Target perceivability and its applications - Signal ... · Target Perceivability and Its...

2588 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 11, NOVEMBER 2001

Target Perceivability and Its ApplicationsNing Li and X. Rong Li, Senior Member, IEEE

Abstract—A concept of target perceivability is introduced,which is related to such concepts as target existence and targetobservability. Its probability provides a basis for an integratedapproach to track initiation, confirmation, termination, andrefinement of track maintenance algorithms. This paper proposesthe concepts of target perceivability and presents a recursion ofits probability based on hidden Markov models (HMMs) andtheir applications to tracker analysis, development, and design,in particular, in the context of the PDA method. Specifically,several important quantities and track life are analyzed; a per-ceivability-based probabilistic approach to track confirmationand termination is proposed; two versions of perceivability-basedPDA trackers are presented. Simulation results are provided todemonstrate their performance.

Index Terms—Aircraft detection and tracking, missile detectionand tracking, radar tracking, sonar tracking, target detection andtracking, track decision, tracking, tracking algorithms, trackingfilters, track initiation, track maintenance.

I. INTRODUCTION

A TARGET may or may not physicallyexistin a surveillanceregion at a given time. If it exists, there is still a possibility

that it cannot be detected due to the limitation of the sensorsused. We are interested only in targets that can be detected. Wesay a target isperceivableif it exists and can be detected by thesensors used. A target is not perceivable if it either does not existor cannot be detected by the sensors used. Target perceivabilityis clearly a fundamental concept.

If perceivable, a target may or may not be actually detected atany given time. When detected, the target-originated measure-ment may or may not fall in a validation region, known as “gate,”which is used to reject false measurements. A target-originatedmeasurement inside the gate is called avalidated target-origi-nated measurement.

For a target that is always perceivable in a clutter-free envi-ronment with perfect target detection, the measurement is al-ways available, unique, and from the target alone at each time.Accordingly, tracking follows from the conventional methodsof estimation and filtering [2].

For tracking a target in the presence of clutter (i.e., false mea-surements), the problem is complicated in that a measurement

Manuscript received September 27, 1999; revised July 5, 2001. Thiswork was supported in part by the Office of Naval Research under GrantN00014-00-1-0677 and the National Science Foundation under GrantECS-9734285. The associate editor coordinating the review of this paper andapproving it for publication was Dr. Alex B. Gershman.

N. Li was with the Department of Electrical Engineering, University of NewOrleans, New Orleans, LA 70148 USA. He is now with Trilogy Consulting Cor-poration, Waukegan, IL 60085 USA.

X. R. Li is with the Department of Electrical Engineering, University of NewOrleans, New Orleans, LA 70148 USA (e-mail: [email protected]).

Publisher Item Identifier S 1053-587X(01)09226-1.

may have originated from the target or clutter [3]. The conven-tional filtering theory cannot be applied directly. Tracking inclutter involves track initiation, confirmation, termination, andmaintenance. Track maintenance deals with target state estima-tion, assuming the target is perceivable. Track initiation, ter-mination, and confirmation are decisions concerning whethera track comes from a particular target. Track initiation is con-cerned with whether or not to begin a new track for a target.In contrast, whether or not to abandon a track is the task oftrack termination. Track confirmation is a decision that a trackis deemed from the target.

A distance-based criterion, albeit reasonable and useful fortrack maintenance, is not necessarily suitable for making deci-sions on tracks, even though it is not uncommon to adopt sucha criterion. A typical example is the popular criterion for trackloss. A target is declared lost if the target state estimation error(i.e., the distance between the track and the true trajectory) ex-ceeds a certain threshold. In fact, track maintenance is differentin nature from track initiation, confirmation, and termination.For track maintenance, the accuracy of state estimation is of theprimary concern, whereas in track initiation, confirmation, andtermination, the focus is on making correct decision as quicklyand reliably as possible. In short, the former is an estimationproblem compounded with the measurement origin uncertainty,whereas the latter are basically decision problems.

What is a suitable basis for these decisions? We propose, inthis paper, the probability of target perceivability as such a basisand, on it, an integrated, probabilistic approach to track confir-mation, termination, initiation, and maintenance. It differs sig-nificantly from the existing techniques, which are heuristic orstatistical in nature [3], [5], [6]. Specifically, we introduce theconcept of target perceivability, present a recursion for com-putation of its probability based on a hidden Markov model(HMM), propose to confirm and terminate a track by comparingits perceivability probability with thresholds, present a perceiv-ability-based tracker analysis and design of key tracker param-eters, and develop two PDA trackers as generalization and re-finement of the probabilistic data association (PDA) filter.

The remainder of the paper is organized as follows. InSection II, we introduce the concept of target perceivability,present a recursive form of its probability, and propose aperceivability-based approach to track confirmation andtermination. As a first application, Section III summarizesthe results of design of several key parameters presented in[15]. A perceivability-based tracker analysis is presented inSection IV. Section V summarizes several recently developedclutter-density estimators in which perceivability probabilityis a key parameter. In Section VI, two PDA trackers aredeveloped as generalization and refinement of the PDA filter.Simulation results are provided in Section VII. A summary of

1053–587X/01$10.00 © 2001 IEEE

LI AND LI: TARGET PERCEIVABILITY AND ITS APPLICATIONS 2589

the paper is given in Section VIII. Mathematical details are leftin Appendix.

II. TARGET PERCEIVABILITY AND ITS PROBABILITY

A. Target Perceivability and Related Concepts

A viable way of making a decision in the presence of uncer-tainties is based on probability. Probabilistic decision makingfor track initiation, confirmation, and termination has been pro-posed in the literature. For example, the “likelihood of track ex-istence” was introduced in [25] to determine the “cost” of eachdecision. The concept of a “weak” track was used in [22], al-though the “probability of track existence” for each track wasnot explicitly computed. The “probability of track existence”was included in the track splitting in [24], where the computa-tion is numerically complex and requiresa priori information.

