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Poisson Modeling and Bayesian Estimation of Low

Photon Count Signal and Noise Components

Prateek Tandon∗, Peter Huggins∗, Artur Dubrawski∗, Karl Nelson†and Simon Labov†

∗Auton Lab, Robotics Institute

Carnegie Mellon University

Pittsburgh, PA 15213

†Lawrence Livermore National Laboratory

Livermore, CA 94550

Abstract—Mobile radiation detectors aim to help identify

sources of radiation. Finding a radioactive source in a man-made

environment such as a city can be challenging because the additive

signal received by the detector contains both photon counts from

the source of interest and from the cluttered and variable ambient

background. Decomposing the overall radiation spectrum into its

background and source signal components is key. When either or

both of the background or source components is low in photon

counts, the estimation of signal components can become especially

challenging.

Gamma-ray spectrometry data is typically presumed to be

created with a Poisson process, though Gaussian-based estimators

are typically used to approximate the truly Poisson-distributed

data. Generally this approximation sufﬁces, but performance loss

can occur when photon counts affecting a sensor are low in

number for any signal component. Low photon count signal

and/or noise components can occur in a variety of real world

scenarios. Photon counts from the source may be low because

the source is very weak or only observable from large standoff

distances. Photon count rate from both source and background

may be low if small sensors (with limited surface area) are used

or if measurement time is limited.

Our study experiments with augmenting established anomaly

detection and match ﬁlter signal component estimators with

Poisson-based models. We apply estimators such as the Poisson

Principal Component Analysis (Poisson PCA) and the Zero-

Inﬂated Poisson (ZIP) models to the source detection problem and

benchmark with respect to popular estimators in the literature.

Finally, we apply Bayesian Aggregation to the Poisson-based esti-

mators to aggregate evidence across multiple spatially-correlated

sensor observations. Our results indicate that the use of such

techniques can aid threat detection when photon counts are low

in signal and/or background noise components.

I. INTRODUCTION

Mobile radiation detectors aim to help law enforcement

effectively identify sources of potentially harmful radiation.

Photons emitted from radioactive sources in the environment

can be measured by radiation spectrometers. Radiation spectra

are denoted as the histogram of counts of photons hitting a

sensor at a set of chosen discrete energy ranges for a particular

measurement time interval. Because various threatening radi-

ation sources have characteristic physical peaks at particular

energy levels, mobile spectrometer systems aim to help quickly

map and characterize possible radiation threat.

The complicated, variable radiation landscapes of most ur-

ban scenes (such as major cities) can, however, make detecting

a radiation source quite challenging. A radiation source being

searched for can be a needle in the haystack of cluttered

background radiation emanated from typical benign radioac-

tive sources present in many urban scenes. Developed threat

detection algorithms are expected to accurately distinguish

threats from non-threats, even when photon count rates from

the source and/or background may be low and possibly limited

in intrinsic information.

The problem of Low Photon Count Signal and Noise

Component Estimation is to succeed in detecting a radioactive

source even when the effective photon count rate received by

a sensor is especially low for either the background or source

components. The low photon count rate scenario may occur

for a variety of reasons due to properties of the source and/or

of the sensor(s) used.

An important class of cases is where the photon count

rate from the source may be low relative to the background,

making estimation of source counts difﬁcult. The radioactive

source being searched for may be very weak, occluded by

signal-attenuating obstacles (e.g. be inside a building), or be

observable only far away in standoff distance from the sensor.

If the source is deliberately engineered to be shielded, it may

be directionally attenuated, leading a loss in photon counts

detected by the sensor. In this class of cases, the photon counts

in the source component are expected to be low, though the

background count rate may be as usual.

Another set of cases is where the photon count rate from

both background and source are low. This can occur due

to limitations of the sensor data collection process and/or

hardware. For instance, a sensor may only have been able to

brieﬂy survey an area, collecting only a handful of photon

counts from both background and source due to lack of data

collection time. Additionally, sensors may also be limited

in their intrinsic material detection capability and sensitivity.

