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Integrated Real-Time Estimation of Clutter Density for Tracking


IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 10, OCTOBER 2000

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Integrated Real-Time Estimation of Clutter Density for Tracking
X. Rong Li, Senior Member, IEEE, and Ning Li
Abstract—The spatial density of false measurements is known as clutter density in signal and data processing of targets. It is unknown in practice and its knowledge has a significant impact on the effective processing of target information. This paper presents in the first time a number of theoretically solid estimators for clutter density based on conditional mean, maximum likelihood, and method of moments, respectively. They are computationally highly efficient and require no knowledge of the probability distribution of the clutter density. They can be readily incorporated into a variety of trackers for performance improvement. Simulation verification of the superiority of the proposed estimators to the previously used heuristic ones is also provided. Index Terms—Bayesian estimation, clutter density, maximum likelihood, method of moments, target tracking.

I. INTRODUCTION ALSE measurements are referred to as clutter in radar/sonar signal and data processing for target detection, tracking and recognition, etc. There are at least three things associated with the term “clutter density”: 1) its spatial density, which quantifies how “dense” the clutter measurements are; 2) its spatial distribution, which describes how the clutter measurements are distributed in space; 3) distribution of the signal amplitude/intensity of the clutter measurements. This paper deals only with the spatial density of the clutter measurements mainly for target tracking. There are two concepts of clutter spatial density: a physical one and a mathematical one. The physical one is the density of the false measurements received by the observation system for signal and data processing. It is in general not constant over time or space. The mathematical one is the parameter of the Poisson model in Assumption A2 below. Clutter density is an important scenario parameter. Its knowledge has significant effect on the performance of a target tracking algorithm. In practice, it is usually not known or at most known partially. The use of a more accurate knowledge of clutter density in a tracker will improve its performance significantly.

F

For example, in the original (parametric) version of the probabilistic data association filter (PDAF) [2], [1], which is one of the most popular and cost-effective algorithms for target tracking in the presence of clutter, the clutter density is assumed to be known perfectly, an assumption not true in most practical situations. Since this density cannot be obtained easily and accurately in practice, shortly after the development of the parametric PDAF, a nonparametric version of the PDAF was introduced as an option [1] based on the use of a noninformative diffuse prior, which is equivalent to assuming a complete ignorance of this clutter density. One consequence of this assumption is that it implies the use of the following estimator of the clutter density: , where is the number of the total measurements in the gate at time generated by the target and clutter and is the volume of the gate. This estimator does not make good sense: The clutter density should be approximately the number of false measurements, rather than the total number, over the gate volume. The following would make much better sense, for , , where is the probability of is the gate probability, i.e., the probatarget detection and bility that the target-originated measurement falls inside the gate, assuming the target is detected. In the Integrated PDAF (IPDAF) [6], the following more appealing formula for the clutter density was used without derivation, explanation, or justification: (1) denotes the sequence of the (sets of) measurements where and is the (predicted) “track through time quality;” that is, the probability that the target exists at time given . This estimator avoids the two extreme assumptions made in the two versions of the PDAF. It is, however, only a heuristic one without a solid theoretical foundation. More such heuristic estimators may be proposed without a theoretically sound justification. For example, the following estimator (2) is needed in the eswould be more appealing than (1) since in the contimator of (1) anyway and the extra information ditioning should improve the estimation accuracy, even though the clutter density assumed in the Poisson model is a purely predicted value. In short, inspired by (1) from [6], it would be much better if estimators with a solid theoretical foundation could be used instead. There is a strong hope that such estimators will improve the performance of a tracker.

Manuscript received March 11, 1999; revised April 19, 2000. This work was supported in part by the NSF under Grant ECS-9734285, the Office of Naval Research under Grant N00014-00-1-0067, and the LEQSF under Grant (1996-99)-RD-A-32. The associate editor coordinating the review of this paper and approving it for publication was Prof. A. M. Zoubir. The authors are with the Department of Electrical Engineering, University of New Orleans, New Orleans, LA 70148 USA (e-mail: xli@uno.edu). Publisher Item Identifier S 1053-587X(00)06685-X.

