Detection and parameter estimation of multiple radioactive sources

Detection and parameter estimation of multiple radioactive sources Mark Morelande

Branko Ristic

Ajith Gunatilaka

Melbourne Systems Laboratory Dept. of EEE The University of Melbourne Australia

Tracking and Sensor Fusion ISR Division DSTO Australia

HPP Division DSTO Australia

[email protected]

[email protected]

Abstract— Given an area where an unknown number of unaccounted radioactive sources potentially exist, and using gammaradiation count measurements collected at known locations within this area, the problem is to estimate the number of sources as well as their locations and intensities. Two approaches are investigated. The first is based on the maximum likelihood estimation and the generalised maximum likelihood rule for multiple hypothesis testing. The second approach estimates the parameters and the number of sources in the Bayesian framework via Monte Carlo integration. Numerical analysis and the performance comparison of both approaches against the Cram´erRao bound are carried out.

Keywords: Gamma radiation, Poisson statistics, Cramer-Rao bound, parameter estimation, Bayesian estimation, Monte Carlo integration I. I NTRODUCTION The risk of terrorist attacks involving improvised nuclear devices has increased sharply in the recent years. A radiation dispersal device (a.k.a dirty bomb), for example, is a radiological weapon which consists of a conventional explosive packaged with radioactive materials, aimed to kill or injure through the initial blast of the conventional explosive and by airborne radiation and contamination. Although this type of threat has not materialised so far, of growing concern are numerous accidents involving a loss or theft of radioactive sources [1]. In this paper we consider the problem of estimating the number and the location of multiple point sources of gamma radiation, using radiation sensors (probes) which record the counts, such as the Geiger-M¨ uller counter. A measured count is directly proportional to intensity (or amplitude) of a source, but inversely proportional to the squared distance between the source and the sensor. In the absence of any source, the measured count can still be non-zero, due to the background radiation (e.g. from cosmic rays). The counts (both from background and from nearby sources) obey Poisson statistics [2]. The assumption is that an area has been identified where the unaccounted radiation sources potentially exist. Furthermore, we restrict ourselves here to a flat area with no obstructions, and to the case of static radioactive point sources. The radiation count sensors acquire measurements at arbitrary

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but precisely known locations. These measurements can be collected either sequentially, if a single moving sensor is used, or instantaneously, if we have a distributed sensor network at our disposal. Special cases of the multiple source detection and estimation problem have been studied in the past. Howse et al. [3] used the least squares criterion to estimate the location of a single, possibly moving, radiological source in a room fitted with four fixed sensors. Detection of possibly moving radioactive sources based on a background dependent threshold of the accumulated sum of counts was considered in [4] and [5]. This approach however assumes that the trajectory of sources is known in advance. Bayesian classification methodology is applied to solve the detection of a single moving source of unknown trajectory in [6]. We consider two approaches in the paper. The first is based on the maximum likelihood estimation (MLE) for parameter estimation and the generalised ML rule for multiple hypothesis testing. This is an extension of [10] where parameter estimation of a single source was considered. In the second approach, estimation of the number of sources and their parameters is considered in the Bayesian framework. The intractable integrals which arise in the computation of the optimal Bayes estimator, the posterior expectation, are approximated using importance sampling. In particular, the idea of progressive correction, originally proposed in the context of particle filtering [7], is used to arrive at an efficient scheme. Selection of the number of sources is performed using the Bayesian information criterion [8]. The rest of the paper is organised as follows. Sec.II introduces formally the problem. Sec.III presents the Cramer-Rao bound analysis of the problem. Sec.IV describes the solution based on the MLE. Sec.V is devoted to the Bayesian solution. Sec.VI presents the numerical results and Sec.VII draws the conclusions. II. P ROBLEM STATEMENT AND MODELLING The problem is illustrated in Fig.1. Assume that r ≥ 0 sources are present in the area of interest, where r is unknown. Let θi = [xi yi αi ]
∈ R2 × R+ , i = 1, . . . , r, denote the parameter vector of the ith source where (xi , yi ) is the

source position in Cartesian coordinates and αi is the source intensity. The source parameter vectors are collected into a stacked vector θ = [θ
1 . . . θ
r ]
. Point sources

