WE consider the detection of particle sources using directional

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1 4472 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 53, NO 12, DECEMBER 2005 Detection of Particle Sources With Directional Detector Arrays and a Mean-Difference Test Zhi Liu, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE Abstract In this paper, the problem of detecting far-field particle sources, such as nuclear, radioactive, optical, or cosmic, is considered This problem arises in applications including security, surveillance, visual systems, and astronomy The authors propose a mean-difference test (T) with cubic and spherical detector arrays, assuming Poisson distributed measurement models Through performance analysis, including computing the probability of detection for a given probability of false alarm, the authors show that the T has a number of advantages over the generalized likelihood-ratio test (GLRT), such as computational efficiency, higher probability of detection, asymptotic constant false-alarm rate (CFAR), and applicability to low signal-to-noise ratio (SNR) For each array, the authors also present an estimator to find the source direction Finally, Monte Carlo numerical examples are conducted that confirm the analytical results Index Terms Directional detector array, generalized likelihood-ratio test, mean-difference test, nuclear detection, particle source, security I INTRODUCTION WE consider the detection of particle sources using directional detector arrays originally proposed in [1] The sources of interest may be nuclear, radioactive, optical, or cosmic, emitting particles such as rays, neutrons, photons, particles, or particles An example of potential application is the detection of nuclear sources, which has increasingly become an important issue in security and defense Another example is the detection of photon illuminants in artificial vision systems (For details on radiation particle detectors, see [2] [6], and on photon detectors see [7] [10]) In [1], we have introduced several directional detector arrays and used them to localize particle sources We presented projection-based sensor models and assumed Poisson distributed data to derive lower bounds on the mean-square angular error [(MSAE), see also [11]] of the source direction estimation In this paper (see also [12]), we consider two detector arrays similar to [1], namely, cubic (in each side of the cube is a detector) and spherical (with a large number of small-size detectors completely covering a sphere surface) These detectors are directional since their surfaces have different orientations with Manuscript received April 19, 2004; revised January 31, 2005 This work was supported by the Air Force Office of Scientific Research Grant F and the National Science Foundation Grant CCR The associate editor coordinating the review of this manuscript and approving it for publication was Dr Feng Zhao The authors are with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL USA ( zliu9@uicedu; nehorai@eceuicedu) Digital Object Identifier /TSP respect to the source direction We employ the measurements of these directional arrays to detect the existence and estimate the direction of the source In contrast, without directional arrays, many existing techniques can detect particle sources only indirectly, such as through image analysis [13], but may not be capable of estimating the source direction Estimating the source direction is important in many applications and should also improve the detection performance We use Poisson distributions to model the detectors measurements of the source and noise particles We first consider the generalized likelihood-ratio test [(GLRT), see [14, Sec 642]] for the cubic array We find that the GLRT has a complex structure in this case (see Section III-A), which makes it difficult to analyze its performance, ie, find analytically the probability of detection ( ) for a given probability of false alarm ( ) [14] Moreover, our numerical examples show that the performance of the GLRT may not be acceptable at low signal-to-noise ratio (SNR) values We then present a new test, the mean-difference test (T), based on the difference between the average of the temporal sample means over two groups of detectors, namely, source-exposed and noise-only detectors We derive the asymptotic distribution of the T for the cubic array and analyze its performance Compared with the numerical results of the GLRT, the T has higher probability of detection, especially at low SNR values In addition, the T requires estimating only one parameter, the source direction, which is of interest in its own right Next, we address the detection using spherical arrays We define a spherical array as multisurface, in each surface corresponds to a detector, the area of each detector approaches zero, and the number of detectors approaches infinity [1] The detectors fully cover the sphere surface For spherical arrays, the GLRT may not be analytically computable due to the complexity of the joint likelihood function In contrast, we derive and analyze the performance of the T and show that it outperforms the cubic array with equal surface area It is computationally simple, applicable at low SNR values, isotropic to the source direction (detection performance remains the same for arbitrary source direction), and has asymptotic constant falsealarm rate (CFAR) [14] The remainder of the paper is organized as follows In Section II, we present the basic mathematical model In Sections III and IV, we derive and analyze the aforementioned tests for both arrays In Section V, we present numerical examples that confirm our analysis In Section VI, we discuss possible extensions of our work X/$ IEEE