1053–587X/00$10.00 ? 2000 IEEE

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If the clutter density is either time-variant or space-variant, its estimation is inherently difficult for the following reasons: A good model for the temporal evolution or spatial distribution of is not available although sometimes a clutter map is available for spatial distribution of clutter, which is however not an accurate model. Consequently, it is not clear how the observations at other times or from other regions can be used to estimate at a particular time over a particular region since most estimation methods for random processes assume the availability of such a model. On the other hand, parameter estimation techniques cannot be directly applied either, because they assume the invariance of the quantity to be estimated. More specifically, a batch estimator in general does not provide benefit over a recursive estimator because data collected at different times or regions may correspond to different clutter densities. A recursive estimator seems more appropriate, but it is at least not clear how the recursion can be obtained without a model for the evolution in time and/or distribution in space. Another difficulty in clutter density estimation with a Bayesian approach is that the required probability distribution of the clutter density, if deemed random, is not available in the literature or in practice, although a variety of models are available for the probability distribution of the intensity of clutter (i.e., the signal strength of the false measurements), such as the Weibull, log-normal and distribution models. As a result, if a Bayesian estimator is desired, the estimator should be distribution free. To the authors’ knowledge, there is no prior work on clutter density estimation from the tracking (data processing) viewpoint, although several heuristic techniques have been used in radar/sonar signal processing. Most trackers assume that the clutter density is available. In practice, however, it is often the case that clutter density is either not estimated or estimated rather crudely by the signal processor and handed over to the trackers. This paper, as a refinement and extension of [4], presents two classes of theoretically solid estimators of the clutter density. They are based on the Bayesian (conditional mean) estimation (Section III) and the maximum likelihood method (Section IV). The method of moments estimators are also presented (Section V). Section II presents the common assumptions used and relevant results on target perceivability. Some common issues of the proposed clutter estimators are discussed in Section VI. Simulation results are given in Section VII to demonstrate the validity and superiority of the proposed estimators. Mathematical details are provided in the Appendix. Further results on this topic are reported in a forthcoming paper, including least-squares and maximum likelihood estimators of the clutter density in a more effective framework of using the number of the validated measurements alone (i.e., without the locations) and the relaxation of some restrictive assumption for the conditional mean estimator. II. ASSUMPTIONS AND RELEVANT RESULTS A true observation is one originated from a target and every other observation, be it from clutter or false alarm, is referred to as a false one. Denote by a (true or false) observation that may include feature (e.g., amplitude or identity) as well as kinematic information of the target. Let

be the sets of validated observations (i.e., those in the validais the total tion gate), where is the th observation, and number of (true and false) observations at time . Denote by the sequence of the sets of observations through time (or the -algebra generated). A. Common Assumptions For clarity, the common assumptions used repeatedly are presented below. A1) The number of validated true measurement per target, , is a binary (0 or 1) random variable at denoted by any given time with the following conditional probability mass functions (pmf)

where is the Kronecker delta function, defined by

undefined and

otherwise

For convenience, we will use to denote either , , or . These three pmfs imply that at any time there is either one or no validated true measurement. The probability of having one validated true measurement depends on the data sets used: given past data, past data and the current number of validated measurements, and past and current data, respectively, there is exactly one validated true measurement with prob, , and , respectively. It is also asabilities is an independent sequence, where sumed that for , or . In addition, the number of measurements originated from a target is also assumed independent of any other target. A2) The number of validated false measurements in any region of the gate at any given time can be described by a suitable Poisson model with a spatial density . of For example, the probability of the total number is given false measurements in the gate of a volume by (3) is an independent seIt is also assumed that quence, where . Also, only random clutter is considered in this paper (i.e., constant clutter is assumed to have been removed). This assumption is reasonable since the validation gate is usually large enough to assure the validity of Poisson approx-

