IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014 193

[16] E. D. Sontag, “Input-to-state stability: Basic concepts and results,” inNonlinear andOptimal Control Theory, P. Nistri andG. Stefani, Eds.Berlin, Germany: Springer–Verlag, 2006, pp. 163–220.

[17] Z.-P. Jiang, A. R. Teel, and L. Praly, “Small-gain theorem for ISS sys-tems and applications,”Mathem. of Control, Signals, and Syst., vol. 7,pp. 95–120, 1994.

[18] A. R. Teel, “A nonlinear small gain theorem for the analysis of controlsystems with saturation,” IEEE Trans. Autom. Control, vol. AC-41, no.9, pp. 1256–1270, Sep. 1996.

[19] Z.-P. Jiang and I. M. Y. Mareels, “A small-gain control method fornonlinear cascaded systems with dynamic uncertainties,” IEEE Trans.Autom. Control, vol. 42, no. 3, pp. 292–308, Mar. 1997.

[20] S. Dashkovskiy, Z.-P. Jiang, and B. Rüffer, “Special issue on robust sta-bility and control of large-scale nonlinear systems,” Mathem. of Con-trol, Signals, and Syst., vol. 24, no. 1, pp. 1–2, 2012.

[21] H. K. Khalil, Nonlinear Systems, third ed. Upper Saddle River, NJ:Prentice–Hall, 2002.

[22] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.:Cambridge University Press, 1985.

[23] W. Ren and R. W. Beard, “Consensus seeking in multiagent systemsunder dynamically changing interaction topologies,” IEEE Trans.Autom. Control, vol. 50, no. 5, pp. 655–661, May 2005.

[24] M. Vidyasagar, Input-Output Analysis of Large-Scale InterconnectedSystems: Decomposition, Well-Posedness and Stability. Berlin/ Hei-delberg/New York: Springer–Verlag, 1981.

[25] M. J. Corless and A. E. Frazho, Linear Systems and Control: An Oper-ator Perspective. New York/Basel: Marcel Dekker, 2003.

Networked Decision Making for Poisson Processes WithApplications to Nuclear Detection

Chetan D. Pahlajani, Ioannis Poulakakis, and Herbert G. Tanner

Abstract—This paper addresses a detection problem where a networkof radiation sensors has to decide, at the end of a fixed time interval, if amoving target is a carrier of nuclear material. The problem entails deter-mining whether or not a time-inhomogeneous Poisson process due to themoving target is buried in the recorded background radiation. In the pro-posed method, each of the sensors transmits once to a fusion center a locallyprocessed summary of its information in the form of a likelihood ratio. Thefusion center then combines these messages to arrive at an optimal decisionin the Neyman-Pearson framework. The approach offers a pathway towardthe development of novel fixed-interval detection algorithms that combinedecentralized processing with optimal centralized decision making.

Index Terms—Decision making, inhomogeneous Poisson processes, nu-clear detection, sensor networks.

I. INTRODUCTION

The physical quantities of interest in many scientific problems canbe captured by random processes characterized by discrete events thatare highly localized in time. Such phenomena can be mathematicallymodeled and analyzedwithin the framework of point processes [1]–[4].

Manuscript received July 26, 2012; revised February 22, 2013; accepted June01, 2013. Date of publication June 10, 2013; date of current version December19, 2013. Recommended by Associate Editor L. Zaccarian.C. D. Pahlajani is with the Department of Mathematical Sciences, University

of Delaware, Newark, DE 19716 USA (e-mail: chetan@math.udel.edu).I. Poulakakis and H. G. Tanner are with the Department of Mechanical Engi-

neering, University of Delaware, Newark, DE 19716 USA (e-mail: {poulakas,btanner}@udel.edu).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2013.2267399