2 LIU AND NEHORAI: DETECTION OF PARTICLE SOURCES WITH DIRECTIONAL DETECTOR ARRAYS AND A T 4473 II BASIC MODELING To simplify the presentation, we consider a single source assumed to be time invariant in the far field Our model is based on the projection-based arrays originally proposed in [1], which employ the fact that the number of particles impinging on each detector surface in the array is proportional to its cross section Denote by the number of detectors in the array Each detector counts the particles impinging on its surface Assume the count of source particles is a random variable with Poisson distribution; in addition, assume the count of noise particles of each detector is also Poisson distributed and independent of the source particle count Further assume each detector in the array has a congruent shape, ie, it is a face of an -surface convex regular polyhedron, and let its surface area be We also follow the assumptions A1 A4 in [1], namely, narrow-band detectors, far-field source, and isotropic efficiency Let be the expected flux of the source particles at the detector array, and be the expected rate of the noise counts on the detector surface More specifically, is the expected count of source particles per unit time crossing a unit area perpendicular to the particles direction of arrival at the detector, as is the expected count of the noise particles per unit time per unit area on the detector surface We assume that and are constant for all the detectors in the same array, due to the time-invariance and far-field assumptions Define the SNR as Consider the detector array measurements of particles over nonoverlapping time-unit intervals The th temporal unit measurement of the th detector is denoted as Under the above modeling and assumptions, can be expressed as and denote, respectively, the source and noise counts of the detector Both are random variables, mutually independent, and have Poisson distributions with expected rates and, respectively, denotes the cross section of the th detector with respect to the source direction Due to the independence of and, the measurement is also Poisson distributed with expected rate [1] Let be the unit vector at the origin pointing toward the source in some reference frame Denote, and are, respectively, the azimuth and elevation angles of Thus, and The unknown parameter vector in this case is defined as The source detection problem of the detector array can now be simply defined as hypothesis test the null hypothesis represents the case there is no source, and the alternative hypothesis represents the case a source is present (1) (2) (3) Fig 1 Cubic array III DETECTION USING CUBIC ARRAYS Consider a projection-based array with a cubical surface in each face is an individual detector s surface In this case, and the area of each face of the cube is In general, if a source is present, three detectors are exposed to the source particles We label the indexes of these detectors as 1, 2, 3, and their opposite detectors as 4, 5, 6, respectively Without loss of generality, we choose the cube s axes such that they coincide with the axes of the reference coordinating system, as shown in Fig 1 To be more specific, denoting the unit vector normal to the cube s th face as ( ), we choose the cube s axes and the reference coordinate system such that,, and are the positive,, and axes, and,, and are the negative,, and axes of this system, respectively Recall the assumption that there are time-unit measurements of each detector, and here ( ; ) denotes one of the temporal measurements Note that these measurements are statistically independent and have Poisson distributions, and the measurements of an individual detector are independent and identically distributed (iid) In this case,,, and is the inner product From (2), it follows that obeys the Poisson distribution with parameter In the unknown source direction case, we do not know which detectors face the source, which causes a problem when labeling the faces of the cube as described above To solve this problem, we first arbitrarily label the six faces with opposite pairs 1 and 4, 2 and 5, and 3 and 6 Then, we employ the following algorithm to estimate the source direction, based on the relationship between the temporal sample means of the opposite detector pairs [1]:, ( ) denotes the temporal sample mean of the th detector, and is the Euclidean norm Finally, using the estimated source direction, we relabel the faces such that the source is in the first octant (4)