One popular approach to data association is the PDA method[3], [4], which weighs the contribution of each measurement bythe probability that it is target originated. A fundamental limita-tion of the PDA method is the assumption that target is alwaysperceivable, and thus, it is not directly applicable to track initi-ation, confirmation, and termination. Initiation and terminationof a track were incorporated into the PDA filter in [8]–[10] basedon the introduction of a concept of “target observability.” Thisis an important introduction, which pioneers integration of trackdecisions and maintenance in the PDA method.

A related but different approach, which is referred to as theinteracting multiple-model PDA (IMMPDA), was proposed in[1] based on the IMM algorithm, where nonperceivability ismodeled by a model with zero probability of detection. Theprobability that the other model(s) are true was defined as theso-called “true target probability.”

Following a similar path as [10], an attempt was made in [23]to fully integrate the estimation and decision tasks of tracking inclutter in the context of the PDA method based on probability oftrack existence, modeled as a hidden Markov chain, as in [10],but slightly more elegantly.

The concept of target perceivability is a refinement of theabove concepts. For example, track existence of [23] doesnot address the possibility that the target cannot be detected,whereas target observability of [10] presumes the presence of atarget. As defined before, a target is perceivable if it is presentand can be detected by the sensors used. This refined conceptprovides a better foundation for not only track decisions butalso to maintain tracks of a temporarily-obscured target. Theprobability that a target is perceivable was introduced in [19]under the term “tracking probability” and revised to the currentterm1 in [14]. It provides a quantitative measure that a targetis present and can be detected, and thus, the track under con-sideration is likely an estimate of its state trajectory. It is moreintegrated, systematic, and economical than the IMMPDA

1The termtarget existenceis not appropriate here since it does not guaranteethat the target can be detected. The termobservabilityis well established in suchareas as estimation theory, system science, and passive tracking, which is con-cerned withuniquedetermination of state by its observations (and input). Thetermdetectabilityhas a similar interpretation in system science. The wordper-ceivableis chosen partly because “not perceivable” could mean either nonexis-tent (at least to some philosophers) or existence but not sensible. The termtrackexistenceis a misnomer since a track is not a physical entity.

approach, as evidenced by the abundant results presented oroutlined in this paper.

B. Assumptions

A1: Target perceivability as a time sequence can be modeledas a first-order homogeneous Markov chain with known transi-tion probabilities

(1)

where

target is perceivable at time (2)

target is not perceivable at time (3)

This assumption is natural and reasonable since knowing theperceivability of a target at present its perceivabilities in thepast and future are, loosely speaking, independent. However,such a Markov chain is not directly observable (e.g., how toobserve directly?) and is thus known as an HMM. HMMsare widely used models in many areas, including tracking. Forexample, they are enabling models for the second generationof multiple-model algorithms [17], including the well-knownIMM algorithm [2], [7]. Albeit assumed known, the actually un-known transition probabilities can be estimated (see, e.g., [12]and the references therein) but are usually treated in practice asdesign parameters. See Section III for more details.

At any time , the set of validated measurements for a giventrack is denoted as . Let be the se-quence of measurement sets through time. We will refer to

and , respectively,as the predicted and updatedprobabilities of target perceiv-ability for a given track.

Under Assumption A1, the total probability theorem gives

(4)

A2: The number of validated target-originated measurementper target per track, which is denoted by at any given time

, is a zero-one binary random variable with the following con-ditional probability mass function:

(5)where

Kronecker delta function with if ,if , , and undefined otherwise;

target detection probability, assuming target is perceiv-able;probability that the target-originated measurementfalls inside the gate assuming the target is detected.

A3: False detections at different times are independent, andthe detection of the target is independent of false detections(for a given detection threshold). It is also assumed that the se-quences and are independent and have zero mean,where , ,and is the number of validated false measurements. The


kinematic components are independent of feature componentsfor both target and false detections.

A4: The number of validated false measurements at anygiven time can be described by a suitable Poisson modelwith a spatial density , that is, the probability of the totalnumber of false measurements in the gate with a volume

is given by

(6)

Note that under this assumption,.

Assumptions A2–A4 are fairly standard in target tracking.

C. Probability of Target Perceivability

Theorem 2.1:With Assumptions A2–A3, the updated andpredicted target perceivability probabilities for a given track arerelated by

(7)

where

(8)with

(9)

denoting the number of total validated measurements from thetarget and clutter.

Proof: See the Appendix.The non-negative scalarsufficient statisticfor perceivability

probability

(10)summarizes all information contained in the validated mea-surements; is the pdf ofkinematic componentsof the th measurement residual conditioned on the event

that the th measurement is target originated,stands for the event that the target-originated measurement

is not in the gate, is the ratio of the pdf oftarget-originatedfeature component to that of a false mea-surement, and is the gate volume. All these quantities arefor the track in question. Note that , is the thmeasurement containing both kinematic componentsandfeature component (intensity, etc.).

Remarks:

• For the number of total validated measurements, both therandom variable and its realization will be denoted as.The unambiguous notation is used only whennecessary.

• Under Assumption A4, (7) and (8) become

(11)

(12)

• Note that only when . In addition,implies . Then, the range of is . Itis clear from (7), (11), and (12) that

if

if

ifor

if

if

if .

(13)

• In the case of with but not always, perceivability probability remains a useful measure.