Sensors can vary in size, shape, and materials which affects

their detection capabilities. Smaller sensors can be more cost-

effective to deploy, but they will receive fewer counts from

both the background and the source than larger sensors due to

their smaller surface area.

In all of these cases, the constraint of fewer photon counts

in radiation spectra makes the detection problem harder. Fewer

photon counts in either a source or background component

generally means that each component is harder to estimate.

An algorithm has less data to reason about the delimitation of

signal components and is subject to greater possible estimation

error. The goal is to develop algorithms that can extract the

most useful information possible out of available photon count

sensor data to succeed in detecting the source.

A. Background Estimation

A radiation spectrum is the histogram of counts of photons

observed at a set of energy levels. Radiation spectra are often

presumed to be composed of two major components: the signal

due to the radioactive source of interest (if any) and the

contribution of characteristic background radiation variation

in the scene. In searching for the presence of source signal in

a spectrum, one of the ﬁrst steps is to perform Background

Estimation on the spectrum.

Background estimators are statistical estimators that seek

to estimate the number of photon counts due to background

(B) and the number of photon counts due to the source (S).

By estimating these two quantities, a background estimator can

thus calculate its estimate of the Signal-To-Noise Ratio (SNR)

of the spectrum as:

SN R =S

√B(1)

When forming the SNR score, the noise term is given by

√B(the square root of total background photon counts) as

opposed to B(total background photon counts) since radiation

data is presumed Poisson where the mean is proportional to

the variance.

Anomaly detectors and match ﬁlters are two approaches to

background estimation commonly used in practice. The goal

of anomaly detection is to ﬂag spectra that are distinct from

typical background. Anomaly detection typically assumes no

knowledge of a source template and simply uses a model

of expected background to measure the background (typical

variation) and source (unusual deviation) components of a

measurement. Match ﬁltering, in contrast, allows the user

to specify a speciﬁc source template of interest to match

against spectrum observations. Both methods allow estimation

of background and source components.

Principal Components Analysis (PCA) is a common tech-

nique of anomaly detection to learn a model for typical

background data. When PCA is run on background radiation

data, it builds a shape model for what typical background

should look like. Additionally, the top principal components

of the model capture the typical background variation of the

data. When new radiation observations are projected onto the

PCA basis learned from training data, the reconstruction error

corresponds to anomalous behavior from typical background

– behavior that is likely to be due to the source. For our

experiments, we retain the top ﬁve components since these

have been empirically shown to capture the principal modes

of variation in the data [1].

The Spectral Anomaly Detector [1] is a variant of PCA that

learns the PCA background model from the correlation matrix

of the background data rather than the covariance. This helps

avoid overly biasing the PCA model towards the most common

background energies. In addition, the resulting compressed

space is used as the null-space model, capturing the expected

types of variability in the background in the top few prin-

cipal components. Like vanilla PCA, the reconstruction error

indicates possible source signal. However, the sum of squared

residuals after background subtraction is normalized by the

sum of measurement counts to form an SNR-like “Spectral”

score.

Match ﬁltering, unlike anomaly detection, assumes knowl-

edge of a source template currently being searched. The

method allows deﬁning of a source-speciﬁc window in the

energy space of the spectrum where source photon counts

are maximally likely to appear. Energy Windowing Regression

[2] is a common technique for match ﬁltering. A regression

estimator is then trained to predict, for a new radiation vector,

the amount of background in the window from the background

photon counts outside the window. Counts left over in each bin

are likely to be related to source signal. Often times, the Least

Squares Estimator (i.e. Linear Regression) is used to facilitate

the prediction. The model typically used is given by:

ˆ

B(y) = (XTX)−1XTy(2)

where Xis the matrix of predictor energy bins and y

is the sum of background counts in the source window.