3 4474 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 53, NO 12, DECEMBER 2005 From Fig 1, we have,, Thus,, and the two hypotheses can be redefined as is unknown is unknown We thus construct a mean-difference () statistic, which is defined as the difference between the temporal sample means averaged over both groups of detectors is the temporal sample mean averaged over source-exposed detectors Therefore, the test for the cubic array is constructed as (7) (8) A GLRT for Cubic Arrays Consider the GLRT for the cubic array GLR (5), and and denote the joint likelihood functions of under and, respectively Through the derivations in Appendix A, we show that the GLRT for the cubic array is equivalent to GLR is the temporal sample mean averaged over noise-only detectors, and is the temporal sample mean averaged over all detectors Observe that it is difficult to analyze the performance of the GLRT, due to its complex structure and non-gaussian nature of the measurements As an alternative, in Section V, we present numerical examples to investigate its empirical performance The results show that the GLRT can detect a source at relatively high SNR values; however, the computations are intensive In contrast to the GLRT, a novel test with the following properties is desirable: simplicity in computation, tractability in finding the distribution of the test statistic, and having high probability of detection for a given probability of false alarm In the remainder of this paper, we will present a new test that has these properties and analyze its performance B Mean-Difference Test for Cubic Arrays The proposed test is motivated by the fact that since the first three detectors measure signal and noise particles, as the last three detectors measure only noise particles, the source can be detected by comparing the averages of the sample means over these two groups of detectors If the two averages differ by more than a prespecified threshold, we decide that a source is present, and vice versa (6) We analyze the performance of the T in (8) as follows, by deriving first the asymptotic distributions of the under both hypotheses Under : Note that both and have the same Poisson distribution with parameter ; by the central limit theorem (CLT) the sample means and should have the same Gaussian distribution, asymptotically (as approaches infinity) Moreover, due to the mutual independence of all the measurements, and should be statistically independent Then, the asymptotic distribution of under is (9) Under : In this case, has the same asymptotic Gaussian distribution as under However, (,2,3; ) is Poisson distributed with expected rate Therefore, has Poisson distribution with parameter Thus, by the CLT, has asymptotic Gaussian distribution Therefore, the asymptotic distribution of under is (10) Using the distributions in (9), we are able to compute the threshold in (7) that guarantees the preset probability of false alarm Since is the probability of an event and ; then is the inverse function of (11) (12)

4 LIU AND NEHORAI: DETECTION OF PARTICLE SOURCES WITH DIRECTIONAL DETECTOR ARRAYS AND A T 4475 Fig 2 Receiver operating characteristics of the T for a cubic array and different source directions u; SNR =0:75 Using the distribution in (10), we derive the asymptotic probability of detection for the T in (7) as (13) is proportional to the cube s cross section with respect to the source direction Since is a function of, we conclude from (13) that the performance of the T varies with the source direction, ie, the performance of the cubic array is not isotropic with respect to the source direction In fact, as shown in (13), is monotonically increasing with respect to Subject to the constraint, reaches its peak when and is minimized when only one entry of is 1 and the other two are 0 (in this case, only one face of the cube is exposed to the source particles) Fig 2 shows the analytical receiver operating characteristics (ROCs) [14] for different values of As can be seen, the T for the cubic array has the optimal detection sensitivity when the source is at the direction IV DETECTION USING SPHERICAL ARRAYS We examine the detection of particle sources using spherical arrays First, we briefly present multisurface arrays as an introduction A Multisurface Array Detection We extend the analysis of the cubic array to a multisurface array with congruent faces, in each face is a detector For symmetry, assume to be an even positive integer Due to the increased complexity of deriving the GLRT, we consider only the T for this array Similar to the cubic array discussed in Section III, if a source is present, detectors would detect both the source and noise particles, and the other would detect only noise particles For convenience, denote the former detectors as source exposed and the latter as noise only Then, the detection procedure consists of two steps: first identify the source-exposed detectors, and then construct the T accordingly In the first step, we define a half-array as a half of the array, and its measurement as the total particle counts of the contiguous detectors in it In principle, the source-exposed halfarray (in all detectors are source exposed) can be identified by choosing the half-array with the highest temporal sample mean when is sufficiently large Due to the finite length of the temporal measurements, the identified source-exposed halfarray may have a small bias from its true version However, the probability of having this bias will vanish as approaches infinity, since the temporal sample mean is a consistent