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imation of the more accurate but less convenient binomial distribution and small enough to assume a constant density in space within the gate. The topic of this paper is estimation of this clutter density . A3) The detection of the target is independent of false deand are tections. As a result, the sequences independent. These assumptions are all more or less “standard” in target tracking. Note that

In other words, all information concerning probabilities of , the number of true/false measurements contained in , and , respectively, is summarized by , and , respectively. B. Expected Number of Target Measurement Various expected numbers of validated target-originated measurement per target can be found [3], [7] to be: using Assumption A1 (4) (5)

ments. There are three expected numbers of false measurements. Denote by the random number of false measurements (in the gate) at time . One is conditioned on only previous data . Since is also a prior quantity (see . The other two are Assumption A2), we have conditioned on the current observations as well as previous data: (i.e., and ) and the other on one conditioned on and ; that is, and . Note that is a more accurate estimator than because the extra kinematic (and feaof that is not in provides ture) information contained in , although it is generally true certain extra information about contained in is also in . that most information about Theorem 1 (Conditional Mean Estimator of Clutter Density): With Assumptions A1–A3, the conditional mean estimator of the clutter density is given by, for (8) (9) Proof: It follows directly from the fact and (5)–(6): Since , we have

(6)

where (7)

and the last equations in (5)–(6) are shown in [3], [7] via a long is the event that the target is perderivation. In the above, ceivable at time . A target is perceivable if it exists and can be detected by the sensors used. It is not perceivable if it either does not exist or cannot be detected by the sensors used. See is a better measure of the ex[3], [7] for details. Note that . pected number of target-originated measurement than is an even better measure. However, and are functions at time while depends only on the of the clutter density clutter densities at previous times. III. CONDITIONAL MEAN ESTIMATION When a Bayesian viewpoint is taken, it is important to note are quantities that the Poisson model (3) and its parameter at time . This is clear from, for “prior” to the observations example, the derivation of the PDAF (see [1, Sec. 3.4.10]). The key to clutter density estimation using conditional mean is to obtain the conditional expected number of false measure-

and (8) thus follows. follows from , which is random in the given, Bayesian viewpoint. Were would be equal to . The proof of (9) is parallel. Remark 1: This conditional mean estimator is distribution free: No matter what distribution the random clutter density is (assumed), the estimator is given as above. Remark 2: Since by Assumption A2 the number of false at measurements obeys a Poisson model with spatial density is, there is a nonzero time , no matter how large (or small) unless probability that (10) is dropped. Thus, it is clear that the where conditioning on does not imply . Unfortunately, the above event conditional mean estimator is not a good one in the degenerated . In this case, it gives the problematic estimate case of . This follows from

where use has been made of the fact that does imply . This result is understandable. Consequently, the con. ditional mean estimate should be avoided in the case of Instead, some other estimators, such as the maximum likelihood

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or least squares estimators, can be used, as presented later and in a forthcoming paper. and depend on the true clutter Remark 3: through and , respectively. See Section VI for density a discussion on how to handle this dependence. Remark 4: Note that because is not known given . is the conditional mean, it is unbiased Remark 5: Since and thus its mean-square error is equal to its variance. Theoretis replaced ically speaking, if the unknown clutter density , the associated error variance should be with its estimate accounted for in the tracker to further improve the performance. This variance is not given here because it appears quite complicated to account for this extra uncertainty and it is believed that the effect of this extra uncertainty on the performance of a tracker is small if not minimal. Remark 6: This estimator can be extended to the multipletarget case in a straightforward way. See Section V for details. Remark 7: The above conditional estimates of the clutter and , which density are always positive if makes sense. is related to somehow. Due to the Clearly, lack of a good model for this relation, it may be acceptable to assume the following model for the prediction of the clutter den, or equivalently sity