Perhaps the most popular performance bound is the Cram´erRao bound (CRB) which places a lower bound on the variance of unbiased estimators. In particular, for an unbiased estimator ˆ of the deterministic parameter θ the CRB for cov(θ) ˆ is J−1 θ where J is the Fisher information matrix [11, p.80], J = −E[∇θ ∇
θ log l(z|θ)]

(x1 , y1 )

(4)

with ∇θ the gradient operator with respect to θ and l(z|θ) the likelihood function of the parameter θ. The CRB holds in the ˆ − J−1 is positive semidefinite. sense that cov(θ) For the model (1), the log likelihood can be written as m  log l(z|θ) = C + [zj log λj (θ) − λj (θ)], (5)

(x2 , y2 )


d1,2

j=1

(ξ1 , ζ1 )

(ξ2 , ζ2 )

(ξ3 , ζ3 ) (ξ4 , ζ4 )

where C is a constant independent of θ. Then, m 
∇θ log l(z|θ) = ∇
θ λj (θ) [zj /λj (θ) − 1]

(ξ5 , ζ5 )

and

Sensors Fig. 1.

∇θ ∇
θ log l(z|θ) =

Problem illustration

For j = 1, . . . , m, measurements zj ∈ Z of radiation dose are taken at locations (ξj , ζj ) ∈ R2 . The following three assumptions are made. First, the radiation sources are isotropic point sources. Second, radiation measurements are independent random variables. Third, for convenience the exposure time for each of the m count measurements is equal. The joint density of the measurement vector z = [z1 . . . zm ]
conditional on the parameter vector θ can then be written as [2], [9], [10] l(z|θ) =

P (zj ; λj (θ)),

(1)

j=1

where P (z; α) = e−α αz /z! is the Poisson probability density function (PDF) evaluated at z ∈ Z+ with parameter α and λj (θ) is the mean radiation count for the jth sensor: λj (θ) = λb +

r 

αi /dj,i ,

(2)

i=1

with

dj,i = (ξj − xi )2 + (ζj − yi )2 .

(3)

being the squared distance between the jth sensor and the ith source. The constant λb in (2) represents the average count due to the background radiation only, and is assumed to be known. The problem is to estimate r and the parameter vector θ using the vector of measurements z. III. C RAM E´ R -R AO BOUND In many estimation problems optimal parameter estimates can be found only through the use of numerical techniques, the reliability of which cannot be guaranteed. Performance bounds provide a way of assessing these numerical techniques.

m  j=1

{∇θ ∇
θ λj (θ) [zj /λj (θ) − 1]

− ∇θ λj (θ) ∇
θ λj (θ) zj /λ2j (θ)}. (7)

+

m 

(6)

j=1

Substituting (7) into (4) and evaluating the expectation (using E(zj ) = λj (θ)) gives J=

m  ∇θ λj (θ) ∇
λj (θ) θ

λj (θ)

j=1

.

(8)

If we let Jk =

k  ∇θ λj (θ) ∇
λj (θ) θ

j=1

λj (θ)

,

k = 1, . . . , m,

(9)

then the FIM up to the kth measurement can be written using the recursive formula, ∇θ λk (θ) ∇
θ λk (θ) , k = 1, . . . , m, (10) Jk = Jk−1 + λk (θ) with initial J0 = 0. The required partial derivatives can be found as, for i = 1, . . . , r, j = 1, . . . , m, ∂λj (θ) = 2αi (ξj − xi )/d2j,i , (11) ∂xi ∂λj (θ) = 2αi (ζj − yi )/d2j,i , (12) ∂yi ∂λj (θ) = 1/dj,i . (13) ∂αi The posterior CRB (PCRB) for random parameters is similar to the CRB for deterministic parameters except the expectation (4) is taken over the joint distribution of the measurements and the random parameter θ and the matrix J0 which begins the recursion (10) is usually non-zero to reflect prior knowledge regarding the parameter [11]. The PCRB is a bound on the mean square error of the estimator rather than the variance.