estimate of the expected rate In the second step, consider the difference between the temporal sample means of the source-exposed and noise-only halfarrays, which is expected to be relatively large under and small under Consequently, we construct the T as (14) When the detectors in a multisurface array become extremely densely distributed, ie, approaches infinity and the size of each detector approaches zero, the array asymptotically becomes spherical We analyze the detection using a spherical array in detail in the following section B Spherical Array Detection We consider a spherical array with all its detectors located on the surface of a sphere, in a large number of small-size detectors completely cover the sphere surface [1] The center of the sphere is chosen as the origin of the reference coordinate system As shown in Fig 3, the sphere surface can be divided into two halves in the presence of a source: the source-exposed hemisphere surface and its complement, the noise-only hemisphere surface Similar to the multisurface array discussed previously, the detection using spherical arrays consists of two steps: searching the source-exposed hemisphere surface and then applying the mean-difference testing 1) Searching the Source-Exposed Hemisphere Surface: Under the previously mentioned assumptions, the measurements for the spherical array are spatial densities of particle counts at all points on the sphere surface, since the area of each detector approaches zero To clarify further, we redefine the model of the spherical array as follows Let be the radius of the sphere, and and be the azimuth and elevation angles of a point on the sphere surface, respectively Therefore, for a given, the doublet defines an arbitrary point on the sphere surface Consider a small area at a neighborhood of a point on the sphere surface, whose area is (approximately) Let be the temporal measurement of in the th time unit, and

5 4476 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 53, NO 12, DECEMBER 2005 rate of should be the maximum among all hemispheres surfaces on the sphere surface (under ), ie, Fig 3 Spherical array be the temporal sample mean of measurements of during time-unit intervals We then define the spatial density of the temporal sample mean at the point as (15) and define the corresponding density of the expected rate at as (16) is the mathematical expectation Further consider an arbitrary hemisphere surface on the sphere surface Observe that it is uniquely identified by its center point, ie, the symmetric center of (on its surface) Denote by the hemisphere surface with center point at Define the measurement of as the sum of measurements of all the detectors in For clarification, let and be the temporal sample mean (stochastic, based on ) and the expected rate (deterministic, based on ) of particle counts of, respectively It is shown in Appendix B that can be expressed as 1 (17) and Obviously, is the only hemisphere surface in all the detectors measure source particles In principle, the expected 1 Without loss of generality, here we assume < 0 A similar expression can be obtained when 0 (18) Note that the center point of lies on the source direction, ie, Furthermore, since the temporal sample mean is a consistent estimate of the expected rate, the source-exposed hemisphere surface should also maximize with high probability as approaches infinity, according to (18) In other words, the identified sourceexposed hemisphere surface by maximizing can be arbitrarily close to the actual one, as approaches infinity Also note that the procedure of finding provides an estimate of the source direction, which is of interest in its own right Observing that is an integration of the stochastic variable, in Appendix C we examine the asymptotic distribution of and show that is unbiased and consistent with when is sufficiently large Below, we present the steps of a gradient-based iterative algorithm to find (hence also the source direction) Other details of this algorithm are given in Appendix D Step 1) Let Denote the point with the maximum as Then, choose the initial value as and Set a small positive quantity as the threshold of convergence Check if If so, stop the iteration; if not, go to Step 2) Step 2) In the th iteration, we do alternate searching on and, respectively First, fixing, find a scalar such that Second, fixing, find a scalar such that Here we employ the Fibonacci method 2 in deciding and Then, let and Step 3) In the th iteration, let and Step 4) If, stop the iteration and claim is the desired point ; otherwise, let and go back to Step 2) 2