Note that there is a subtle but important difference between and . It is currently under investigation to replace this assumption with a more reasonable one, such as one based on a Markov assumption. Once the prediction model is assumed, the key to clutter density estimation is the update step; that is, from to . This can be and done using the result of Theorem 1 by replacing with and , respectively. A potential weakness of this recursive estimator is that it requires proper initialization (i.e., to set a proper initial clutter density ). In practice, may be available from the estimator signal processor that provides data for tracking. Otherwise, a batch estimator may be used, such as the one based on the maximum likelihood or least squares estimation, proposed later and in a forthcoming paper. IV. MAXIMUM LIKELIHOOD ESTIMATION In the maximum likelihood estimation (MLE), the clutter density at each time is assumed unknown but nonrandom. For convenience, the Bayesian notation of putting on the right instead side of the conditioning line is used, such as , although it is not the of the non-Bayesian notation authors’ intention to advocate the Bayesian viewpoint here. One significant advantage of the maximum likelihood estimation over the conditional mean estimation is that it does not need initialization. For clutter density estimation, it can be used

either as the primary estimator or to provide the initial estimate required by another estimator whenever initialization or reinitialization is deemed appropriate. Two different classes of maximum likelihood estimators can be developed for clutter density. The first one maximizes the marginal likelihood and the second one maxi, where mizes the joint likelihood is the vector of clutter densities through time . More specif, ically, we have , where and denotes the estimated clutter density at time based on through time . the joint likelihood Clearly, it is also possible to develop maximum likelihood estimators that maximize the marginal or joint likelihood of the clutter density given only the number of observations without other information of the observations: , , , where . Theoretically, the joint maximum likelihood estimator is superior to the marginal maximum likelihood estimator . However, except in some very special cases, such as when is independent, the joint the observation-set sequence maximum likelihood estimator is not appropriate for real-time applications because it is a batch estimator unless a recursion is found: The estimates of the clutter densities at all time instants need to be recalculated from scratch once a new observation is available. That is the reason for the notation , as opposed to the simpler one , which is reserved for the marginal MLE. In addition, to carry out the joint maximum likelihood estimation, it is usually necessary to make additional assumptions on the and . Nevertheless, it is still relationship between useful in some special cases, as is clear later. and will be presented in this paper beNo results of cause they are theoretically inferior to and not significantly simpler than the ones presented. In view of the above, the primary maximum likelihood estimators of this paper belong to the class based on the marginal likelihood. They are presented in the following theorem. Theorem 2 (Marginal MLE of Clutter Density): With Assumptions A1–A3, the marginal maximum likelihood estimaare, for tors of

(11) (12) and are given by (7). where Remark 1: There is a singular case for these two estimators: and (or ), then (or ). If is This is another limitation of the estimators: Although a strong indication that the only validated measurement is from the target and thus there is no false measurement, it is not a guarantee. Furthermore, the fact that there is no false measurement at a particular time does not exclude the possibility of a nonzero

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clutter density. We explain below why this is the case, dropping

sense: The indication of a possible positive in this case comes . If , the marginal MLE only from and thus the joint MLE since the fact that forces . In the , the fact that case of forces because is small enough. V. METHOD OF MOMENTS The method of moments (MOM) is actually a heuristic approach since it does not optimize any optimality criterion and in general does not possess any optimality property. Nevertheless, it is one of the most popular estimation methods because of its simplicity. The most natural and simplest clutter density estimation by method of moments is to use the sample mean of the number of false measurements. A. Single-Target Case For a single-target case, the sample of the number of false measurements always has a size 1 (i.e., only one realization/observation of the number is available) for any given time if the clutter density is not invariant with respect to the gate (i.e., if it is not a constant for different gate either because itself is time-varying or because gate varies). In other words, the sample mean is simply equal to the sample itself (since it has a size of 1). Thus, we have

Thus, the likelihood is a product of a and a straight line with decreasing exponential function of , whose peak for is a positive slope passing through if or at 0 if located either at . , the above estimators Remark 2: In the case of and both lead to since by (10)

As explained before, such an estimate is problematic. Unfortunately, this is an inherent limitation of the marginal MLE because it relies entirely on the information contained in the current observation. In other words, the marginal MLE ignores the information prior to the current time , which is the only hope to prevent the clutter density estimate from being zero. In this case, it is much more appropriate to use joint maximum likelihood estimation. This is presented in Theorem 3 below for the more general case in which there is no validated measurements through time . from time Theorem 3 (Joint MLE of Invariant Clutter Density): Assume that but . With Assumptions A1–A3, and the assumption , the joint maximum that likelihood estimators of is given by

where It leads to

has been replaced by its sample mean

.