Applications include nuclear detection [5]–[7], queueing networks [1],[8], optical communications [9], neuroscience [10], and others. Our pri-mary focus in this work is the detection of illicit radioactive (nuclear)materials in transit. Of special interest in this regard are Poisson pro-cesses, which provide the natural models describing the emission andmeasurement of radiation.The problem of detecting (moving and stationary) radioactive

sources using networks of sensors has received a fair bit of attention inthe literature. In situations where the parameters (location, trajectory,activity) of the source are unknown, Bayesian methods are frequentlyused [6], [11]–[13], embedding the issue of detection in a parameterestimation problem. While powerful, Bayesian methods for sourceparameter estimation exhibit computational complexity exponentialin the number of parameters estimated, posing challenges for theirimplementation in real time for networks with more than ten nodes[6], [11]. An important insight—and one that serves as the startingpoint for our analysis—is that in many cases of interest, the problemof source localization can be decoupled from the problem of sourcedetection. Indeed, there are improved methods [14]–[16] for trackingthe carrier of a potential radioactive source using sensor modalitiesother than a Geiger counter. Armed with this observation, sourcedetection reduces to the problem of deciding whether the countsobserved by a spatially distributed network of radiation sensors cor-respond solely to background radiation, or whether they also includeemission from a radioactive source with known parameters. In thissetting, [6] explores the Signal-to-Noise Ratio (SNR) resulting fromthe combination of data from a network of radiation sensors, allowingfor spatially varying background rates. The analysis is restricted,however, to uniform linear source motion and does not provide adecision test. The costs and benefits of using networked sensors formoving sources, together with a threshold test (based on the totalnumber of recorded counts) are addressed in [17], assuming uniformbackground and constant geometry between source and sensor.1 Forthe case of a stationary source and correlated sensor measurements, adistributed detection scheme is developed in [18] using the theory ofcopulas. The work in [11] studies detection (via Bayesian estimation)for a moving source, but the motion is required to be linear withconstant velocity. Detection and parameter estimation for an unknownnumber of static radioactive point sources are treated in [12], [13].Evidently, the networked detection problem for general source motionwith spatially varying background intensity has yet to be studied.Motivated by the above, we pose the following binary hypothesis

testing problem: a spatially distributed network of radiation sensorsrecords impinging photons over a fixed time interval, and has to de-cide at the end of the interval whether the registered counts correspondsolely to background radiation, or whether the counts are the superpo-sition of background radiation with the emission from a moving sus-pected radioactive source with known intensity. If a source is in factpresent, the relative motion between the source and the sensors leadsto a time-inhomogeneous Poisson arrival process at the sensors. Underthe assumption of conditionally independent sensor observations, weidentify an optimal Neyman-Pearson decision scheme that combinesdecentralized processing (local processing at each individual sensor)with centralized decision making via a fusion center. The method re-lies on the sensors communicating processed information in the formof locally-computed likelihood ratios to the fusion center. The fusioncenter then combines these messages to arrive at a decision, without theneed for any additional information such as the location or the raw dataof individual sensors. This approach combines the significantly lower

1Our analysis indicates that the optimal test involves comparing the likelihoodratio against a threshold, rather than the total number of counts.

0018-9286 © 2013 IEEE

194 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014

Fig. 1. Architecture of the detection scheme as applied to radiation detection.There is a network of sensors each counting the total number of rays that havearrived at them. Thick dashed arrows represent rays emitted from a movingradioactive source and thin dashed arrowsmark rays from background radiation.Based on the number and timing of arrival of those counts, each sensor computesa likelihood ratio which is then transmitted to a fusion center. The fusioncenter combines this information to make a decision regarding the presence ofthe radioactivity in the target moving in front of the sensors.

communication cost of decentralized processing (not decision) with theenhanced accuracy of centralized decision making. For our problem ofradiation detection, the framework accommodates arbitrary continuoussource motion in any number of dimensions, allowing for sensor mo-bility and spatially varying background rates.The paper is organized as follows. In Section II, we state the problem

and our technical assumptions. The main result (Theorem 1), which in-dicates how a global likelihood ratio test can be formulated based onlocal computation of sensor-specific likelihood ratios, is presented inSection III. The proof of this result and the supporting technical ma-terial are found in Section IV. Based on this analysis, we offer con-servative lower and upper bounds on the probabilities of detection andfalse alarm, respectively, in Section V. Finally, a numerical exampleof a one-dimensional case of networked nuclear detection is developedin Section VI, highlighting the benefits of using multiple sensors. Theresults provided in this paper can be viewed as a building block towarda general decision-making framework that leverages networks of mo-bile sensor platforms to enhance detection capability in problems thatinvolve time-inhomogeneous point processes.