D. Perceivability-Based Probabilistic Approach to TrackDecisions

As explained before, a distance-based criterion is, in general,not suitable for making decisions concerning tracks. Rather,target perceivability probability is more suitable. The perceiv-ability probability tells us how likely we are perceiving a targetand estimating its state. In view of this, we naturally propose toconfirm and terminate a track using the following rules:

A track is confirmed if (14)

and a track is terminated if (15)

where and are confirmation and termination thresholds,respectively.

In a cluttered environment, there could be more than one trackper target. There is sometimes a need to identify the best ofall tracks on a real-time basis. The one that is closest to thetrue trajectory of the target cannot be identified because of theneed for the unknown truth. In the context of perceivability-based approach, we propose to use the so-calledmost probableconfirmed track(MPCT). The MPCT is defined as the track withthe highest perceivability probability over a given time periodamong the tracks that have been confirmed at least once. It isthe best track on average in terms of perceivability probability.It can be expected that an accurate evaluation of perceivabilityprobability will lead to an MPCT that is closer to the (unknown)true trajectory of the target. In other words, the more accuratethe evaluation of perceivability probability, the more crediblethe MPCT.

III. PERCEIVABILITY -BASED TRACKER DESIGN

Application of the perceivability probabilistic approach pro-posed above requires knowledge of thresholdsand andtransition probabilities , among other things. Theoretical for-mulas for the design of these parameters have been presentedin [15] for enhancing perceivability probability based on some


common-sense rules. In this section, we describe briefly repre-sentative design formulas presented in [15].

A. Design of Confirmation Threshold and TransitionProbabilities

Rule 1: The confirmation threshold should be chosen toguarantee that a track at time is always confirmed if theperceivability probability and never confirmedif there is no validated measurement atand .

The rationale of this rule is the following. If ,we still want to confirm the track at time, even if there is novalidated measurement, which we believe is due to imperfecttarget detection , finite gate , or some othernonideal conditions. On the other hand, the event that there isno validated measurement atstrongly suggests that the trackis dubious and, thus, should not be confirmed if prior informa-tion is not overwhelming; otherwise, too many false tracks maybe confirmed. In the case where and there is atleast one validated measurement at, confirmation depends onthe relative values of and . The relationship betweenand is given in the next section. Note that this rule is pre-sented here using the limit , although the limitcan never be reached in a cluttered environment. Note also thataccording to this rule, a track without a validated measurementat a given time will not be confirmed for that time, but it maybe confirmed at other times when there is one or more validatedmeasurements.

Design 1 (Relationship Between Confirmation Thresholdand Transition Probabilities):With Rule 1 and AssumptionsA1–A4, the confirmation threshold and transition probabili-ties and should be selected such that

(16)

Simulation results presented in [15] indicate that tracking per-formance is not insensitive to the values of , and thus, as thekey parameter of the HMM for perceivability, it should be care-fully designed. Clearly, it should be close to unity in most situ-ations because it is a rare case that a perceivable target becomesnot perceivable.

B. Design of Termination Threshold and Initial PerceivabilityProbability

Rule 2: The termination threshold should be chosensuch that a track at time is terminated if and only if either a)

and there is no validated measurement at, orb) and the sufficient statistic is small enoughrelative to , where is a design parameter.

This rule makes sense. A decision to terminate a track shouldrely on botha priori anda posteriori information. The eventthat there is no validated measurement atis a strong indicationthat the track is probably not a good one. Therefore, the trackshould be terminated unless thea priori information stronglysuggests the opposite (i.e., ). If the a priori infor-mation is not very indicative (i.e., ), then the termi-nation decision should rely on thea posterioriinformation, thatis, the value of relative to . As shown in the next section,

, and when there is and there is no vali-dated target-originated measurement, respectively, in a clutteredenvironment when feature information is not used. A reasonablechoice is , that is, the track should be maintained if itsperceivability probability exceeds the perceivability probabilityassigned to an initial track (otherwise, it would be initiated rightafter it is terminated). With this choice, we have the followingdesign.

Design 2 (Termination Threshold and Initial PerceivabilityProbability): With Rule 2 and Assumptions A1–A4, the termi-nation threshold and the initial perceivability probabilityshould be selected such that

(17)

For more information about the perceivability-based trackerdesign, see [15].

IV. PERCEIVABILITY -BASED TRACKER ANALYSIS

A. Expected Value of Sufficient Statistic

For the perceivability probability update, all information inthe measurements at time is summarized in the scalar suf-ficient statistic of (10). Thus, a perceivability-based trackeranalysis can be carried out by analyzing.

Note first that the statistic is obtained pertaining to a gate, which is usually given by [3]

where is the predicted measurement (kinematic compo-nent) with covariance .

In general, it can be seen from Theorem 2.1 that the greater, the better. Its expected value is a key parameter for tracker

analysis. Note also that if (and only if) .Theorem 4.1:Under Assumptions A3–A4, the expected

value of the sufficient statistic without feature information is

if

if

if

(18)

where

Proof: See the Appendix.Remarks:

• It follows from (13) and (18) that speaking on average,if , ,

, and thus, . In other words, statis-tically speaking, the proposed track termination schemeguarantees that all false tracks will be terminated eventu-ally in a cluttered environment. More precisely, perceiv-ability probability decreases statistically every time the


track does not have the support from the target-originatedmeasurement (i.e., ), although the decrement maybe small (see Section IV-D).

• When the track has the support from the target-originatedmeasurement (i.e., ), implies

on average, meaning loosely that the per-ceivability probability increases on average.

• When feature information is used, increases if, but it does not change if . See [3] for details for

the benefit of using feature information.• Under the fundamental Gaussian assumption of the PDA

filter [3] , ; with, we have

where is the incomplete gammafunction, is dimension of measurement space, and

.