This estimator is identical to (1) as used but not justified in [6]. B. Multiple-Target Case tracks with In the multiple-target case, suppose there are nondisjoint gates associated with targets. The th track has an and a gate expected number of target measurement of volume at time ; is the number of the measureand ments in its gate at time . Let be the sets of validation gates and the expected numbers of target-originated measurements, respectively. If it can be reasonably assumed that , that is, the clutter density over the gates of these tracks are the same (e.g., in the case where these gates have a large overlap), then the joint MOM estimator is based on the following relation, with Assumption A2

where . Remark 1: Theorem 2 is a special case of Theorem 3 with . , it is always the case that either Remark 2: Since or , where is the marginal MLE using at time . This is intuitively appealing: only the observation beThe clutter density estimate should be lowered from cause there is no validated measurements (and thus no false meathrough time . Likewise for . surements) from time Remark 3: The assumption seems reasonable since and there is no information to indicate the opposite. Remark 4: Unfortunately, there is still a singular case for this if and only if and ; joint MLE: if and only if and . This makes

where , and stands for the volume of the union of all gates and are the number of total measurements, total false measurements and total target-originated measurements in all gates.

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Under Assumption A1, the total number of target-originated measurements is the sum of independent binary random , where subscript variables with parameters denotes quantities pertaining to the th target. The distribution is in general complicated unless ’s are equal to each of . The other, in which case it is binomial with parameter expected value can, however, be obtained easily: . The associated variance by Assumption A1 is . var Consequently, the joint MOM estimator of the clutter density is given by

A practical solution is to start with and to obtain or and then use or . An iteration could be used to in if desirable. In principle, the superiority of performance is probably not significant because the extra inforcontains in that is not in is usually not mation about requires more computation. much. In addition, depends on through , , , which are the probabilities of target perceivand ability, a fundamental concept for target tracking (see [3], [7] for more details). These probabilities can be calculated recursively by [3], [5], [7]

Consider next the marginal MOM estimation of the clutter density. Note that even if the elements of the sample are correlated, the formula for the sample mean is valid, although the variance of the sample mean differs from the true variance of the population divided by the sample size because of the correlation. Thus

where

where subscript denotes quantities pertaining to the th target, which leads to the estimator

It is clear that the marginal MOM estimator is never larger than . the joint estimator since , like other clutter density Note that in the case where estimators, the above MOM estimators are also problematic in that they will lead to negative estimates and thus has to be set to zero. In such a case, it is recommended that the ML estimator of Theorem 3 be used assuming the clutter density is invariant over the period of several scans. VI. DISCUSSIONS If an estimator presented in this paper is a function of , it , and ; that is, can has three versions be replaced with , or . As the expected number of target-originated measurements, is the most accurate and is the least accurate. Thus is in general most accuand , however, require a knowledge of while rate. requires only to know . Note that to be more rigorous, of (8) should be replaced by the solution of

In the above, are the transition probabilities of the assumed is the probability Markov chain for target perceivability ; is the of target detection given that the target is perceivable; gate probability, that is, the probability of the true measurement is falling inside the gate given that the target is detected; is a scalar the number of total validated measurements; and sufficient statistic for perceivability that summarizes all inforat time (see [3], [5] mation contained in the observations and [7] for details). Note that the last two perceivability probabilities above at at . On the other hand, time depend on the clutter density it should not be surprised that the clutter density estimate depends on the perceivability. Fortunately, all recursive estimators of the clutter density presented rely on the perceivability probaor , bility only through either which can be calculated from the clutter density estimate at the previous time. Most real-world problems are too intricate to have a precise mathematical formulation with an exact and tractable solution. A simplification similar to the above parameterization of clutter is inevitable for density estimators by the probabilities most practical problems. All estimators based on the current observation only are . doomed to be problematic in the case Clutter density estimation is actually a constrained optimization problem where the clutter density is subject to a nonnegativity constraint. Since is a scalar, this constrained optimization problem can be solved as if it were an unconstrained problem and then the nonnegativity constraint was imposed. If the solution does not satisfy the constraint, then the boundary is checked to see if it is a solution to the constrained point problem.