II. PROBLEM STATEMENT AND ASSUMPTIONS

Consider a collection of radiation sensors deployed over some spa-tial region of interest. A moving suspected radioactive source is identi-fied at . In addition to recording incident photons and their timesof arrival, we assume that each sensor , , has access tothe distance between itself and the suspected source for all times inthe interval , . The sensors are configured in a parallel fu-sion architecture (Fig. 1). The problem is to decide, at time , whetheror not the target is in fact a source, based on measurements from the-sensor array.Radiation sensors always record background radiation due to cosmic

radiation and also due to naturally occurring radioactive isotopes in theenvironment. In the absence of illicit nuclear material (call this hypoth-esis ), the sensors simply measure background. If radioactive ma-terial is present (call this hypothesis ), the sensors record the sumof the photons coming from background and the photons coming fromthe source. The two sources of radiation (when hypothesis is true)act independently, and one can treat each sensor as observing a single

Poisson process whose intensity is the sum of intensities due to back-ground and source. Let , , be the intensity of backgroundradiation at sensor , . With denoting the distance be-tween sensor and the potential source, the intensity at sensordue to the source (under hypothesis ) is modeled in [6] by

(1)

where is the activity of the potential source2 and is thesensors’ cross-section coefficient.3 Sensor thus observes a Poissonprocess whose intensity is under , and under. The detection problem can thus be summarized as follows: given

a single realization of a -dimensional vector of Poisson processes overthe time horizon (the components corresponding to the sen-sors), decide whether the intensities are given by the collection ,or by the collection , .We now state precisely our modeling assumptions. We start withAssumption 1: Conditioned on hypothesis , , the ob-

servations at distinct sensors are independent.Regarding the background radiation, we assumeAssumption 2: For , is a

bounded, continuous function with , ,independent of .

Finally, we insist that no sensor is ever closer than a pre-specifieddistance to the target, ensuring thatAssumption 3: For , is a

bounded, continuous function with , ,independent of .Note that by Assumptions 1–3, the sensor observations are inde-

pendent but not identically distributed. This is in agreement with thephysics of the problem, since a gamma ray emitted from the sourcecannot pass through more than one sensor simultaneously (see Fig. 1),and the time varying nature of the distance between source and sensorchanges the arrival statistics on the sensor side [6].

III. THEORETICAL FRAMEWORK

For the network in Fig. 1, Theorem 1, stated below, gives a proce-dure for locally processing sensor information and transmitting com-pressed summaries (at a single time) to the fusion center to enablenetworked decision making that recovers the performance of a cen-tralized Neyman-Pearson scheme. We start with a measurable space

, on which a -dimensional vector of counting processes, is defined. In our problem, is

the number of counts registered at sensor up to (and including) time. The two hypotheses and regarding the state of the en-

vironment correspond to two distinct probability measures on .Hypothesis corresponds to a probability measure , with respectto which the , , are independent Poisson processesover with intensities , respectively. Hypothesis cor-responds to a probability measure , with respect to which the ,

, are independent Poisson processes over with in-tensities , respectively. To avoid cases of singular detection[19], [20], one requires that be absolutely continuous with respect to, denoted , i.e. whenever with , we have

. This property is built into our construction of (Propo-sition 1 in Section IV). The decision problem is now one of identifyingthe correct probability measure ( versus ) on based on arealization of the -dimensional process .We will keep track of the flow of information using the filtration

generated by the process ; here, for ,

2More accurately, is the minimum intensity of the suspected source.3For the case of heterogenous sensors, each will have its own .