6 LIU AND NEHORAI: DETECTION OF PARTICLE SOURCES WITH DIRECTIONAL DETECTOR ARRAYS AND A T 4477 The definition of is given in (D1), and gradients and are expressed in (D2) and (D5) We denote the searching result of the sourceexposed hemisphere surface as, an estimate of, is the corresponding estimate of the source direction Let be the complement of 2) T for Spherical Arrays: We construct the T for the spherical array as then Thus, the asymptotic probability of detection is (25) (19) and are the temporal sample means of particle counts of and, respectively In the following part of this section, we analyze the performance of the T for the spherical array in two cases, as follows a) Performance analysis of the test for known source direction: The performance of the T in this ideal case can be used as an upper bound of its performance in general cases Let and be the temporal sample means of particle counts of and Since in this case and, the T in (19) becomes (20) Under, since only noise particles are measured, within a time-unit interval, the measurements of and obey the same Poisson distribution with parameter Then asymptotically, by the CLT it follows (21) Note that and are statistically independent Hence, under, the asymptotic distribution of in this case is (22) Under, since measures only noise particles, the asymptotic distribution of remains the same as in (21) On the other hand, the cross section of with respect to the source direction is Taking into account the noise particles, within a time-unit interval, the measurement of obeys a Poisson distribution with parameter Thus, by the CLT, has the asymptotic Gaussian distribution In addition, and are statistically independent Therefore, under, the asymptotic distribution of in this case is (23) From (22), it follows that (26) Observe first from (22) that the asymptotic distribution of the statistic under has as the only unknown parameter, which makes it difficult to compute in (25) Hence, if we replace in (22) with its maximum likelihood estimator (MLE), this distribution will not depend on any unknown parameters Moreover, we can expect that the performance of the T after this replacement would not differ much from that in (24) when is large, since is a consistent estimate of Then, the threshold required to maintain a constant can be found from (25) with replaced by, ie, the T for the spherical array is asymptotically CFAR, which illustrates the capability of the T to detect the source in unknown environments Second, observe that as expected, the detection with the spherical array is isotropic since does not depend on the source direction b) Performance analysis of the test for unknown source direction: In this case, we take into account the possible small error in searching for Denote by the error angle between the actual source direction and its estimate We assume that is stochastic and has probability density function (PDF) From the isotropic nature of the spherical array, we further assume that does not depend on or Now consider the conditional distribution of in (19) given Under, since no source is present, for arbitrarily chosen hemisphere surface and its complement, the statistic has the same distribution as in (22) Note that this distribution does not depend on In this case, the estimation error has no effect on the distribution of Under, the cross section of, the estimated, with respect to the source direction is Therefore, the expected count of source particles that impinge on is, as that of the noise particles is According to the Poisson data model, the PDF given is Since (24) particle count of the spherical array

7 4478 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 53, NO 12, DECEMBER 2005 the conditional PDF of given is shown in the equation at the bottom of the page Observe that the conditional distribution of is also Poisson with parameter By the CLT, as approaches infinity, the asymptotic conditional distribution of given is (27) and In the following, we analyze the performance of the T in this case, according to the distributions in (22) and (27) Since depends only on the distribution of under, which is the same in both known and unknown source direction cases, the threshold in (19) and (20) should also be the same, which is given in (25) Then, for a given, we calculate as (28), shown at the bottom of the page Fig 4 Numerical histograms of the log-glr for a cubic array for various SNR values: (a) SNR =0:5; (b) SNR =0:75; (c) SNR =1; and (d) SNR =2 (28) In practice, however, usually it is difficult to know thoroughly the distributions of the estimation errors, which makes it difficult to evaluate in (28) in this case Despite the difficulty in analytically computing, we observe that since the distribution of the statistic remains the same as the known source direction case, the T for the spherical array is also CFAR when the source direction is unknown Moreover, due to the fact that is independent of and,we claim that does not depend on the source direction due to the nonappearance of and in (28), ie, the T for the spherical array also has isotropic detection performance in this case V NUMERICAL EXAMPLES In this section, we first employ Monte Carlo (MC) simulations to examine the experimental performance of the GLRT for the cubic array Then, we compute the analytical performance of the T for both arrays and compare them with the experimental performance In these MC simulations, we conduct a total of experiments; in each single experiment, 100 time-unit measurements for each detector are generated ( 100) In the following numerical examples, we choose and We also let the Fig 5 Numerical receiver operating characteristics of the GLRT for the cubic array for various SNR values cubic and spherical arrays have equal surface area to make their performance comparable A Numerical Examples for Cubic Arrays Consider first the numerical performance of the GLRT for the cubic array We select the threshold in (5) using MC simulations for a given Figs 4 and 5 show the experimental histograms and ROCs of the GLRT, respectively, for various SNR values: 05, 075, 1, and 2 As shown in Fig 4, the GLRT statistic has non-gaussian numerical distributions under both hypotheses, as expected We further observe that the distributions of the GLRT statistic under both hypotheses are too similar to enable good detection when SNR From Fig 5, we