The ˆ

Bestimator, after training, predicts, for a new radiation

spectrum, how many background counts are estimated in the

source window given the photon counts present outside of the

source window. The remainder is assumed to be the source

signal. Energy Windowing Regression can signiﬁcantly boost

detection when the expected source type is known [2].

B. Poisson Modeling

Spectrometry data is often presumed to be created with

a Poisson Process. However, the available anomaly detection

and match ﬁltering methods typically rely on Gaussian models

of the data. In anomaly detection, both the vanilla PCA

and Spectral Anomaly Detector methods assume a Gaussian

distribution of the data and primarily ﬁnd linear directions of

maximal variance. Likewise, Energy Windowing Regression is

typically based on a Linear Regression (also Gaussian) model

of the data.

When the photon count rate for both source and back-

ground components is sufﬁciently high, a Poisson distribu-

tion can be approximated well by a Gaussian distribution.

In these cases, the Gaussian modeling assumption does not

invoke much loss. When the photon count rate for either or

both components is low, however, the Gaussian approximation

ceases to work as a sufﬁcient model of the data.

The remedy is to augment the estimation process with

Poisson models to boost performance. Our study experiments

with using Poisson Principal Component Analysis (Poisson

PCA) [3] instead of standard Gaussian PCA to estimate source

SNR using anomaly detection. Similarly, we experiment with

using Zero-Inﬂated Poisson (ZIP) Regression [4] instead of

Linear (Gaussian) Regression when estimating SNR with a

match ﬁlter. Our results suggest that these Poisson-based

methods can mine additional useful information from the data,

unavailable in a pure Gaussian model, to help detect potentially

harmful sources of radiation.

In addition, we use our Bayesian Aggregation (BA) algo-

rithm to further improve detection performance. BA leverages

spatial aggregation of multiple correlated radiation observa-

tions to robustly test source location hypotheses [5]. By

aggregating evidence statistically over multiple observations,

detection performance is made more robust against possible

spurious noise in the data that can cause false alarm.

II. ME TH OD S

A. Data and Point Source Simulator

Our experiment data set contains real ﬁeld data collected

over a period of ﬁve consecutive days using a double 4x16”

NaI planar spectrometer installed on a van driving in an urban

area. The data contains 70,000 measurements, reﬂective of

background and any existing nuisance sources, collected at 2

second intervals while the vehicle was in motion. Annotation

data recorded for each measurement include timestamp, lon-

gitude and latitude obtained from GPS, and the current speed

of the vehicle.

The radiation spectra in the data are 128-dimensional

vectors of counts of particles hitting the sensor where each

dimension corresponds to a speciﬁc energy interval. Each

spectrum is a histogram of counts at 128 energy intervals,

quadratically spaced across the 80 keV to 2800 keV energy

range. The data can be down-sampled to simulate different

background count rates, including relatively low ones.

A source simulator is used to inject user-supplied synthetic

radiation proﬁles into collected background radiation data to

simulate the presence of point sources. Given a geographic

subregion, the simulator chooses a random location within

proximity to the collected trajectory, and injects additional

source photon counts to the pre-existing background measure-

ments according to a Poisson distribution. In injecting the

source, the simulator takes into account the expected exposure

of the source to the sensor, the velocity of the mobile detector,

and the duration of the measurement interval. The simulator

also allows speciﬁcation of source type and source intensity

(e.g. count rate).

B. Poisson PCA

Poisson PCA is an extension of PCA that allows a Poisson-

type loss function instead of the sum of squared error loss

function used by traditional PCA [3]. The sum of squared error

loss function in traditional PCA imposes a Gaussian model

on the data. Poisson PCA uses a Poisson-based loss function

instead.

Standard PCA ﬁnds key linear directions of variation in

the data and allows for negative components. Poisson PCA,

in contrast, gives a space of typical background spectra which

are always non-negative. Accumulations of measurement de-

viations in unlikely energy bins are much less likely to be

over-ﬁtted. We use the Poisson PCA formulation in [6] for

our experiments.

