Robust Bayesian Aggregation for Radioactive Source DetectionJack H. Good, Ian Fawaz, Kyle Miller, and Artur DubrawskiThe Auton Lab, The Robotics Institute, Carnegie Mellon University

IntroductionSource detection is the problem of determining thepresence or absence of certain radioactive materials,usually with gamma-ray spectroscopic data. It hasapplications in•medicine• industrial safety• national securityThe source detection problem, especially in busy anddynamic urban environments, a likely target for radi-ological dispersion devices, presents challenges:• variation in background radiation• uncertain sensor position due to GPS error• dynamic gamma-ray occlusions•moving sources• nuisance sources, such as medical isotopes• anisotropic shieldingThis work focuses on adapting the Bayesian Aggrega-tion (BA) framework [1] to overcome uncertaintyin background, position, and occlusion. Futurework will focus on moving sources, nuisance sources,and anisotropic shielding.

Gamma-ray Spectroscopic DataA gamma-ray sensor measurement is a vector of pho-ton counts at several energy bins over a fixed time in-terval. These observations are the input to our model.

0 20 40 60 80 100 120KeV bin

0

50

100

150

200

250

300

350

phot

on c

ount

Gamma Ray Spectrum

backgroundbackground with source

A typical spectroscopic measurement, with andwithout source injected.

Bayesian Aggregation FrameworkBayesian Aggregation is a framework for leveragingevidence from multiple observations xini=1. Forsource detection, it is defined as

BA(xini=1) =n∏i=1

P (xi|source)P (xi|no source).

We define a vector ζ that captures uncertain physicalfactors of interest such as source intensity, distance tosource, and gamma-ray attenuation due to occlusion.With the assumption that the components of ζ areindependent given source intensity I , we proposetwo novel variations of the BA score: one thatmaximizes over ζ and one that marginalizes over ζ.

BAmax(X) = maxI∈R

P (I)n∏i=1

maxζi∈Ω

P (ζi|I)P (xi|source, ζi)P (xi|no source, ζi)

BAmarg(X) =∫RP (I)

n∏i=1

∫Ω

P (ζi|I)P (xi|source, ζi)P (xi|no source, ζi)

dζi dI

Matched Filter Likelihood ModelWe use matched filter [2], the linear model that max-imizes signal to noise ratio for known source tem-plates, as the basis of our likelihood model. It is de-fined as MF (x) = sTΣ−1x, where s is the sourcespectrum and Σ the background covariance fromtraining data. Since this is a weighted sum of Pois-son random variables, it is approximately normallydistributed; we thus define a likelihood modelP (x|source, ζ)P (x|no source, ζ)

=√√√√√ σ2b

σ2b + ζσ2

s

exp(x− µb − ζµs)2

2(σ2b + ζσ2

s)− (x− µb)2

2σ2b

,where

µs = w · s σ2s = (w w) · s

µb = w ·B σ2b = (w w) ·B

with w = Σ−1s and B the background rate vector.

PriorsWe let P (I) be a very weak L1 regularization cen-tered at 0, and we let P (ζi|I) be a L1 regularizationcentered at some estimate ζ based on a noisy esti-mate of sensor position, e.g. a GPS reading. Thereis much flexibility in altering P (ζi|I) to suit the avail-able auxiliary data.

Boundary of ImprovementWe derive a first order asymptotic approximation ofthe maximum difference between the true positiverates of the oracle (fully informed model) and baselineBA at a false positive rate F0. It is given as

max T − T = erferf−1(1− 2F0)

1− cos θ1 + cos θ

,where θ is a measure ofthe difference between thetrue and estimated param-eters of the scene andis approximately the anglebetween ζ and the true ζ. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

θ

0.0

0.2

0.4

0.6

0.8

1.0

max||α||T−T

Upper Bound in Improvement Over Baseline

FPR

1e-011e-021e-031e-041e-05

ResultsThe following plots show improvement over the base-line at a false positive rate of 10−3 for various scenesand source and background specifications.Each model with several regularization penalties:

The oracle, the maximization model with penalty512, and the marginalization model with penalty 256:

With a tuned regularization, both models have in-creased performance over the baseline under variouslevels of uncertainty, and even demonstrate true pos-itive rates close to the oracle.

The difference between the two models:

ConclusionWe find that our models are very effective in over-coming targeted sources of uncertainty; both showimprovement over baseline BA and come close to theperformance of the oracle. We recommend themarginalization approach, since it shows com-paratively better performance and is less sen-sitive to regularization tuning, with the caveatthat it is more computationally expensive than max-imization. Future work will integrate with video andcomputer vision techniques to reduce uncertainty andprovide a narrower hypothesis space.

References[1] P. Tandon, P. Huggins, R. Maclachlan, A. Dubrawski, K. Nelson, and

S. Labov. Detection of radioactive sources in urban scenes usingbayesian aggregation of data from mobile spectrometers. Inf. Syst.,57(C):195–206, Apr. 2016.

[2] G. Turin. An introduction to matched filters. IRE Transactions onInformation Theory, 6(3):311–329, June 1960.

AcknowledgementsJack Good is funded by the Carnegie Mellon UniversityRobotics Institute REU Site (NSF Grant #1659774). Hethanks Kyle Miller and Artur Dubrawski for their mentorship.