TABLE I

IV. MLE APPROACH

MLE ERROR PERFORMANCE FOR TWO SOURCES . Y

A. Estimation of source parameters The MLE is widely used for parameter estimation because, if an asymptotically unbiased and minimum variance estimator exists for large sample sizes, it is guaranteed to be the MLE [11]. The MLE is determined as the vector θ which maximises the likelihood function l(z|θ): ˆ ML = arg max l(z|θ) θ

(14)

θ

which is equivalent to finding θ for which the gradient (6) equals zero. In the absence of an analytical solution we carry out maximisation in (14) using the Nelder-Mead numerical method [12] implemented as the MATLAB built-in routine fminsearch. In order to illustrate the performance of the MLE and, at the same time, verify the theoretical CRB derived in the previous section, we simulate the scenario shown in Fig.2. There are two sources, indicated by asterisks (∗) in the figure, the first at (−20, 10)m in the Cartesian coordinate system, and the second at (125, −65)m. The dots represent the sensor locations, placed equidistantly on a circle with radius R = 200m, centred at (0, 0)m. The number of measurements is m = 60; the background radiation mean count is λb = 1. The intensities of the two sources are equal, α1 = α2 = α, and are selected to give a certain value of signal-to-noise ratio (SNR). The SNR is defined as the mean count received at the sensors from a source with intensity α located at the origin. Thus, the relationship between the SNR and the intensity α is SNR[dB] ≈ 10 log10 (λb + α/R2 )

(15)

This definition is based on the approximation of the Poisson distribution by a Gaussian distribution with both the mean and variance equal to λj ([0, 0, α]
). Note that the mean count due to a source located away from the origin will vary with sensor position about the given SNR.

SNR [dB]

Source 1

Source 2

5 10 15 20 5 10 15 20

Position√[m] STD CRB 31.29 17.94 16.72 10.30 9.24 5.40 5.17 8.86 4.84 4.87 2.93 2.72 1.48 1.52

5] Intensity [×10 √ STD CRB 0.36 0.64 0.62 1.20 1.08 2.11 1.91 0.29 0.48 0.50 0.94 0.89 1.69 1.57

errors of the MLE were obtained by averaging over 50 realizations. The MLE search is initialised by a random parameter vector which assumes that both sources are placed within the circle shown in Fig.2. Observe from Table I how the RMS error of the MLE and the CRB show a remarkable agreement if the SNR is 10 dB or higher. For smaller values of the SNR, the MLE is unreliable and in many cases does not converge. Note also that the position error decreases as the SNR increases this result is in agreement with our intuition. However, the estimation error for intensity parameters increases with the SNR! This is due to the nature of the Poisson process in which the variance is equal to the mean count and our definition of the SNR. For the high SNR case, the mean count is high and therefore the variance of the measurements is also high. This results in higher estimation error in the intensity parameter. Finally comparing the results for sources one and two, we note that one can localise the second source a lot more accurately than the first source. These results reflect the fact that the sensors closest to a source provide the bulk of the information regarding the source parameters. It can be seen from Fig. 2 that, although many sensors are closer to source 1 than to source 2, no sensors are located as close to source 1 as some of the sensors are to source 2. B. Estimation of source number

200 150

y−position [m]

100 50

1

0 2

−50 −100 −150 −200 −200

−100

0 100 x−position [m]

200

Fig. 2. Scenario for performance analysis and comparison: ∗ indicate source locations; circles indicate sensor locations; K = 60

Table I shows the error performance of the MLE against the theoretical CRB for SNRs from 5 dB to 20 dB. The RMS

Estimation techniques such as the MLE assume that the number of sources is known a priori. In practice, this number needs to be estimated too from the data (this type of a problem is referred to as the model order selection). The conventional solution to this problem is the so called generalised maximum likelihood (ML) rule [13, p.223]. Let M denote a set of candidate source numbers. For example, if we wish to test for all source numbers up to some maximum rmax , then M = {0, . . . , rmax }. It is desired to determine which of the source numbers in M best fits a given collection of measurements z. This is done by first computing for r ∈ M the following quantity: ˆ ML,r ) − βr = log p(z|θ

1 ˆ ML,r )|, log |J(θ 2

(16)

ˆ ML,r is the MLE under the assumption that r sources where θ are present and J(θ) is the Fisher information matrix derived in Section III evaluated for the parameter value θ. The estimate

of the number of sources r is then: rˆ = arg max βr .