B. Expected Numbers of Validated True and FalseMeasurements

The expected numbers of validated measurements is a keyquantity that reflects the quality of a tracker. A good trackershould have an expected number of target-originated mea-surements close to detection probability and a small expectednumber of false measurements.

Clearly, implies , and thus, we areinterested only when .

Theorem 4.2:With Assumptions A2 and A3, the expectednumber of validated target-originated measurements with var-ious conditioning is

(19)

(20)

where .Proof: The proof of (19) is given in (47), whereas (20)

follows from

and (30).Corollary: The expected numbers of validated false mea-

surements are, since

(21)

Fig. 1. Improvement threshold for� =� .

C. Track Improvement

A track is improved in terms of perceivability at timeif (and only if) . By (13), the above inequalitybecomes , where

Clearly, whenever , which is almostalways true. This indicates that perceivability probability atincreases only if . In other words,

is not sufficient to guarantee the increasein perceivability probability, although by Remark 2 of Theorem4.1, it guarantees the increasein an averagesense. Recall that

if and only if the track has the supportfrom the target-originated measurement (i.e., ).

Fig. 1 illustrates several typical curves (for ,, ) of this improvement threshold versus

. A track is improved at if and only if is abovethe corresponding curve for a given .

D. Track Confirmation

Assume that a track has been confirmed at time, i.e.,. A sufficient condition for the track to remain to be confirmed

at time ( is an arbitrary positive integer) is. By (12), this is equivalent to for ,

where

The second equation above follows by setting .Note that it is possible that , in which case, the track

will continue to be confirmed at time no matter whether ithas the support from the true measurement or not. This is reason-able because only when is very large or is quitesmall, indicating either a strong prior perceivability probability


or poor detection probability. In any case,is required for the track to remain confirmed. When is largeenough (i.e., ), it is possible for the track to con-tinue to be confirmed at , even if ; other-wise, continual confirmation requires .

E. Lifespan of a False Track

Consider a pure false track that is supported during a periodof several scans by validated false measurements alone (i.e., novalidated true measurement). In this case, , which leadsto and . If a track has been ini-tialized with initial probability , then .Denote by the perceivability probability of the track termi-nated at . Then

or , where is the terminationthreshold. Therefore, the expected lifespan of a pure false trackis

(22)

For , , (these numbers arefrom the optimal design of [15]), the expected lifespan of sucha “pure” false track is 136 scans. It is 82 scans for ,

, and . The lifespan is quite long, butsuch a false track is (almost) the worst situation possible sinceit is always supported by some false measurements. It makessense that the lifespan of a track without true measurements ismuch longer than the life of the track without the support of anyfuture measurement.

F. Lifespan of a Confirmed Track Without Future MeasurementSupport

When a track has been confirmed, say , howmany scans can it survive without support from any future mea-surement? It follows from (4) and (12) that for ,

, we have

We see from Fig. 2 that the confirmed track can only last twoto three scans. Consequently, from a perceivability probabilitypoint of view, a track should be terminated if it does not havesupport from any measurement for three consecutive scans. Thistheoretical result thus provides a justification for the heuristictrack-loss criterion that a target is deemed lost if there is novalidated measurement in three consecutive scans.

Fig. 2. Expected track life without future measurement support (confirmed atk).

V. ONLINE CLUTTER DENSITY ESTIMATION

Several online estimators of clutter density have been devel-oped recently in [20] and [21]. A key parameter needed in theseestimators is perceivability probability, although no knowledgeabout the spatial distribution or temporary evolution of theclutter density is needed. In other words, these estimators,being the first with a solid theoretical foundation, could nothave been developed without the concept and computation ofperceivability probability. In this section, we summarize verybriefly the main results presented in [20] and [21] that are nottoo involved to demonstrate their dependence on perceivabilityprobability.

It follows from (21) that theconditional mean estimators ofclutter densityis given by, for

(23)

where .The clutter density estimator by the method of momentsis,

for

which follows from replacing with its samplemean .

Themaximum likelihood estimators of clutter densityis, for

where and are given above.


It is shown in [20] that theleast square estimators of time-invariant clutter densityis given by

wheregate volume at time ;given by (23);beginning time for the measurement of.

It is clear that the predicted perceivability probabilityis needed in all these estimators of clutter density. For more in-formation about these and some other clutter density estimatorsand their dependence on perceivability probability, see [20] and[21].

VI. TWO REFINED PDA TRACKERS

A. Perceivability-Based PDA Tracker

The well-known PDA filterimplicitly assumes that a target isalways perceivable and is thus valid only for track maintenance.

In the perceivability-based PDA (PB-PDA) tracker, the stateestimate is updated by

where

with

is the residual of theth measurement, is the filter gain,and the probabilistic weights are defined by

(24)

When there is no validated measurement or none of the val-idated measurements are target originated, the PDA filter, likeall other tracking filters using a gate, uses the predicted covari-ance as the error covariance associated with the corre-sponding state estimate (i.e., ). This is, how-ever, incorrect. As shown in [16], the correct covariance for theboth cases is given by2

(25)

2The event that the target-originated measurement is not in the gate carriesthe information that the state prediction actually has an error covariance greaterthanC (i.e., without this event). The incrementb K S K reflects thisfact.

where

(26)

and

where is the incomplete gamma function defined in

Section IV-A. Then the error covariance of isfor , and for

where

is the spread of the means.The weights are identical to those of the well-known

PDA filter (parametric version, incorporating feature informa-tion) [3]

(27)

where , and was definedbefore.