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014 195

is the smallest -field on with respectto which all the ( -dimensional) random variables , aremeasurable. The interpretation is: for any event , an observerof the sample path , , knows at time whetheror not the event has occurred. The -field thus represents theinformation generated by the totality of sensor observations up to; to wit, the information on which the decision must be based.A test for deciding between hypotheses and on the basis

of observations can be thought of as a set with thefollowing significance: if the outcome , decide ; if

, decide .4 For a test , two types of errorsmight occur. A “false alarm” occurs when the outcome (i.e.decide ) while is the correct hypothesis. A “miss” occurs when

(i.e. decide ) while is the correct hypothesis. Clearly,the probability of false alarm is given by , while the probabilityof a miss is given by . The probability of detection is givenby .In the Neyman-Pearson framework, one is given an acceptable upper

bound on the probability of false alarm , and the problem isto find an optimal test: a set which maximizes the proba-bility of detection over all tests whose probability of false alarm is lessthan or equal to . The following result provides an optimal test thatemploys local information processing at the sensor level, to enable de-cisions at the fusion center that recover the optimal performance of acentralized Neyman-Pearson test. The underlying probabilistic setup isas described above.Theorem 1: Consider a network with sensors and a fusion center

connected in the parallel configuration of Fig. 1. For ,let , denote the observation at sensor over the timeinterval and let be the jump times of .Assume that at decision time , sensor transmits to the fusion centerthe statistic

computed on the basis of its observation , . 5

Then, the test performed at the fusion center, with, and satisfying , is

optimal for -observations in the sense that for any with, we have .

Remark: We emphasize that if sensors are informed of the target’strajectory by means of a tracking algorithm (e.g. [22]), the motion ofthe potential source does not need to be known in advance. Indeed, ifsensor , , records the times at which countsare registered, and computes the distance function to the source,then (1) implies that can be locally computed at sensor at time, without need for any additional information (including knowledgeof the measurements at other sensors).Remark 2: Since is the optimal Neyman-Pearson test for-observations (the latter comprising the totality of information in

the waveforms , ), it yields the performance of acentralized framework, where each sensor transmits its entire observedsample path to the fusion center. As noted in Remark 1, however, thetest can be realized by having each sensor transmit its lo-cally computed to the fusion center at a single time, with thecomparison of the product against the thresholdperformed at the fusion center. We thus retain the accuracy of central-ized decision making while decentralizing most of the data processing,thereby accruing significant savings in communication costs.

4We restrict attention here to tests without randomization; see [19], [21].5By convention, .

IV. PROOFS

We start with Proposition 1, which adapts to our problem, a measuretransformation technique for point processes [1, Theorem VI.2.T3].This provides the probabilistic setup for the statement and proof of The-orem 1.Proposition 1: Suppose is a probability space, on which

, , is a vector of independent-Poisson processes whose components admit intensities, , with satisfying Assumption 2. Define the

process by

(2)

Then,

(3)

defines a probability measure on with , with re-spect to which , , are independent -Poisson pro-cesses over with intensities , with satis-fying Assumption 3.

Proof: Lemma 1 in the Appendix assures us that is a nonnega-tive random variable with .6 Hence, defined through (3)is indeed a probability measure on which is absolutely contin-uous with respect to . In order to show that with respect to ,for , are independent Poisson processes over withintensities , we use time rescaling to apply [1, TheoremVI.2.T3].Let be a rescaled time variable taking values in [0, 1].

For , , let . Letand let for

. By Lemma 2, in the Appendix, each , isa -Poisson process with intensity . Theindependence of and , followsfrom the independence of and .Denote by the sequence of jump times of . Note

that . Letting for ,define a process by

(4)

It is easily checked that for all , which im-plies that . We now apply [1, Theorem VI.2.T3], usingthe rescaled time variable , filtration , with and

in place of and respectively, to infer thathas intensity with respect to . Using Lemma

2 in the “reverse” direction (i.e. interchanging and , replacingby ), it follows that the ’s, , have -intensities

, respectively. To complete the proof, it remains to showthat the , , are independent and Poisson under .Assertion of [1, Theorem II.3.T8] (with ) implies that for

, is a -martingale. Bythe Multichannel Watanabe Theorem [1, Theorem II.2.T6], it now fol-lows that with respect to , the ’s are independent -Poissonprocesses over with intensities , respectively.