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VII. SIMULATION VERIFICATION To test the proposed estimators for clutter density, the scenario as used in [6] was simulated. The area under surveillance was 1000 m long and 400 m wide. The number of false measurements satisfied a Poisson distribution with some spatial density . A single target was moving in the area at a constant velocity perturbed by a zero-mean noise which accounts for small target maneuvers. The target motion was modeled in Cartesian coor, where is the target state vector dinates as and consists of position and velocity in each of the two coordi, the transition matrix is nates

with sample period . The plant noise is zero-mean white Gaussian noise with the known covariance , where is the Kronecker delta function and

The sensor introduced independent errors in and coordinates m. with root mean square value Since our task was to evaluate clutter density estimators rather than track formation schemes, the track was initiated based on the first three true measurements in the first three scans without any false measurements as follows. The first two true measurements were used to form a preliminary track by the two-pointdifferencing technique. The third true measurement was used to update the preliminary track to an initial track. False measurements were introduced from the fourth scan on. Measurements at the fourth scan were classified into two groups, one for updating tracks and the other as new initiators. A measurement was treated as an initiator if and only if it was not used for updating tracks. From the fifth scan on, measurements were classified into three groups with the following priority: for updating tracks, for forming new preliminary tracks if it fell in the initial square gate of an initiator at the previous scan, and as initiators. Each measurement belongs to one and only one group. Newly initialized tracks might be formed successively at each scan after the fourth. If there were new initial tracks formed from preliminary tracks, the predicted probability . of target perceivability was set to Fig. 1 gives the evaluation and comparison of the following six estimators using 100 Monte Carlo runs, each with 21 scans: 1) heuristic estimator (denoted as Musicki), which uses (1), as proposed in [6]; 2) heuristic estimator (Huris1) of (2); 3) heuristic estimator (Huris2), given by (13) 4) conditional mean estimator (MMSE1) of (8); 5) conditional mean estimator (MMSE2) of (9); 6) maximum likelihood estimator (MLE) of (11). In each run, the target would reappear with m/s .
Fig. 1. (a) True clutter density  (c) True clutter density  = 1:5

f

= 0. (b) True clutter density  = 10 2 10 . (d) True clutter density 

=

.

Clearly, the estimators proposed in this paper outperform the heuristic ones (Musicki, Huris1 and Huris2) substantially in all the four cases. Note that the six estimators considered are all computationally efficient and have virtually the same computational complexity.

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VIII. CONCLUSIONS A number of theoretically solid estimators of the spatial density of false measurements, known as clutter density, have been presented. They include the conditional mean, maximum likelihood, and method of moments estimators. This is the first effort known to the authors for clutter density estimation in the tracking community that does not rely on heuristics. These estimators have some nice features: They are distribution free in the sense that they are valid no matter what probability distribution the clutter density is (assumed); they are computational efficient because they use only quantities already obtained in a tracking filter, such as the IPDAF. These estimators are complementary to each other. They are good in different situations. Some of these estimators can be applied to multiple target case as well as single target case. It is verified through computer simulation of the integrated PDAF in a generic tracking example that estimators have substantially better performance than the heuristic ones used before. APPENDIX A. Proof of Theorem 2 Based on Assumptions A1–A3, it is shown in Proposition 3 of [3] via a long derivation that the likelihood is, for

This shows that MLE is really the maximum point of the likelihood function (14). Similarly, the marginal MLE of the pdf

gives

B. Proof of Theorem 3 Based on Assumptions A1–A3, when , , but ; the observations that can . By chain rule, defining be made use of is

. . . one has the joint likelihood function (14) is the pdfs of the amplitude of valwhere idated false measurements. The marginal MLE is the estimator that maximizes

where

By setting lihood equation

, we have the like. . .

