta.2018.0048
Fuel cask monitoring.
Exposure times of about 7 hours.
Effective detection of missing fuel.
https://doi.org/10.1098/rsta.2018.0052 Deep convolutional neural networks applied to muon tomography images 46