Proof of Theorem 1: Let , be the restrictions of ,respectively to . Since on , the restrictions of andto the smaller -field inherit the absolute continuity, i.e.. Hence, the Radon-Nikodym derivative exists; i.e. there exists a

6 denotes expectations with respect to .

196 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014

nonnegative, -measurable random variable , denoted ,such that for any ,

(5)

Moreover, thisRadon-Nikodymderivative is unique in the sense that ifis anynonnegative, -measurable randomvariable satisfying (5)withreplacing , then , -a.s. Since is a nonnegative, -mea-surable randomvariable (byLemma1)which satisfies (5), it follows that

, -a.s. A direct application of the Neyman-PearsonLemma, [1, Theorem VI.1.T1] completes the proof.

V. PERFORMANCE ANALYSIS

Here we provide a lower bound on the probability of detection andan upper bound on the probability of false alarm when the proposed de-tection scheme is used. It turns out that bounds on both these probabil-ities involve the tails of (different) Poisson distributions. If

, , denotes the Poisson distribution with parameter, the left and right tail probabilities [23] are defined, respec-

tively, by , ,for . Note that .The following quantities will also be of interest:

(6)

It follows from (2) that . Let

denote the integer ceiling function, which assigns to a real numberthe smallest integer greater than or equal to .For , consider the test . A lower bound

on the probability of detection , and an upper bound onthe probability of false alarm can now be obtained asfollows. Recalling Assumptions 2 and 3, define

(7a)

(7b)

Note that if one can find , such that , then for, . Letting

, , it can be verifiedthat . Next, note that

, . With respect to ,the ’s, , are independent Poisson random variableswith parameters , implying that is Poissonwith parameter . Under the probability measure, the ’s, , are independent Poisson random variables

with parameters , implying that isPoisson with parameter . Hence,

(8a)

(8b)

VI. EXAMPLE

In this section, we illustrate the results of this paper in the context ofa concrete example. The setting here is simple, but representative of afrequently encountered class of scenarios. Our method is not restricted,however, to such simple settings. Indeed, the results in the previoussections apply whenever the intensity of the suspected source is known

Fig. 2. Distribution of background intensity (2(a)) and integrated perceivedsource intensity (2(b)) over the considered ten-sensor array.

and the motion of the source relative to the sensors can be tracked, withminimal restrictions on the geometry.The specific assumptions for this problem are as follows: The

workspace is the horizontal plane, . We have sensors uni-formly spaced along the positive -axis at locations , ,

in a configuration as that shown in Fig. 1.To span a length of approximately 100 m, we choose ,

, and for simplicity, we assume that the sensors are identical.Let denote time, where corresponds to the instantthe count recording is initiated, and is the final time at which adecision regarding the existence of a source is to be made. Letbe the intensity of background radiation at the location of sensor ,

, which does not have to be uniform and in general canbe time-dependent. For simplicity, we assume in this example thatbackground intensity is time-invariant, so , whereis assumed to be varying between locations, from a minimum of

counts per second (cps) to a maximum of ,with the maximum appearing at the first and last sensor and theminimum occurring at the sensor in the middle (Fig. 2(a)). We assumethat a target is passing at a distance (the equivalent of 141/4 inches) from the -axis, namely with a constant coordinate ,appearing first at some initial location ,and moving with constant speed (roughly 38 mph) in thedirection of the positive -axis. To illustrate the derivation process, letus for the sake of argument assume that the acceptable probability offalse alarm in this scenario is (see (8b)). Recall (1), where

is the activity of the potential source (in cps) and (in )is the sensors’ cross-section coefficient. We assume a numerical valuefor equal to what has been used in [6], but shielded in 3 cm of lead,

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014 197

Fig. 3. Bound on the probability of false alarm (PFA). The Poisson tail thatupper bounds the probability of false alarm decreases monotonically with itssecond argument (Fig. 3(a)), and a value for the latter can be identified for whichthe former falls below a desired value. Without resetting the threshold constant,we see that the upper bound on the PFA decreases with the addition of newsensors. For sensors and a PFA is at most 20%, while fortwo additional sensors one can guarantee a one-in-a-million chance for a falsealarm.

dropping the source’s perceived intensity by one order of magnitudeto . We also assume that is always bounded.Since the location of the potential source at time is, the distance between the potential source and sensor ,is given by Recalling

(1) with for the decision time, we get, for .