The proposedPDA trackerdiffers from the well-knownPDAfilter (parametric version, incorporating feature information) inthree aspects. First, the unknown clutter densityis replacedwith its estimates that were obtained by an estimator presentedin Section V. This is also superior to the nonparametric versionof the PDF filter, which amounts to assuming total ignorance ofclutter density.3 Second, the incorrect error covariance

is replaced by the correct one and by . Fi-nally, and most importantly, with the built-in perceivability, thetracking algorithm proposed in this paper, which includes therecursion of perceivability probability (Theorem 2.1) and com-parison with corresponding thresholds, provides an integratedtool for track initiation, confirmation, and termination, as wellas track maintenance. For example, a track will be maintainedif and only if , where is the termination threshold;otherwise, no state estimation is carried out. Therefore, we referto what is proposed here as a PDAtracker, as opposed to thecommonly used PDA (tracking)filter, which is valid only fortrack maintenance (i.e., filtering in the presence of measure-ment-origin uncertainty). The state estimation (i.e., track main-tenance) part of PB-PDA tracker will be referred to as PB-PDAfilter, or PB-PDAF for short.

3An even better alternative is to estimate weights� directly from data,which is, however, not a simple job.


B. Existence-Based PDA Tracker

The above PB-PDA tracker provides state estimates onlywhen a perceivable target is deemed present. One may arguethat estimating the state of a target that is present is of interestno matter whether it can be detected or not. In other words, stateestimation of a target that is around but may be nonperceivableover some time periods (e.g., temporarily obscured) may stillbe of interest.4

For this purpose, one may be tempted to define the state es-timate as . In fact, this definition is meaningless. Thiscan be seen well after we decompose it into

(28)

where stands for the event that a target is present. Note thatcannot be defined since the concept of the state

of a target is meaningless in the absence of a target. Thus, thedefinition is meaningless.

Nevertheless, is meaningful even if the targetcannot be detected at time. With the above in mind, we define

. Then

where and are state estimates, assumingthe th measurement is target originated and computed in thesame way as in the PDA filter. The probabilistic weights aredefined similarly by

(29)

and

because the prediction is the best estimate knowing(i.e., the target-originated measurement is either nonexistent oroutside the gate). As a result, the formula for is identical

to that of , but the corresponding quantities are actually dif-

ferent because, e.g., the weights differ from , amongother things.

It should be emphasized that although , itserror covariance is neither nor of (25). In fact,since

4This does not imply that PB-PDAF is not applicable to a temporarily-ob-scured targetin practice.

and noting that the error covariances of andare and , respectively, the

error covariance of is

where

with . As a result, the error covariance of

is, for

and if .Following a similar procedure as the derivation of the proba-

bilistic weights in the well-known PDA filter, it can be derivedthat

(30)

where

and is the predicted perceivability probability.The filter presented above, which is referred to as target-ex-

istence-based PDAF (EB-PDAF), is equivalent to what is pre-sented in [11]. In fact, Dezert and the authors developed thisfilter independently.

To calculate recursively, we assumethat perceivability conditioned by target presencecan be adequately described by an HMM similar to the onefor perceivability alone. Specifically, we assume thattarget perceivability as a time sequence given that targetis present is a first-order homogeneous Markov chain withknown transition probabilities and

. Then, the recursion of is exactlythe same as that of perceivability probability , except that

and are replaced by and , and it is assumed inthe calculation of the sufficient statistic that the target ispresent.

C. Comparison of the Two PDA Trackers

Some simulation results of PB-PDA and EB-PDA algorithmswere given in [11]. We provide a more thorough and complete


comparison between the two versions under the common basisthat a target is perceivable. Note that this comparison cannotbe done if the target is not perceivable, in which case, and

in PB-PDA are meaningless.Since for , we have,

dropping subscript for ,

with equality holds only if (which is the case whenor ), i.e., the target is perceivable

at . Similarly

(31)

Note that implies . Wesee that state estimate puts a heavier weight on the com-

bined residual than does. This indicates thatPB-PDAFplaces more trust in the current measurements and less trustin state prediction than EB-PDAF. This makes sense since theformer presumes target perceivability, which is more than targetpresence, as assumed in the latter.

From (27) and (30), we have, when

A clear relationship thus follows:

In a cluttered environment, since the PB-PDAF places aheavier weight on validated measurements than the EB-PDAF[because ], we can expect that it performs betterthan the EB-PDAF if the target-originated measurement is inthe gate, and worse otherwise. A natural question is “which oneis superior if we only know the total number of validated mea-surements ?” This question can be answered by examiningtheir error covariances.

The difference of their error covariances is, for

This makes sense. As shown in [16], the event that a target isdetected but its measurement is outside the gate implies thatwith has a larger error than . This error increaseis larger in the case when the above event is less likely, whichis indeed the case when the target is perceivable compared withthe case when it is just present and not necessarily perceivable.

For , we have

Note that

where use has been made of sincefor almost all practical situations. We conclude that

when there are one or more validated measurements, PB-PDAFprovides better state estimates than EB-PDAF, which shouldcome as no surprise since PB-PDAF only estimates the stateof a perceivable target, whereas EB-PDAF is for a not-neces-sarily perceivable target. This is the price paid by the EB-PDAFto have a guaranteed ability to track a target that may be tem-porarily obscured. However, this is not to say that PB-PDAFcannot be used to estimate the state of a nonperceivable targetin practice (see Scenario 2 results in Section VII).

These conclusions are based on an analysis of the proba-bilistic weights used in the two trackers. Caution should betaken when they are applied. For example, it is not clear howthe choice of the weights affects track life, which relies on per-ceivability probability , on which the impact of theweights is not clear.