C. Zero-Inﬂated Regression Poisson Match Filter

The standard Energy Windowing Regression match ﬁlter

uses the Least Squares Estimator to derive source and back-

ground components, imposing a Linear (Gaussian) Regression

model on the data. One can use a Poisson Regression model

for the expectation instead:

E(y|x) = eλTx(3)

where yis the predicted sum of counts in the energy

window, xis counts in the predictor energy bins, and λis

the Poisson mean. Using Poisson Regression instead of Linear

Regression in match ﬁltering can help improve the model.

One hurdle is that extremely low photon count data can also

be sparse and contain excess zeros that occur by chance in no

signal cases. These excess zeros, if not accounted for, may

lead to over-dispersion in the estimates of the λparameters in

the Poisson model when signal is expected. Since the mean

and variance are given by the same parameter in a Poisson

distribution, ﬁtting a Poisson distribution to data with very

sparse number of counts may not lead to a good model of

the data. An effective numerical trick is to use a Zero-Inﬂated

Poisson (ZIP) model:

P(yj= 0) = π+ (1 −π)e−λ(4a)

P(yj=hi) = (1 −π)λhie−λ

hi!, hi≥1(4b)

The ZIP model on the radiation data uses a two-step

hierarchal approach. Logistic regression is used to ﬁrst classify

the presence of non-zero counts from predictor energy bins.

If the spectrum is predicted to have non-zero background

counts in the source window, then Poisson Regression is run

to predict the amount. The separate probability densities for

the zero count and non-zero count cases help prevent over-

dispersion when estimating the mean parameter of the Poisson

distribution for the case of expected non-zero source signal.

III. EXP ER IM EN TS A ND DISCUSSION

Our experiments consist of benchmarking different

anomaly detection methods with respect to each other as well

as comparing different match ﬁlter methods. Match ﬁlters typ-

ically outperform anomaly detectors when the expected source

type is known. In all of our experiments, we compare methods

mostly using a single shielded ﬁssile material source type. It is

generally established that anomaly detectors will outperform

match ﬁlters when a particular source type is expected [5].

Thus, this comparison was not particularly interesting and is

omitted. There are, however, interesting signiﬁcant differences

that emerge internally within the category of anomaly detection

methods and internally within the category of match ﬁlter

methods.

A. Benchmarking Poisson PCA vs. Gaussian PCA

We benchmarked the detection capability of Poisson PCA

with respect to other anomaly detectors in the literature such

as the vanilla Gaussian PCA and Spectral Anomaly Detector

methods. A training set of 13,278 un-injected background

measurements was subsampled from our urban data set. All

methods built null models for the training data, storing their

top Nprincipal components.

A testing set of 13,015 background spectra was subsampled

from the data. The un-injected data was used as a negative

example set. The point source simulator was used to inject

synthetic point sources into the testing data to form a posi-

tive example set containing source-injected measurements at

different distances to the source, from 1to 20m.

All methods assigned scores to all positive and negative

examples based on their learned models from training. Distri-

butions of positive and negative scores were assembled, and

discriminative capability was compared between methods. The

key success metric used was the Symmetric Kullback-Leibler

(SKL) divergence between positive and negative point score

distributions:

KL(P||Q) = X

i

P(i) ln P(i)

Q(i)(5a)

SK L(P||Q) = 1

2[KL(P||Q) + K L(Q||P)] (5b)

where Pindicates the distribution of positive scores and

Qindicates the distribution of negative scores for a partic-

ular method. Typically, KL divergence can be used to non-

parametrically compare distributions. Standard KL divergence,

however, is not symmetric, so the deﬁnition of the SKL

divergence helps to ﬁx this.

The SKL formula estimates the difference (average log-

odds ratio) between two distributions. A higher SKL indicates

that the distribution of positive scores is increasingly distinct

from the negative score distribution, and thus source is more

detectable against background. The top-performing method for

the low count detection problem should produce the greatest

divergence between the positive and negative score distribu-

tions.