(17)

r∈M

Next we demonstrate the performance of the generalised ML rule for estimation of the number of sources. The candidate source numbers are M = {0, 1, 2}. The results, shown in Table II, indicate fairly accurate source detection performance. TABLE II P ERCENTAGE OF TIMES EACH SOURCE NUMBER IS SELECTED FOR A SCENARIO WITH r = 0, 1, 2 SOURCES . (100 REALIZATIONS ) True r

SNR=10dB SNR=15dB SNR=20dB

0 1 2 1 2 1 2

Estimate 0 1 92 8 0 100 0 32 0 100 0 5 0 100 0 3

attributed to their excellent performance in many situations and the relative ease with which they can be implemented. One such technique, importance sampling, will be used here. Approximation of (19) via importance sampling involves drawing samples of the parameter vector from an importance density q and approximating the integral by a weighted sum of the samples. In particular, the posterior mean (19) is approximated as n  ˆB ≈ θ wi θ i , (20) i=1 i

where, for i = 1, . . . , n, θ ∼ q and wi = C π(θ i )/q(θi ) with C such that the weights sum to one. Although the weights are written here as the ratio of the posterior density to the importance density it is possible to replace the posterior density by a quantity f such that π(θ) = B f (θ) with B a constant independent of θ. From (18), we can use f (θ) = l(z|θ)π0 (θ), so that, for i = 1, . . . , n,

rˆ 2 9 0 68 0 95 0 97

wi = C l(z|θi )π0 (θi )/q(θi ). In summary, both the MLE (for the estimation of source parameters) and the generalised ML rule (for the determination of the number of sources) are accurate and reliable techniques at moderate-to-high SNR. However, they have a drawback: the numerical complexities involved in the maximisation over the large parameter spaces become prohibitively expensive as the number of sources is increased. We were unable to implement and run in reasonable time the MLE for more than two sources. This motivates the need for an alternative algorithm suitable for detection and estimation of larger numbers of sources. V. BAYESIAN APPROACH A. Estimation of source parameters In the Bayesian framework the source parameter vector θ is a random variable. The availability of a prior distribution for θ, which represents our knowledge of the parameters before any measurements are processed, is assumed. Thus, initially θ ∼ π0 . The information contained in the measurements is combined with the prior information to give the posterior PDF, π(θ) ∝ l(z|θ)π0 (θ).

(18)

Quantities of interest related to θ can be computed from the posterior PDF. For instance, a point estimate of θ can be obtained by computing the posterior mean,  ˆ B = E(θ|z) = θ π(θ) dθ. θ (19) The posterior mean has the desirable property of being the minimum mean square error (MMSE) estimator of θ. Since the posterior PDF π, and hence the posterior mean, cannot be found exactly for the signal model of Section II, it is necessary to consider approximations. This is a common problem in Bayesian estimation and many numerical methods exist for approximating integrals of the form (19). Of particular interest in recent times has been the class of Monte Carlo techniques [14]. The popularity of the Monte Carlo methods can be

(21)

This is important since exact computation of the normalization factor for the posterior density is not possible. The simplest importance density for this problem would be the prior, i.e., q = π0 . However, straightforward use of the prior as the importance density would not work well because it can be expected that the prior will often be far more diffuse than the likelihood. As a result many, or even all, samples will be drawn in undesirable parts of the parameter space and estimates of the source parameters will be poor. Instead it is proposed to use a multi-stage procedure in which samples are obtained from a series of posterior distributions which becomes progressively closer to the true posterior distribution. The idea is that the posterior distribution approximations used in the early stages should be simpler to obtain samples from than the true posterior distribution. For our problem this can be achieved by adopting an approximate likelihood which is somewhat more diffuse than the true likelihood. In the context of particle filtering this is referred to as progressive correction [7]. Annealed importance sampling, in which samples from the importance density are drawn via a specially constructed Markov chain, is a similar idea [15]. The progressive correction procedure works as follows. Let lk , k = 1, . . . , s denote the likelihood used for the kth stage with ls = l. The likelihood at the kth k stage will take the form lk (z|θ) = l(z|θ)Γk with Γk = j=1 γj , γj ∈ (0, 1] and Γs = 1. Note that Γk is an increasing function of k which is bounded by one. Thus the likelihood used for k < s will be broader than the true likelihood, particularly in the earlier stages, making it more probable that samples drawn from the diffuse prior will have a high likelihood. In the later stages the likelihood approximation sharpens so that the samples gradually concentrate in the area of the parameter space suggested by the true likelihood. Let πk (θ) ∝ lk (z|θ)π0 (θ), k = 1, . . . , s, (22)