8 LIU AND NEHORAI: DETECTION OF PARTICLE SOURCES WITH DIRECTIONAL DETECTOR ARRAYS AND A T 4479 Fig 6 Numerical and analytical distributions of the mean-difference statistic for the cubic array for various SNR values: (a) SNR =0:5; (b) SNR =0:75; (c) SNR =1; and (d) SNR =2 Fig 8 Probability of miss P of the GLRT and the mean-difference test as functions of the SNR for the cubic array; P =0:05 Fig 7 Receiver operating characteristics of the mean-difference test for the cubic array for various SNR values: analytical and numerical results observe that the GLRT has a relatively poor performance for an SNR smaller than 1 These experimental results illustrate that the performance of the GLRT for the cubic array may not be satisfactory, especially at low SNR values The analytical and numerical performance results of the T for the cubic array are shown in Figs 6 and 7 Fig 6 shows a close match between the numerical and theoretical distributions of, which verifies the asymptotic Gaussian distributions given in (9) and (10) Through the ROC curves at different SNR values shown in Fig 7, we can see that the T for the cubic array performs quite well even at low SNR values such as 05 and 075 By depicting the probability of miss ( ) versus SNR curves for a given, Fig 8 compares the performances of the T and the GLRT, in the difference of the detection performances between both tests can be observed Fig 9 Numerical and analytical distributions of the mean-difference statistic for the spherical array and known source direction, for various SNR values: (a) SNR =0:5; (b) SNR =0:75; (c) SNR =1; and (d) SNR =2 We then conclude from the above results that the T obviously outperforms the GLRT, especially at low SNR values For instance, as SNR and, of the GLRT is about 06, as that of the T is 045 (The measurement data used for the above analysis are the same) B Numerical Examples for Spherical Arrays 1) Known Source Direction Case: Consider the performance of the T for the spherical array in the known source direction case We first generate temporal measurements, and, Then, the temporal sample means are and Construct the T according to (20) Fig 9 shows the analytical and experimental distribution of, in which we can see that the numerical histograms

9 4480 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 53, NO 12, DECEMBER 2005 Fig 10 Receiver operating characteristics of the T for the spherical array and known source direction, for various SNR values: analytical and numerical results match the analytical distributions (given in (22) and (23)) closely, under both hypotheses It can be further observed that at relatively low SNR values such as 075, the distribution of the T statistic under and are separate enough to allow a relatively high As shown in Fig 10, the analytical and numerical ROCs of the T for the spherical array match closely to each other 3 We also observe that increases quickly with and becomes close to one when The high of the T at low SNR values illustrates its applicability to detection in a noisy environment 2) Unknown Source Direction Case: We present the numerical performance results of the T for the spherical array with unknown source direction Here, we approximate a spherical array with a multisurface array with 60 faces [15] Fig 11 presents the numerical histogram of the statistic for the spherical array in this case, as Fig 12 shows the ROCs of the T at different SNR values In Fig 13, we compare the probability of miss of the T between known and unknown source direction cases, for a fixed Aswe can see, compared with the T in the known source direction case, the T with unknown source direction performs similarly well when SNR Also observe that when SNR, it does not perform as well due to the increase in the error of the source direction estimation VI CONCLUDING REMARKS We presented mean-difference tests for detecting particle sources using cubic and spherical arrays and analyzed their performance For the cubic array, we showed that the T outperforms the GLRT by its higher probability of detection for a fixed at low SNR values For the spherical array, as the GLRT may be too complex to derive, we have shown that the T has desired properties such as computational simplicity, 3 The deviation between the numerical and analytical ROC curves as P is less than 01 is due to the quantization error in the numerical analysis This deviation can be reduced by increasing N Fig 11 Numerical histograms of the mean-difference statistic for the spherical array in the unknown source direction case, for various SNR values: (a) SNR =0:5; (b) SNR =0:75; (c) SNR =1; and (d) SNR =2 Fig 12 Numerical receiver operating statistics of the mean-difference test for the spherical array in the unknown source direction case, for various SNR values being asymptotically CFAR, applicability to low SNR cases, and isotropic detection performance As part of the detection, we also estimated the source direction, which is of interest in many cases and should improve the detection performance In the performance analysis of the T for spherical arrays, we compared the analytical with the experimental results and confirmed that they match very well Further research will include analyzing the performance of the proposed source direction estimators, investigating arrays with lenses, and optimally designing arrays by maximizing their detection performance Some practical aspects, such as including the effect of a specific shape of a mounting part, will