1) Detecting a far away source: An experiment was set up

that allocated each anomaly detection method some number,

from N= 2 to 5of their top Principal Components (PCs), and

the SKL performance was compared at each distance (from

1-20m) to a source. Figures 1(a)-1(d) compare the Gaussian

PCA, Spectral Anomaly Detector, and Poisson PCA methods

for different numbers of PCs. Figure 1(e) shows the maximum

SKL performance plotted over the best combination of PCs at

each distance for each method.

The experiment dealt with the case of low photon count rate

from the source but normal background photon count rates.

The source count rate was 65 counts/sec at 10m standoff from

the source, while the background count rate was 1263 ±267

counts/sec. The source is thus well within the tolerance of the

background, though the background rate is not considered low.

The SKL metric peaks near the source for all methods as

all methods succeed in close-range detection. It falls off for

all methods as standoff distance is increased to the source,

but falls off slower for Poisson PCA. The results suggest that

Poisson PCA outperforms other methods in detecting far away

sources at distances where source-originating photon counts

are very low.

(a) N=2

(b) N=3

(c) N=4

(d) N=5

(e) Max SKL

Fig. 1. SKL comparison of methods as a function of distance to the source.

(a) N=2

(b) N=3

(c) N=4

(d) N=5

(e) Max SKL

Fig. 2. SKL comparison of methods when detecting a faint source.

2) Detecting a faint source: The experiment was repeated

with a weaker source (i.e. lowered count rate) injected into the

background data. For this experiment, the overall counts from

the source were scaled down by half from the ﬁrst experiment.

The source count rate was 33 counts/sec at 10m standoff, and

the background count rate remained unchanged.

The results indicate that Poisson PCA outperforms other

methods in detecting low intensity sources that have half

the photon counts from the source intensity used in the ﬁrst

experiment. Interestingly, there is even more possible utility to

Poisson PCA for detecting sources at large standoff distances

to the sensor. As can be seen in Figure 2(e), the performance

gain from Poisson PCA comes much earlier on the distance

scale than it did in the ﬁrst experiment. In the ﬁrst experiment

we saw Poisson PCA outperform Gaussian methods at about

13m standoff to the source, while with the lowered intensity,

the performance gain can come at even 9m standoff from the

fainter source. More beneﬁt manifests in the top four and ﬁve

principal components (Figures 2(c)-2(d)).

3) Detecting a source with shorter livetime: We also

performed an experiment where measurements had reduced

sensor livetime. The counts in the measurement (for both

background and source) were scaled to simulate halved sensor

measurement intervals. Figure 3(a) shows the maximum SKL

performance. The results indicate that Poisson PCA can help

detect sources at shorter measurement interval. The net effect

is that sensors can travel faster, covering more area while still

providing useful levels of source detection.

(a) Max SKL

Fig. 3. SKL comparison of methods when detecting a source with shorter

measurement livetime.

B. Benchmarking Poisson Match Filters vs. Gaussian Match

Filters

An experiment was designed to test different match ﬁlter

background estimators in their capability to estimate source

and background counts from spectra that were low in both

background and source counts. The average background count

rate was reduced to only 22 counts/sec with the source count

rate being even less, making it a very challenging detection

scenario.

We compared three different match ﬁlters: the vanilla

Linear (Gaussian) estimator, a Poisson regression method, and

the Zero-Inﬂated Poisson (ZIP) method. Each estimator was

trained on a set of 1,527 radiation observations consisting

of background radiation to match against a shielded ﬁssile

materials source template. The estimators were subsequently

(a) Source Component Estimation Error

(b) Background Component Estimation Error

(c) SNR Estimation Error

(d) Signed Error

(e) Weighted Mean Method

Fig. 4. Experiments comparing match ﬁlter estimators.

tested on a set of 23,389 radiation background observations

with source-injected counts at a range of different distances

and exposures to a point source. Each estimator estimated

the background and source count amounts in each of the

observations, and the errors were plotted.