be the posterior PDF of the source parameters according to the kth likelihood. Note that πs = π is the posterior PDF under the true likelihood. Progressive corrections works by successively drawing samples from π1 , π2 and so on up to πs . In the first stage samples are to be drawn from π1 . This is done in four steps. First, samples are drawn from the prior, θ0,i ∼ π0 , i = 1, . . . , n. Second, weights are computed for each sample, w1,i = C1 l1 (z|θ 0,i ), i = 1, . . . , n, where C1 is such that the weights sum to one. In order to have a reasonable number of samples with a significant weighting the likelihood approximation l1 should be reasonably diffuse. This can be done by selecting a small value for γ1 . The third step is to perform resampling according to the weights, i.e., select indices j 1,1 , . . . , j 1,n such that P(j 1,i = l) = w1,l . 1,1 1,n The collection of samples θ 0,j , . . . , θ0,j forms a discrete approximation to the distribution π1 in the sense n    1,i , δ θ − θ 0,j

π1 (θ) ≈ 1/n

(23)

i=1

where δ is Dirac’s delta function. Proceeding to the next stage 1,1 1,n with the collection of samples θ0,j , . . . , θ0,j is of little use since it contains only sample values originally drawn from the prior distribution π0 . This collection of samples will become increasingly depeleted in later stages until very few, even only one, distinct sample values are present. A more useful collection of samples from π1 can be obtained by sampling from the regularised approximation π1 (θ) ≈ 1/n

n 

  1,i , g1 θ − θ0,j

(24)

i=1

where g1 is a suitably chosen kernel density. The kernel density can be selected using results in [16]. Sampling from (24) is the fourth step of the first stage. The samples are found 1,i as θ 1,i = θ0,j + 1,i where 1,i ∼ g1 , i = 1, . . . , n. These samples are passed onto the second stage in which the aim is to draw a collection of samples from π2 . This is achieved in three steps. First, noting that θ 1,1 , . . . , θ 1,n are sampled from π1 and using (22), the second-stage weights are computed as w2,i = C2

π2 (θ 1,i ) l2 (z|θ 1,i ) = C2 = C2 l(z|θ1,i )γ2 . (25) 1,i 1,i π1 (θ ) l1 (z|θ )

The second step is to select the sample indices j 2,1 , . . . , j 2,n according to the weights w2,1 , . . . , w2,n and the third step is to sample from the regularised approximation π2 (θ) ≈ 1/n

n 

2,i

g2 (θ − θ1,j ).

(26)

i=1

These three steps produce a collection of samples θ2,1 , . . . , θ2,n from π2 which is passed to the third stage. Note that the kernel densities used in the regularised approximations will usually be parameterised differently at each stage. This is reflected in the notation by indexing on the stage. The remaining stages are performed in the same way as the second stage.

The complete procedure is summarised in Table III. The performance of the procedure depends somewhat on the number s of steps and selection of the expansion factors γ1 , . . . , γs . The adaptive scheme proposed in [7] can be used to select both the number and size of the corrections. In this adaptive scheme the expansion factors are selected after each step rather being selected a priori. TABLE III I MPORTANCE SAMPLING WITH PROGRESSIVE CORRECTION .

1) Select γ1 , . . . , γs and let

ρk (θ) =

m  j=1

P (zj ; λj (θ))γk .

2) For i = 1, . . . , n, draw θ0,i ∼ π0 . 3) For k = 1, . . . , s: a) Compute the weights, for i = 1, . . . , n, w k,i = Ck ρk (θk−1,i ), with Ck such that the weights sum to one. b) Resample according to the weights w k,1 , . . . , w k,n and let j k,1 , . . . , j k,n denote the indices of the retained samples. k,i c) For i = 1, . . . , n, compute θk,i = θk−1,j + k,i where k,i  ∼ gk .