Since

from (6) we obtain

and for the ten-sensor array we have .

It is seen that and

.Thus, for the case of the ten-sensor array, (7) evaluates to

,. With reference to (6), we have counts, and

with this we can attempt to numerically compute a threshold forthe likelihood ratio test using (8b). It can be verified that the Poissontail on the right hand side of (8b) falls below when the secondargument of increases to 338 (see Fig. 3(a)). We thus compute

the value of for which , and obtain thatwith , the bound on the probability of false alarm fallsat , which is below the acceptable error rate. The decisionrule therefore is based on the test: which if true, suggeststhat the target is indeed a radioactive source. It should be mentionedthat there is conservatism built in the bounds (7), which renders theprobability of detection using rather impracticallysmall for the given false alarm rate. In addition, it is acknowledgedthat the illustrated method for obtaining a threshold makes the solutionfor very sensitive to changes in the underlying parameters , ,

and . Improving the bounds in (8) is part of ongoing work.Nevertheless, the analysis still gives insight into the effect of differentparameters on the probability of detection. To illustrate that point, letus consider the possibility of using more sensors with the same spacingas before. Without changing the decision rule (i.e.keeping the same threshold), the analysis shows (Fig. 3(b)) how theupper bound on the probability of false alarm (PFA) estimated in (8b)not only falls monotonically with the addition of new sensors, but thatthere is a clear transition between the regime where the sensor networkdecision is unreliable, and one where an alarm should be taken intoaccount seriously. Such information can be useful for determiningthe minimum number of functional sensors needed to ensure reliabledecision making.

VII. CONCLUSIONS

A network of sensors can be deployed to optimally decide betweentwo hypotheses regarding the statistics of a time-inhomogeneous pointprocess in a way that preserves the accuracy of centralized decisionmaking without incurring the increased communication cost. The sen-sors collect their measurements over a fixed-time interval, at the end ofwhich a processed summary is communicated to a fusion center. In par-ticular, each sensor transmits a locally computed likelihood ratio to thefusion center, which then compares the product of the sensor-specificlikelihood ratios against a threshold to arrive at a decision. The anal-ysis is based on the Neyman-Pearson formulation. A set of conserva-tive performance bounds on the error probabilities is provided and theframework is applied to the problem of detecting a moving radioactivesource using an array of sensors. The work here supports the develop-ment of a general decision-making framework that leverages networksof mobile sensor platforms to enhance detection capability in problemsthat involve time-inhomogeneous point processes.

APPENDIX

Lemma 1: The process defined by (2) is a nonneg-ative -martingale. Thus, has constant mean, i.e.

for all .Proof: The non-negativity of is evident from (2). By [1, Equa-

tion VI.2.4], we have

(9)

198 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 1, JANUARY 2014

where . To complete the proof, it sufficesto show that each of the integrals on the right in (9) is a martingale. By[1, Theorem II.3.T8], for , is a

-martingale whenever for .7

By Assumptions 2, 3, we get that , where. Since and the , are independent

with each non-decreasing in , we get

where the last line follows from the fact that under , for , eachis a Poisson random variable with parameter .

Note that Lemma 2, stated next, is formulated in terms of a generalprobability space, not necessarily identical to the one in Proposition 1.Lemma 2: Let be a -Poisson process with

intensity . Fix . Let . Let ,

for . Define by for. Then, is a -Poisson process with intensity.Proof: The -measurability of for all implies

that is -adapted. The independence of and forfollows from the independence of and .

Finally, for ,

The change of variables yields that is a Poissonrandom variable with parameter .

REFERENCES

[1] P. Brémaud, Point Processes and Queues. Berlin: Springer-Verlag,1981.

[2] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of PointProcesses, 2nd ed. Berlin: Springer, 2002, vol. I.

[3] ——, An Introduction to the Theory of Point Processes. Berlin:Springer, 2007, vol. II.