Figures 4(a)-4(b) show mean absolute error in estimation

of source and background photon counts for each method.

Poisson Regression has a slight gain over standard Linear

(Gaussian) Regression, while ZIP Regression has a major

advantage in the match ﬁlter. Figure 4(c) shows mean absolute

error in estimation of Signal-to-Noise Ratio (SNR). Both

Poisson and ZIP Regression outperform Linear (Gaussian)

Regression in reducing SNR estimation error.

Interestingly, the Gaussian and ZIP Regression Match Filter

estimation errors were only weakly correlated. The background

estimation errors exhibited a linear correlation value of only

0.5519, while the SNR estimation errors exhibited only a

0.2745 linear correlation value. This indicates that the two

methods are extracting different information from spectra to

ﬂag sources.

Figure 4(d) plots the signed error of the Gaussian and

ZIP models. The direction of the signed error is opposite.

Gaussian models tend to systematically overestimate back-

ground components while ZIP models systematically appear

to overestimate the source component. Using this observation,

we created a mixed method that would fuse the Gaussian and

ZIP component estimates in a Weighted Mean. The idea was

to regress the Gaussian and ZIP components against the true

amounts to attempt to cancel out signed estimation error as

much as possible. This method had some success in lowering

the SNR estimation error even further as shown in Figure 4(e).

C. Incorporation into BA Framework

To further boost performance detection for these challeng-

ing signal estimation problems, we leveraged the Bayesian

Aggregation (BA) algorithm [5]. BA is a framework for

spatially aggregating multiple noisy, low Signal-to-Noise Ratio

(SNR) observations to score a geographic location as possibly

containing a point source.

Training BA consists of building empirical sensor mod-

els based on collected ﬁeld data (though threats are often

simulated). Distributions of SNR scores are assembled as a

function of exposure on the detector’s surface. Two sensor

models are built: one for the “null” hypothesis (H0) that is

assembled from raw background radiation data and one for

the “alternate” hypothesis (H1) model that is assembled from

data containing injection of a particular source intensity and

type. These empirical models establish a testable expectation

of what new measurements are supposed to look like if they

contain (or do not contain) source signal.

Figures 5(a)-5(b) show example BA sensor models using

the Spectral Anomaly Detector background estimator. Figures

5(c)-5(d) show example BA sensor models using Poisson PCA

as the underlying spectrum SNR estimator.

Given new radiation measurements, BA scores the mea-

surements using the statistical sensor models. BA leverages

the following Bayesian update rule to aggregate probabilities

of data observations Dj∈Dto test a hypothesis H:

(a) Spectral Method Null Model (b) Spectral Method Alternate Model

(c) Poisson PCA Null Hypothesis (d) Poisson PCA Alternate Model

Fig. 5. Example BA sensor models for Spectral Anomaly Detector and

Poisson PCA methods.

P(H|D)∝P(H)Y

Dj∈D

P(Dj|H)(6)

where P(H)is the prior probability (belief) assigned to

hypothesis Hand P(Dj|H)is the empirical data likelihood.

The general strategy is to maintain a set of source hy-

potheses (including possible source parameters such as source

location, intensity, type, etc) about the data. Each hypothesis

is tested for data support using the Bayesian update formula

shown above. For each possible source hypothesis conﬁgura-

tion, we test both the alternate hypothesis (H1) that a source

of a particular conﬁguration is present and the null hypothesis

(H0) that a source of the particular conﬁguration is not present

in the data.

Fig. 6. Example BA Test Scenario.