ˆ B = 1/n  θ s,i . θ n

4) Compute the parameter estimate

i=1

B. Source number estimation Let M denote a set of candidate source numbers. It is desired to determine which of the source numbers in M best fits a given collection of measurements z. This is a model selection problem in which each candidate r ∈ M corresponds to a different model and we seek the most suitable candidate. In a Bayesian framework, the selected model can be taken as that which maximises the posterior probability, i.e., rˆ = arg max π(r|z), r∈M

(27)

where the posterior probability of the source number can be written as π(r|z) ∝ p(z|r)π0 (r). (28) with π0 (r) the prior distribution of the source number and p(z|r) the marginal likelihood,  p(z|r) = l(z|θ, r)π0 (θ|r) dθ. (29) It is evident from (29) that computation of the source number posterior probabilities requires specification of a source parameter prior distribution π0 (·|r) for each r ∈ M. In estimation problems it is possible to obtain satisfactory results by choosing a vague or improper prior to reflect a lack of a priori knowledge. However, the same does not apply to model selection where the prior distribution can influence the selected model to the extent that no number of measurements will result in selection of the correct model [17]. Numerous methods have been proposed to address this problem including fractional [18] and intrinsic Bayes factors [19].

The approach taken here is to adopt the Bayesian information criterion (BIC), also called the Schwartz criterion, as a measure of model quality [8]. The BIC is an asymptotic, in the number of measurements, approximation to the posterior model probability. Although the BIC was originally formulated for a particular family of exponential distributions [8], it has been suggested that it has a much wider validity [20]. According to the BIC, the estimated number of sources is ˆ ML,r , r) + 3r log K], rˆ = arg max[2 log l(z|θ r∈M

Although the MLE was able to achieve similar estimation errors for higher SNRs, the Bayesian estimator is able to provide accurate estimates for SNRs down to 5dB while the MLE failed for SNRs below 10dB. TABLE IV BAYESIAN ESTIMATOR ERROR PERFORMANCE FOR TWO SOURCES . (50 REALIZATIONS )

(30)

ˆ ML,r is the MLE of θ under the assumption that where θ r sources are present. Due to the difficulties involved in computing the MLE, the posterior expectation, approximated using the procedure of Table III, will be used in place of the MLE in (30). VI. N UMERICAL RESULTS In this section the performance of the Bayesian parameter and source number estimators are examined. A. Parameter estimation Recall that the PCRB for random parameters involves taking the expectation (4) over both the joint distribution of the parameters and the measurements. A valid comparison between this bound and the performance of the Bayesian estimator can be obtained if Monte Carlo simulations to compute the mean square error are performed with parameters and measurements drawn randomly for each realization. However, here it is desired to assess the performance of the Bayesian estimator for a particular scenario. Thus Monte Carlo simulations will be performed in which the parameters are fixed and new measurements are drawn for each realization. Estimator performance will be compared to the CRB. Since the CRB is a lower bound on the variance of unbiased estimators of deterministic parameters, it does not bound the performance of the Bayesian estimator. It is used here as a performance benchmark which allows comparison of the Bayesian estimator with a wellknown standard. We begin by examining the performance of the Bayesian parameter estimator for the two source scenario depicted in Fig. 2. The prior distribution for the source locations is uniform within the circle of radius 200m described by the sensor positions. The initial source intensities are Gamma distributed with shape parameter κ = 0.5 and scale parameter β = 107 . This prior distribution is sufficiently vague that the amount of prior information is small relative to the information contained in the measurements. The RMS error results for the Bayesian estimator, averaged over 50 realizations, are shown in Table IV. The posterior expectation is approximated using the numerical procedure of Table III with a sample size of 10 000. The adaptive algorithm of [7] is used to determine the size and number of corrections with the maximum number of corrections set to 25. On average, the required number of corrections increases with the SNR. The accuracy of the numerical approximation is demonstrated by the closeness of the RMS errors to the CRB.