[4] D. L. Snyder and M. I. Miller, Random Point Processes in Time andSpace, 2nd ed. Berlin: Springer-Verlag, 1991.

[5] R. Byrd, J. Moss, W. Priedhorsky, C. Pura, G. W. Richter, K. Saeger,W. Scarlett, S. Scott, and R. W. , Jr., “Nuclear detection to prevent ordefeat clandestine nuclear attack,” IEEE Sensors J., vol. 5, no. 4, pp.593–609, 2005.

[6] R. J. Nemzek, J. S. Dreicer, D. C. Torney, and T. T. Warnock, “Dis-tributed sensor networks for detection of mobile radioactive sources,”IEEE Trans. Nuclear Sci., vol. 51, no. 4, pp. 1693–1700, 2004.

[7] L. Wein and M. Atkinson, “The last line of defense: Designing radia-tion detection-interdiction systems to protect cities form a nuclear ter-rorist attack,” IEEE Trans. Nuclear Sci., vol. 54, no. 3, pp. 654–669,2007.

7Actually, Theorem II.3.T8 in [1] also requires that be -predictable.This follows from the left-continuity and -adaptedness of , by [1, The-orem I.3.T5].

[8] H. Chen andD. D. Yao, Fundamentals of Queueing Networks. Berlin:Springer-Verlag, 2001.

[9] R. M. Gagliardi and S. Karp, Optical Communications, 2nded. Hoboken: Wiley-Interscience, 1995.

[10] H. C. Tuckwell, Stochastic Processes in the Neurosciences. Philadel-phia: SIAM, 1989.

[11] S. M. Brennan, A. M. Mielke, and D. C. Torney, “Radioactive sourcedetection by sensor networks,” IEEE Trans. Nuclear Sci., vol. 52, no.3, pp. 813–819, 2005.

[12] M. Morelande, B. Ristic, and A. Gunatilaka, “Detection and parameterestimation of multiple radioactive sources,” in Proc. IEEE Int. Conf.Information Fusion, 2007, pp. 1–7.

[13] B. Ristic, M. Morelande, and A. Gunatilaka, “Information drivensearch for point sources of gamma radiation,” Signal Processing, vol.90, pp. 1225–1239, 2010.

[14] R. Kaestner, J. Maye, Y. Pilat, and R. Siegwart, “Generative object de-tection and tracking in 3d range data,” inProc. IEEE Int. Conf. RoboticsAutomation, 2012, pp. 3075–3081.

[15] D. Held, J. Levinson, and S. Thrun, “A probabilistic framework for cardetection in images using context and scale,” in Proc. IEEE Int. Conf.Robotics Automation, 2012, pp. 1628–1634.

[16] N. Wojke and M. Haselich, “Moving vehicle detection and trackingin unstructured environments,” in Proc. IEEE Int. Conf. Robotics Au-tomation, 2012, pp. 3082–3087.

[17] D. L. Stephens and A. J. Peurrung, “Detection of moving radioactivesources using sensor networks,” IEEE Trans. Nuclear Sci., vol. 51, no.5, pp. 2273–2278, 2004.

[18] A. Sundaresan, P. K. Varshney, and N. S. V. Rao, “Distributed detec-tion of a nuclear radioactive source using fusion of correlated deci-sions,” in Proc. IEEE Int. Conf. Info. Fusion, 2007, pp. 1–7.

[19] H. V. Poor, An Introduction to Signal Detection and Estimation, 2nded. Berlin: Springer-Verlag, 1994.

[20] T. Kailath and H. V. Poor, “Detection of stochastic processes,” IEEETrans. Information Theory, vol. 44, no. 6, pp. 2230–2259, 1998.

[21] E. L. Lehmann, Testing Statistical Hypotheses, 2nd ed. Hoboken:Wiley, 1986.

[22] N. Dahal and B. Triggs, “Histograms of oriented gradients for humandetection,” in Proc. IEEE Conf. Computer Vision Pattern Recognition,2005, pp. 886–893.

[23] P. W. Glynn, “Upper bounds on poisson tail probabilities,”OperationsResearch Lett,, vol. 6, no. 1, pp. 9–14, 1987.