Often times, a rectangular grid is overlaid on the trajectory

to test environment locations as potentially containing the

radiation source. Figure 6 shows an example gridding of a

collected trajectory into a hypothesis space of possible source

locations. To speed up computation, KD-trees help efﬁciently

ﬁnd grid points close to the trajectory and trajectory points

close to the gird points. It is assumed that trajectory points far

away from a grid point provide negligible evidence for that

grid point as a source hypothesis.

1) Use of Poisson PCA information in BA: We experi-

mented with the use of Poisson PCA information in the BA

framework. We ﬁrst built BA sensor models using SNR scores

estimated using Poisson PCA.

As can be seen in Figure 5(d), the SNR trend as a function

of exposure for Poisson PCA for the alternate hypothesis

can differ in shape from the expected SNR trend for a top-

performing Gaussian PCA method like the Spectral Anomaly

Detector (shown in Figure 5(b)). The differences in trend

indicate that Poisson PCA and the Spectral (Gaussian) PCA

may be extracting different information from the same spectra

in ﬂagging spectra as containing source-injection or not.

This conjecture is further supported by a correlation anal-

ysis of Poisson PCA and Spectral Anomaly Detector scores.

TABLE I. SP ECT RA L AND PO ISS ON PCA SNR SC ORE CO RR ELAT ION

COMPARISON

Data Set Spearman Rank Correla-

tion

95% Conﬁdence Intervals

in Rank Correlation

Null Scores 0.1621 [0.1202,0.1862]

Alternate

Scores

0.1410 [0.1018,0.1671]

As can be seen in Table I, Spectral and Poisson PCA scores

are only weakly correlated for both the null and alternate SNR

score distributions. This indicates the methods are detecting

different features of source signals in the same spectra. The

correlation analysis suggested using Poisson PCA and Spec-

tral Anomaly Detector methods in a Combined Aggregation

scheme. The Combined Aggregation method uses a weighted

combination of Poisson PCA and Spectral Anomaly Detector

BA scores after BA is applied to each method.

An experiment was conducted comparing Spectral

Anomaly Detector in BA (“Spectral Aggregation”), and a com-

bined aggregation scheme that used a weighted combination

of Poisson PCA and Spectral Anomaly Detector BA scores

(“Combined Aggregation”).

1,000 bootstrap replicants of a geographic subset of the

data were prepared. Each subset of data was injected with

exactly one radioactive point source using our point source

simulator and a shielded ﬁssile materials source template. The

source count rate was 65 counts/sec at 10m standoff from

the source, while the background count rate was 1263 ±267

counts/sec. A 2m grid of hypothetical source locations was

overlaid on the trajectory and both methods scored each grid

point as possibly being the source location. The region was

scored by both methods with and without source injection.

Fig. 7. Detection results for using Poisson PCA information in BA.

Figure 7 shows the detection accuracy for these methods

with respect to the scores of the top 30 false positives. As

can be seen, the Combined Aggregation method that used

information from both Poisson PCA and Spectral Anomaly

Detector (“Combined Aggregation”) outperformed the method

that just used information from the Spectral Anomaly Detector

in BA (“Spectral Aggregation”). The optimal weights for

the Combined Aggregation were [0.85,0.15] on Spectral and

Poisson PCA scores, respectively. The result suggests that

using Poisson PCA information in BA can help improve threat

detection capability.

2) Incorporation of ZIP Regression Match Filter into BA:

We performed a similar experiment benchmarking the ZIP

Regression Match Filter in BA against other match ﬁlters in

BA. We compared the Linear (Gaussian) Regression and ZIP

Regression match ﬁlters both on their own and in the BA

framework. We also experimented with marginalizing over the

posterior scores of all of these methods in a combined scheme

(BA-Marg).

1,000 simulated worlds were prepared, each with an in-

jected shielded ﬁssile material point source. The background

had an average count rate of 22 counts/sec with the source

count rate being fewer than 10 counts/sec at 10m standoff. A

2m grid was overlaid onto the world, and all methods scored

the grid points as being the likely location of the source.