Since

, we have the solution of the above equation and, similarly to (14)

By taking second derivative, we further have, by evaluating the logarithm at

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Thus the joint likelihood is

[3] N. Li and X. R. Li, “Target perceivability and its applications,” IEEE Trans. Signal Processing, 2001, to be published. [4] X. R. Li and N. Li, “Integrated real-time estimation of clutter density for tracking,” in Proc. 1998 SPIE Conf. on Signal and Data Processing of Small Targets, vol. 3373, Orlando, FL, Apr. 1998. , “Intelligent PDAF: Refinement of IPDAF for tracking in clutter,” [5] in Proc. 29th Southeast. Symp. System Theory, Cookeville, TN, Mar. 1997, pp. 133–137. [6] D. Musicki, R. Evans, and S. Stankovic, “Integrated probabilistic data association,” IEEE Trans. Automat. Contr., vol. 39, pp. 1237–1241, June 1994. [7] N. Li and X. R. Li, “Target perceivability and its applications,” in Proc. 1st Int. Conf. Information Fusion 1998, Las Vegas, NV, July 1998, pp. 174–181.

(15) where tion of Theorem 2, by setting . Similar to the deriva-

the MLE is

The second derivative is given by

X. Rong Li (S’90–M’92–SM’95) received the B.S. and M.S. degrees from Zhejiang University, China, in 1982 and 1984, respectively, and the M.S. and Ph.D. degrees from the University of Connecticut, Storrs, in 1990 and 1992, respectively. He joined the University of New Orleans, New Orleans, LA, in 1994. From 1986 to 1987, he did research on electric power at the University of Calgary, Calgary, AB, Canada. From 1992 to 1994, he was an Assistant Professor at the University of Hartford, Hartford, CT. His current research interests include signal and data processing, information fusion and target tracking, statistical inference, stochastic systems, and electric power. He has published three books: Multitarget-Multisensor Tracking (Storrs, CT: YBS, 1995), Estimation and Tracking (Norwood, MA: Artech House, 1993), and Probability, Random Signals, and Statistics (Boca Raton, FL: CRC, 1999); four book chapters; more than 30 articles in leading journals; and more than 80 conference papers. He has consulted for several companies Dr. Li has served as an Editor for IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS since 1995. He received a Career award and an RIA award from National Science Foundation; a 1996 Early Career Award for excellence in Research from the University of New Orleans; served as Steering Chair, General Vice-Chair, and Steering Chair for the 1998, 1999, and 2000 International Conferences on Information Fusion, respectively; given numerous seminars and short courses in the United States, Europe and Asia; and won several outstanding paper awards. He is listed in Marquis’ Who’s Who in the World, Who’s Who in America and Who’s Who in Science and Engineering.

which shows that MLE likelihood (15).

is really the maximum point of the

REFERENCES
[1] Y. Bar-Shalom and X. R. Li, Multitarget-Multisensor Tracking: Principles and Techniques. Storrs, CT: YBS, 1995. [2] Y. Bar-Shalom and E. Tse, “Tracking in a cluttered environment with probabilistic data association,” Automatica, vol. 11, pp. 451–460, 1975.

Ning Li received the B.S. and M.S. degrees from Northwestern Polytechnical University (NPU), China, in 1983 and 1988, respectively, both in applied mathematics. He is currently pursuing the Ph.D. degree in the Department of Electrical Engineering, University of New Orleans, New Orleans, LA. He is a Statistical Consultant with Trilogy Consulting Corporation. From 1983 to 1995, he was with the NPU faculty as an Assistant Lecturer, Lecturer, and Associate Professor. He has published more than ten journal papers. His research interest was on electromagnetic scattering theory and array antennas. Since August 1995, he has been engaged in statistical estimation/decision/computation study and research for target tracking methods. Mr. Li received the first- and second-class prizes of Science and Technology Progress Awards of Shaanxi Province, China, in 1992 and 1993.


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