10 LIU AND NEHORAI: DETECTION OF PARTICLE SOURCES WITH DIRECTIONAL DETECTOR ARRAYS AND A T 4481 (A4) (A5) From (A5), we have (A6) Substituting (A6) into (A2), we get the maximized likelihood function under as Fig 13 Probability of miss P as a function of SNR, of the mean-difference test for the spherical array with both known and unknown source direction; P =0:05 (A7) Then, the GLRT follows according to (5), (A1), and (A7): be considered in the future The results of this paper can serve as a benchmark of the best performance when this part does not affect the performance GLR (A8) APPENDIX A DERIVATION OF THE GLRT FOR THE CUBIC ARRAY To derive the generalized likelihood-ratio test (GLRT) for cubic arrays, we maximize the likelihood function under both hypotheses As assumed, each unit-time measurement of the th detector is Poisson distributed with parameter, ie, A Under All the measurements are iid The likelihood function is Letting, we get, Then, the maximized likelihood function under becomes B Under (A1) (A2) Taking the derivative of with respect to,,,,, we get Further simplification of (A8) leads to the GLRT-equivalent test for the cubic array given in (6) APPENDIX B DERIVATION OF Here, we derive the expression of in (17) As shown in Fig 14, under the assumption that, there is a point on the surface of which has the largest coordinate, whose azimuth angle is Denote the point as Itis easy to show that For simplification, we project the boundary of onto the plane, and let denote the resulting ellipse, as shown in Fig 15 In a two-dimensional (2-D) polar coordinate system, let points and be the projections of the point and an arbitrary point on onto the plane, respectively Denote by one of the points of intersection between s boundary circle and the plane Obviously, and are the semimajor and semiminor axes of, respectively In addition,, and From the property of the orthogonal projection, we have,, and Let be the coordinate of point in a 2-D polar system, is the distance from the origin and is the azimuth angle It then follows that The two solutions of are (B1) (A3)

11 4482 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 53, NO 12, DECEMBER 2005 or and (B3) Integrating over the region subject to the constraints in (B3), we get the time-averaged measurement of hemisphere surface in (17) APPENDIX C ASYMPTOTIC DISTRIBUTION OF Fig 14 Hemisphere surface H with center point at ( ; ) We derive the asymptotic distribution of to investigate the conditions under which it is consistent with For a small area on the source-exposed hemisphere surface, its measurement in a time-unit interval obeys a Poisson distribution with parameter, is the angle between the source direction and the radius vector pointing to Note that is a function of and for fixed and By the CLT, the temporal sample mean of measurement of during time-unit intervals has the following asymptotic distribution: From (15), the asymptotic distribution of is (C1) Fig 15 and Projection of the boundary of H onto the XY plane Then, the two solutions of are (B2) From (B2), for any point that belongs to the hemisphere surface,wehave and Also observe from Fig 14 that when In summary, a point belongs to the hemisphere surface if and only if it satisfies the following constraints: and And the expected particle counting density at is ; here, belongs to the source-exposed hemisphere surface and thus As can be seen in (C1), is asymptotically unbiased to, while the variance of depends on the product of and Assume, is a positive real quantity, then we find the following a) :, is a constant has a finite variance b) : has an infinite variance, which is not desired c) : The variance of is 0, asymptotically This is the case that we desire From the above discussion, we conclude that if is large enough such that, the density of the temporal sample mean is unbiased and consistent with the density of the expected rate