Finally, a discussion with the IMMPDA approach is in order.Notwithstanding the more artificial nature of the IMMPDAF,one may think that the IMMPDA approach [1] is superior toPB-PDA and EB-PDA trackers because it is based on the moreappealing IMM mechanism that includes mixing of the pre-vious estimates. The fact is that while it is appealing for manyother applications, mixing does not necessarily help here be-cause what we are dealing with is whether the target is perceiv-able or not—it is controversial how the estimate of a perceivabletarget can take advantage of the estimate of a nonperceivabletarget. One may still argue that state prediction can be used asthe estimate of a nonperceivable target, but this has been takeninto account in the PDAF already. Nevertheless, in view of thefact that PDAF, as shown in [18], is a GPB1 algorithm with aspecial design of the transition probabilities, the PB-PDAF andEB-PDAF can indeed be improved by replacing the inherentGPB1 mechanism with the corresponding IMM mechanism.Note, however, that what results is not the IMMPDA algorithm.On the other hand, it is clear that the perceivability-based ap-proach proposed in this paper is more systematic, integrated,and economical than the IMMPDA approach.


VII. SIMULATION RESULTS

To compare the performance of PB-PDA and EB-PDAtrackers, Monte Carlo simulations were carried out. The scenarioas used in [23] was chosen. The area under surveillance is 1000m long and 400 m wide. The number of false measurementssatisfies a Poisson distribution with a (possibly time-varying)density that is not known to the tracker. A single target ismoving in the area with a constant velocity perturbed by azero-mean plant noise that accounts for small target maneuvers.The target motion is modeled in Cartesian coordinates as

where is thetargetstatevectorat timeandconsistsofpositionand velocity in each of the two coordinates

with the transition matrix

where is the sample period. The noise and arezero-mean white Gaussian sequences with known covariances

where is the Kronecker delta function, and

The sensor introduced independent errors inand coordinateswith root mean square value m.

The track initiation was done as follows [13]. At the firstscan, every single measurement is treated as aninitiator. At thesecond scan, aninitial (square) gateis formed around each ini-tiator. If a number of measurements at this time fall in the ini-tial gate, the same number ofpreliminary tracksare formed bytwo consecutive measurements. At the third scan, if no measure-ments fall in the gate, no new tracks are initiated, and the initialgate will be terminated. If there are measurements in it, the ini-tial predicted perceivability probability [15] is used,and a new initial track is formed from the corresponding pre-liminary track.

During tracking, new tracks might be formed successivelyat each scan, beginning from the third scan. Measurements areclassified into three groups and in the following order:

1) for updating tracks;2) for forming initial tracks;3) for initiators.

Each measurement can only belong to one group.The first estimator of (23) was used in both trackers. The ex-

periment consisted of 100 runs, each run with 21 scans. In eachrun, the target would reappear with probability at initial state

, 35m/s, and would existin every scan unless stated otherwise, where standsfor a zero-mean Gaussian random variable with variance.

, , , , ,and from (16) and from (17) were used.

A. Scenario 1

In the first scenario, , for andm for , both PB- and EB-PDA trackers produced50 MPCTs. The flops per run were 3.6410 and 3.24 10 ,and the cpu times per run were 15.07 and 13.60 s, respectively.

Fig. 3 shows the comparison results for Scenario 1. Fig. 3(a)and (b) gives the percentages of the runs in which the MPCTsdefined in Section II-D were confirmed at the given time andthe root-mean-square (RMS) values of the position estimationerrors of the MPCTs. Fig. 3(c) gives the average number of con-firmed tracks (not necessarily MPCTs), and Fig. 3(d) gives theaverage number of maintained tracks.

Note first that if and , thenand . Both trackers would have the same be-havior in theory if the unknown truth should have beenused. In reality, they differ slightly because only estimatescan be used. This is reflected in Fig. 3 in the clutter-free period(i.e., ) where both trackers had virtually the same per-formance (confirmation percentages, RMS errors, and averagenumbers of tracks).

As shown in Section VI-C, in a cluttered environment,since the PB-PDAF places a heavier weight on validatedmeasurements than the EB-PDAF [because ], it hasa smaller error covariance (at the price of a larger number ofconfirmed/maintained tracks) than the EB-PDAF when thereare measurements in the gate. This is verified by the curves inFig. 3 (a)–(d) during the cluttered period (i.e., )because one or more measurement was often validated (since

). However, the clutter densitym is not heavy, the performance difference between the twotrackers is not very significant.

It can be seen from Fig. 3(c) and (d) that during the clut-tered period, the PB-PDA tracker had more tracks because ittrusts relatively more in the measurements for each track thanthe EB-PDA tracker. As a result, it is computationally more in-tensive than the EB-PDA tracker.

Fig. 4 shows the average lifespan of terminated tracks. As ex-plained in Section VI-C, the above analysis based on the proba-bilistic weights is not applicable to track life. The tracks duringthe clutter-free period ( ) are all true tracks. The av-erage lifespan of terminated tracks for this period is that ofthe true tracks that were terminated because of the poor luckof missed detections. Their average lifespan was short becausetrue tracks of a high age will have a large perceivability prob-ability and, thus, will not be terminated by occasional misseddetections. The terminated PB-PDAs true track has a slightlylonger lifespan than those of EB-PDAs since the former pre-sumes target perceivability rather than just presence, as in thelatter. In the cluttered period, almost all terminated tracks werefalse ones, and their average lifespan was very short (close to2). Note that the minimum age of a track is two scans, or they


(a) (b)

(c) (d)

Fig. 3. Comparison of PB-PDA and EB-PDA trackers for Scenario 1.

Fig. 4. Average life span of terminated tracks for Scenario 1.

are not treated as tracks. This explains why the average lifespansuddenly drops at scan 15 (because ).

B. Scenario 2

In the second scenario, m for all scans,and the target is perceivable in every scan except forand 14 (i.e., obscured for the period and 14). An averageconfirmation threshold was used to determine MPCTsfor trackers. The PB- and EB-PDA trackers produced 70 and 67MPCTs, respectively. The flops per run were 7.4010 and6.07 10 , and the cpu times per run were 38.22 and 30.77 s,respectively.