Figure 8 shows the Receiver Operating Characteristic

(ROC) detection results of the methods that indicate true

positive rate as a function of the false positive rate.

Fig. 8. Detection results for using ZIP Regression Match Filter information

in BA.

The ZIP Regression Match Filter BA outperforms the Lin-

ear (Gaussian) Regression Match Filter BA. BA-Marg provides

a slight advantage and is technically the method of choice. The

results indicate that using ZIP Regression information in BA

can lead to a more powerful match ﬁlter method than simply

using the Linear (Gaussian) Regression Match Filter when the

photon counts in spectra are low.

IV. CONCLUSION

Our study experimented with augmenting the Bayesian

Aggregation (BA) framework with Poisson-based background

estimators to aid analysis of gamma-ray radiation spectra

where the photon count rates of either or both the source and

background are relatively low.

Typically, Gaussian assumptions sufﬁce for estimating the

background and source components of a radiation spectrum

to calculate its SNR when the spectrum contains a sufﬁcient

number of photon counts in its additive components. When the

photon counts in the spectrum from either the background or

source are relatively low, however, the Gaussian approximation

to the Poisson distribution is not effective. In these cases,

background estimators such as the Poisson PCA anomaly

detector or ZIP Regression match ﬁlter may be useful in

improving source detection capabilities of radiation threat

detection systems.

The results of our experiments suggest that the use of

Poisson models in background estimation can help improve

threat detection when photon counts from the source (and also

potentially from the background) can be relatively low. Our

study illustrated the use of such Poisson models for a variety

of real world scenarios where this may occur. For instance,

we showed Poisson PCA can help detect a radioactive source

that is faint or far away or when a sensor moves fast by

it. ZIP Regression can help detect a source when smaller,

less sensitive sensors are used for data collection. In all of

these tested cases, the Poisson-based background estimators

appear to improve detection capability. Exploiting the spatial

correlation between measurements in BA can further improve

performance for these challenging source ﬁnding problems.

We hope that our results will enable more sensitive and

accurate threat detection as well as possibly provide new

functionalities at the absolute boundaries of detectability with

spectrometers. For example, if mobile radiation detectors need

fewer photon counts to locate a radioactive source, they may

be able to move faster, monitoring wider areas in the same

interval of time. Similarly, if sources can be detected at fainter

SNRs and at larger standoff, there would be substantial impact

on the sensitivity of threat detection systems. The net result is

increasing the safety of the population at risk.

ACKNOWLEDGMENT

This work has been partially supported by the U.S.

Department of Homeland Security, Domestic Nuclear De-

tection Ofﬁce under competitively awarded grant 2014-DN-

0770ARU087-01, by the U.S. Department of Energy under

grant DE-NA0001736 and by the National Science Foundation

under award 1320347. Lawrence Livermore National Labo-

ratory is operated by Lawrence Livermore National Security,

LLC, for the U.S. Department of Energy, National Nuclear

Security Administration under Contract DE-AC52-07NA27344

REFERENCES

[1] K. Nelson and S. Labov, “Detection and alarming with sords unimaged

data: Background data analysis,” Lawrence Livermore National Lab

Technical Report, 2009.

[2] ——, “Aggregation of Mobile Data,” Lawrence Livermore National Lab

Technical Report, 2012.

[3] M. Collins, S. Dasgupta, and R. E. Schapire, “A generalization of

principal component analysis to the exponential family.”

[4] M. Xie, B. He, and T. Goh, “Zero-inﬂated poisson model in statistical

process control,” Computational statistics & data analysis, vol. 38, no. 2,

pp. 191–201, 2001.

[5] P. Tandon, P. Huggins, A. Dubrawski, S. Labov, and K. Nelson, “Simulta-

neous detection of radioactive sources and inference of their properties.”

IEEE Nuclear Science Symposium, 2013.

[6] N. Roy, G. J. Gordon, and S. Thrun, “Finding approximate pomdp

solutions through belief compression,” 2005.