12 LIU AND NEHORAI: DETECTION OF PARTICLE SOURCES WITH DIRECTIONAL DETECTOR ARRAYS AND A T 4483 APPENDIX D OTHER DETAILS OF THE ITERATIVE ALGORITHM TO FIND In Section IV-B-1), we present the steps of an iterative algorithm to find by maximizing 4 Here, we discuss other details of the algorithm, including initializing the parameters, deriving the gradient of, and how to prevent the algorithm from converging to local extrema First note that is a single peak function and reaches its maximum when From the consistency property, is likely to have a single peak as becomes large Therefore, we can maximize it by searching for the zero of the gradients of Since, as,we choose the initial value of as the point with the maximum value, since is a consistent estimate of when is sufficiently large Assume the temporal sample mean is a sufficiently smooth function of and, such that its first partial derivatives exist almost every on the sphere surface This assumption is valid when is large enough Define,,, and Hence, it follows from (D3) that (D1) Obviously (D2) To derive the partial derivative of with respect to,we recall that (D3) and are differentiable functions of Then, we rewrite (17) into the following form: (D4) 4 This algorithm depends on the assumption of <0, to be consistent with the assumption of < 0 in (17) This will not cause loss of generality since we can first assume <0 and apply the following algorithm, and then assume 0 and apply a similar algorithm Combination of both results provides a thorough search of the source direction (D5)

13 4484 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 53, NO 12, DECEMBER 2005 The notations and indicate that they are functions of and To avoid converging to local extrema (which is the case has multiple peaks), we repeat Steps 1) 4) with multiple initial values of, and then compare the results and find the point with the maximum ACKNOWLEDGMENT The authors would like to thank Dr E Paldi for his help in simplifying (6) as well as helpful comments on presentation REFERENCES [1] A Nehorai and E Paldi, Localization of particle sources with detector arrays, in Proc 27th Asilomar Conf Signals, Systems, Computers, Pacific Grove, CA, Nov 1993, pp [2] J B Bricks, The Theory and Practice of Scintillation Counting Oxford, UK: Pergamon, 1964 [3] E Fenyves and O Haiman, The Physical Pronciples of Nuclear Radiation Measurements New York: Academic, 1969 (German) [4] Semiconduct Detectors, G Bertolini and A Coche, Eds, Elsevier North Holland, Amsterdam, The Netherlands, 1968 [5] K Kleinknecht, Detectors for Particle Radiation Cambridge, UK: University Press, 1986 [6] G F Knoll, Radiation Detection and Measurement, 3rd ed New York: Wiley, 2000 [7] R H Kingston, Detection of Optical and Infrared Radiation Berlin, Germany: Springer-Verlag, 1978 [8] R W Boyd, Radiometry and the Detection of Optical Radiation, 2nd ed New York: Wiley, 1983 [9] B E Saleh and M C Teich, Fundamental of Phontonics New York: Wiley, 1991 [10] A Yariv, Optical Electronics, 4th ed New York: Holt, Rinehart and Winston, 1991 [11] A Nehorai and E Paldi, Vector-sensor array processing for electromagnetic source localization, IEEE Trans Signal Process, vol 42, no 2, pp , Feb 1994 [12] Z Liu and A Nehorai, Detection of particle sources with directional detector arrays, in Proc 3rd IEEE Sensor Array Multichannel Signal Processing Workshop, Sitges, Spain, Jul 2004, p 5 [13] P C Schaich et al, Automatic image analysis for detecting and qualifying gamma-ray sources in coded-aperture images, IEEE Trans Nucl Sci, vol 43, no 4, pp , Aug 1996 [14] S M Kay, Fundamentals of Statistical Signal Processing, Vol II: Detection Theory Englewood Cliffs, NJ: Prentice-Hall, 1998 [15] Matlab 65 Help File: The Bucky Ball [Online] Available: mathworkscom/access/helpdesk/help/techdoc/math/sparse12html Zhi Liu (S 04) received the BSc degree in electrical engineering from the North China University of Technology (NCUT), Beijing, China, in 1996 and the MSc degree in information and control engineering from Tongji University, Shanghai, China, in 2001 He is currently working toward the PhD degree in the Electrical Engineering Department at the University of Illinois at Chicago (UIC) His research interests are in statistical signal processing and its application to sensor-array processing He is also interested in image processing, medical imaging (eg, PET and SPECT), and biological vision Mr Liu has been a member of the Phi Kappa Phi honor society by election of the UIC Chapter since April 2004 He received the Siemens Prize for outstanding graduate study in 2000 Arye Nehorai (S 80 M 83 SM 90 F 94) received the BSc and MSc degrees in electrical engineering from The Technion Israeli Institute of Technology, Haifa, and the PhD degree in electrical engineering from Stanford University, Stanford, CA From 1985 to 1995, he was a Faculty Member with the Department of Electrical Engineering at Yale University In 1995, he joined the Department of Electrical Engineering and Computer Science at the University of Illinois at Chicago (UIC), as a Full Professor From 2000 to 2001, he was Chair of the department s Electrical and Computer Engineering (ECE) Division, which is now a new department In 2001, he was named University Scholar of the University of Illinois Dr Nehorai has been a Fellow of the Royal Statistical Society since 1996 He is Vice President (Publications) of the IEEE Signal Processing Society (SPS) He is also Chair of the Publications Board and a Member of the Board of Governors and the Executive Committee of this Society He was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 2000 to 2002 and is the Founding Editor of the special columns on Leadership Reflections in the IEEE SIGNAL PROCESSING MAGAZINE He was co-recipient of the IEEE SPS 1989 Senior Award for Best Paper with P Stoica, as well as coauthor of the 2003 Young Author Best Paper Award and of the 2004 Magazine Paper Award with A Dogandzic He was elected Distinguished Lecturer of the IEEE SPS for the term 2004 to 2005