SNR [dB]

Source 1

Source 2

5 10 15 20 5 10 15 20

Position√[m] RMSE CRB 27.03 31.29 16.12 16.72 9.34 9.24 5.21 5.17 9.17 8.86 4.02 4.87 2.58 2.72 1.46 1.52

5] Intensity [×10 √ RMSE CRB 0.30 0.36 0.47 0.62 0.95 1.08 1.65 1.91 0.29 0.29 0.41 0.50 0.86 0.89 1.57 1.57

We consider now parameter estimation of three sources. The scenario is that of Fig. 2 with a third source at (80, 90)m. Once again all sources have the same intensity. The prior distribtuons for the parameters are the same as those used for the two source scenario. The simulation results from 50 realizations are shown in Table V for SNRs from 5 dB to 20 dB. Accurate estimation is achieved for SNRs down to 5 dB. Comparison of Tables IV and V shows that the addition of the third source makes the source one parameter estimates considerably less accurate while estimates of the source two parameters are only moderately less accurate. To understand this, consider the mean count from each source as a function of the sensor index. These radiation profiles are plotted for the three sources in Fig. 3 with the jth sensor at (200, 2π(j − 1)/m), j = 1, . . . , m in polar coordinates. Let λj,i = αi /dj,i denote the mean count due to the ith source at the jth sensor and ji = arg maxj λj,i denote the index of the sensor closest to the ith source. Then, the ratio r  τi = λji ,i λji ,k , (31) k=1

has a large influence on the accuracy with which the ith source can be estimated. This can be inferred from the CRB derivation of Section III. Since λj1 ,1 is quite small, as can be seen from Fig. 3, the addition of an extra source greatly reduces τ1 and therefore significantly reduces the accuracy of the source one parameter estimates. On the other hand, λj2 ,2 is large so that τ2 , and the accuracy of the source two parameter estimates, are not greatly reduced by the addition of the third source. B. Source number estimation The performance of the Bayesian technique for estimating the number of sources is analysed in one, two and three source scenarios for several SNRs. The setup is the same as that used for the parameter estimation analysis with sources 1, . . . , r used in the r source scenario. The Bayesian estimation procedure is used with 10 000 samples and the adaptive

TABLE V

TABLE VI

BAYESIAN ESTIMATOR ERROR PERFORMANCE FOR THREE SOURCES .

P ERCENTAGE OF TIMES EACH SOURCE NUMBER IS SELECTED FOR A SCENARIO WITH r

(50 REALIZATIONS ) SNR [dB] 5 10 15 20 5 10 15 20 5 10 15 20

Source 1

Source 2

Source 3

Position√[m] RMSE CRB 57.58 69.68 35.30 37.77 17.98 20.98 12.82 11.75 15.40 10.86 8.54 6.00 3.40 3.35 2.00 1.88 33.48 22.89 22.73 12.59 7.23 7.03 4.06 3.95

5] Intensity [×10 √ RMSE CRB 0.47 1.09 0.97 1.88 2.32 3.31 6.14 5.86 0.27 0.39 0.62 0.68 1.30 1.21 2.41 2.14 0.74 0.78 1.86 1.35 1.97 2.38 4.41 4.23

= 1, 2, 3 SOURCES . (100 REALIZATIONS ) True r

SNR=10dB

SNR=15dB

SNR=20dB

0 1 2 3 1 2 3 1 2 3

Estimated source number 0 1 2 3 100 0 0 0 0 100 0 0 0 0 100 0 0 0 12 81 0 100 0 0 0 0 100 0 0 0 0 93 0 100 0 0 0 0 100 0 0 0 0 93

rˆ 4 0 0 0 7 0 0 7 0 0 7

R EFERENCES 400

Mean count due to source

350 300 250 200 150 100 50 0 0

10

20

30 40 Sensor index

50

60

Fig. 3. Received mean count plotted against sensor index for source 1 (solid), source 2 (dashed) and source 3 (dotted).

scheme of [7] is used for selection of the correction factors. The set of candidate models is M = {0, 1, 2, 3, 4}. The results from 100 realizations are shown in Table VI. Source numbers are estimated with excellent accuracy for all SNRs, particularly for the one and two source scenarios. VII. C ONCLUSIONS The problem of radiological source estimation was considered. Here both the source number and source parameter estimation problems were considered. First, a maximum likelihood algorithm was studied for estimation of the number of sources and their parameters. This algorithm demonstrated good performance for a two source scenario but is not feasible for scenarios containing three or more sources. Another drawback of the maximum likelihood algorithm is its relatively high threshold SNR. Both of these deficiencies are remedied by the Bayesian estimation algorithm proposed here. Importance sampling with progressive correction is used to approximate the intractable integrals which arise in the Bayesian approach. Simualtions showed that the Bayesian approach is capable of excellent performance in two and three source scenarios with reasonable computational expense.

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