Fig. 5 shows the comparison results for Scenario 2. It can beobserved that during the nonperceivable period, although both

were quite low, the EB-PDA tracker had a higher confirma-tion percentage, whereas the PB-PDA tracker outperforms theEB-PDA tracker when the target is perceivable. This verifies theconclusions of our theoretical comparison in Section VI-C.

Fig. 6 shows a comparison of average lifespan of terminatedtracks. Almost all terminated tracks should be false ones. Theterminated tracks of PB-PDA and EB-PDA trackers have virtu-ally the same life spans.

VIII. C ONCLUSIONS

Target perceivability is a fundamental concept for targettracking. Inspired by previous work, this paper establishes/re-fines this important concept. It is shown here that the probabilityof target perceivability plays a central role in many aspectsof target tracking, including tracker design, analysis, anddevelopment. A recursion for the calculation of this probabilityhas been derived based on a hidden Markov model. It enablesquantitative applications of target perceivability to varioustracking tasks, in particular, decision oriented ones, such astrack confirmation and termination. Several such applicationshave been presented or summarized in this paper, includingtracker analysis, tracker design, development of two refinedPDA trackers, and clutter density estimation.

APPENDIX

This Appendix also includes some results that belonglogically here but are needed only in some other papers. Allprobabilities used are conditioned on , which is, however,


(a) (b)

(c) (d)

Fig. 5. Comparison of PB-PDA and EB-PDA trackers for Scenario 2.

Fig. 6. Average life span of terminated tracks for Scenario 2.

dropped for brevity unless otherwise required for clarity. Somecommon variables used in this Appendix are defined by

(32)

(33)

(34)

(35)

(36)

(37)

A. Proof of Theorem 2.1

where is expressed in (8) and calculatedby, when


when

Propositions 1 through 4 used for the above derivation arepresented next.

B. Propositions on Perceivability Probability

Proposition 1: With Assumptions A2 and A3, the par-tial-predicted/partially-updated perceivability probability

can be expressed by

(38)

Remarks: P1-1: An equivalent form of (38) is

(39)

where

(40)

This expression is easily obtained by converting (38) to

Specifically

P1-2: Another equivalent form of (38) is

(41)which is easily obtained by plugging (36) and (40) into (39).

Proof: Based on Assumption A2

(42)

First, considering the case of the special clutter densityat time , based on Assumption A3, we have

(43)

(44)

(45)

As a result, we have the equation shown at the bottom of thenext page. Second, considering at time , we have

As a result

where is given by (40).


Proposition 2: With Assumptions A2 and A3, the eventprobability can be expressed by

(46)Remarks: P2-1: When , an equivalent form of (46)

is

(47)

This is easily obtained by converting (46) to

P2-2: Another equivalent form of (46) is

(48)

which is obtained by combining (39) and (47)

Proof: Based on Assumptions A2 and A3

(49)

No matter whether or not

(50)

First, consider the case of . , is equallylikely to be true kinematic component if the target-originatedobservation falls in the gate. now becomes , which is de-fined right after (10). Then, we have

(51)

(52)


As a result

(53)

Second, considering the case of , we have, noting thatnow is ,

(54)

(55)

As a result

Proposition 3: With Assumptions A2 and A3, the pdfcan be expressed, for , by

(56)where

(57)


(58)

This is easily obtained by converting (56) to

P3-2: One important by-product is

(59)

Proof: It is shown in [3] that

(60)

(61)

Note that by Assumptions A2 and A3, the kinematic compo-nent is independent of feature componentfor both targetand false detections. Considering the case of , we have[3]

(62)

(63)

As a result

In the case of , we may simply use as anotation of pdf for a diffuse prior distribution.

Proposition 4: With Assumptions A2 and A3, the updatedperceivability probability can be expressed by

(64)


(65)


where was given by (40). This is easily obtained byconverting (64) to

Specifically

P4-2: The change rate of with respect to is, when

Further, we define

P4-3: The change rate of with respect to is, when

and undefined when .Proof: By total probability theorem

First, in the case of , the above is

Second, in the case of , the above is

If we further define when , we have uniformly

(66)

or

It is just (64).

C. Proof of Theorem 4.1

For a target measurement, the kinematic informationis summarized by sufficient statistic .The feature informationis summarized by sufficient statistic

.When

(67)

When ,

(68)

where

(69)


When ,

(70)

can be evaluated given distribution of, which is equal to[16] under a Rayleigh

distribution assumption for the signal amplitudes, whereisthe expected SNR of the target signals, and is false alarmprobability.

ACKNOWLEDGMENT

The authors are grateful to the anonymous referees for thecareful reviews and numerous constructive comments thathelped the authors improve the quality of the presentation.

REFERENCES

[1] Y. Bar-Shalom, K. C. Chang, and H. A. P. Blom, “Automatic track for-mation in clutter with a recursive algorithm,” inMultitarget-MultisensorTracking: Advanced Applications, Y. Bar-Shalom, Ed. Norwood, MA:Artech House, 1990.

[2] Y. Bar-Shalom and X. R. Li,Estimation and Tracking: Principles, Tech-niques, and Software. Boston, MA: Artech House, 1993. (Reprintedby YBS, 1998).

[3] , Multitarget-Multisensor Tracking: Principles and Tech-niques. Storrs, CT: YBS, 1995.

[4] Y. Bar-Shalom and E. Tse, “Tracking in a cluttered environment withprobabilistic data association,”Automatica, vol. 11, pp. 451–460, 1975.

[5] S. S. Blackman,Multiple Target Tracking With Radar Applica-tions. Norwood, MA: Artech House, 1986.

[6] S. S. Blackman and R. F. Popoli,Design and Analysis of ModernTracking Systems. Norwood, MA: Artech House, 1999.

[7] H. A. P. Blom and Y. Bar-Shalom, “The interacting multiple model al-gorithm for systems with Markovian switching coefficients,”Automat.Contr., vol. AC-33, pp. 780–783, Aug. 1988.

[8] S. B. Colegrove, “Multiple tracking in a cluttered environment,” inProc.ISSPA, Brisbane, Australia, Aug. 1987.

[9] S. B. Colegrove and J. Ayliffe, “An extension of probabilistic data as-sociation to include track initiation and termination,” inConv. Dig. 20thIREE Int. Conv., Melbourne, Australia, Sept. 1985, pp. 853–856.

[10] S. B. Colegrove, A. W. Davis, and J. K. Ayliffe, “Track initiation andnearest neighbors incorporated into probabilistic data association,”J.Elect. Electron. Eng.—Australia, vol. 6, Sept. 1986.

[11] J. Dezert, N. Li, and X. R. Li, “A new formulation of IPDAF for trackingin cutter,” inProc. Euro. Contr. Conf., Karlsruhe, Germany, Sept. 1999.

[12] V. P. Jilkov and X. R. Li, “Adaptation of transition probability matrixfor multiple model estimators,” inProc. 2001 Int. Conf. Inform. Fusion,Montreal, QC, Canada, Aug. 2001, pp. ThB1–3–ThB1–10.

[13] N. Li, “Development, analysis, and design of intelligent probabilisticdata association filter for target tracking in clutter,” M.S. thesis, Univ.New Orleans, New Orleans, LA, May 1997.

[14] N. Li and X. R. Li, “Target perceivability: An integrated approach totracker analysis and design,” inProc. Int. Conf. Inform. Fusion, LasVegas, NV, July 1998, pp. 174–181.

[15] , “Tracker design based on target perceivability,”IEEE Trans.Aerosp. Electron. Syst., vol. 37, pp. 214–225, Jan. 2001.

[16] X. R. Li, “Tracking in clutter with strongest neighbor measure-ments—Part I: Theoretical analysis,”IEEE Trans. Automat. Contr., vol.43, pp. 1560–1578, Nov. 1998.

[17] , “Engineer’s guide to variable-structure multiple-model estima-tion for tracking,” in Multitarget-Multisensor Tracking: Applicationsand Advances, Y. Bar-Shalom and D. W. Blair, Eds. Boston, MA:Artech House, 2000, vol. III, ch. 10, pp. 499–567.

[18] X. R. Li and C. He, “2M-PDAF: An integrated two-model probabilisticdata association filter,” inProc. SPIE Conf. Signal Data Process. SmallTargets, vol. 3809, Denver, CO, July 1999, pp. 384–397.

[19] X. R. Li and N. Li, “Intelligent PDAF: Refinement of IPDAF for trackingin clutter,” inProc. 29th Southeastern Symp. System Theory, Cookeville,TN, Mar. 1997, pp. 133–137.

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Ning Li received the B.S. and M.S. degrees fromNorthwestern Polytechnical University (NPU),Xi’an, China, in 1983 and 1988, respectively, bothin applied mathematics, and the M.S. and Ph.D.degrees in electrical engineering from University ofNew Orleans, New Orleans, LA, in 1997 and 2001,respectively.

From 1983 to 1995, he was with the NPU facultyas an assistant lecturer, lecturer, and associateprofessor. He is currently with Trilogy ConsultingCorporation, Waukegan, IL. He has published

more than ten journal papers. His research interest was on electromagneticscattering theory and array antennas. Since August 1995, he has been engagedin statistical estimation/decision/computation study and research for targettracking methods.

Dr. Li received the first- and second-class prizes from the Science and Tech-nology Progress Awards of Shaanxi Province, China, in 1992 and 1993.

X. Rong Li (S’90–M’92–SM’95) received the B.S.degree and the M.S. degree from Zhejiang Univer-sity, Zhejiang, China, in 1982 and 1984, respectively,and the M.S. and Ph.D. degrees from University ofConnecticut, Storrs, in 1990 and 1992, respectively,all in electrical engineering.

He joined University of New Orleans, New Or-leans, LA, in 1994. From 1986 to 1987, he did re-search on electric power at the University of Calgary,Calgary, AB, Canada. He was an Assistant Professorwith the University of Hartford, Hartford, CT, from

1992 to 1994. His current research interests include signal and data processing,information fusion and target tracking, statistical inference, stochastic systems,and electric power. He has authored and coauthored four books:Estimationand Tracking(Norwood MA: Artech House, 1993),Multitarget-MultisensorTracking(Storrs, CT: YBS, 1995),Probability, Random Signals, and Statistics(Boca Raton, FL: CRC, 1999), andEstimation with Applications to Trackingand Navigation(New York: Wiley, 2001); four book chapters; 30 journal arti-cles; 100 conference papers. He has consulted for several companies.

Dr. Li has served as an Editor for IEEE TRANSACTIONS ONAEROSPACE AND

ELECTRONIC SYSTEMS since 1996. He received a Career award and an RIAaward from National Science Foundation and the 1996 Early Career Award forexcellence in Research from University of New Orleans. He has served as VicePresident for Technical Activities of the International Society of InformationFusion since 2000, as General Chair, Steering Chair, or General Vice-Chair forthe 1998, 1999, 2000, and 2002 International Conferences on Information Fu-sion and has given numerous seminars and short courses in the United States,Europe, and Asia. He has won several outstanding paper awards and is listed inWho’s Who in AmericaandWho’s Who in Science and Engineering.

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