Metrics for Feature-Aided Track Association

Transcription

1 Metrics for eature-aided rack Association Chee-Yee Chong BAE Systems Los Altos, CA, U.S.A. Shozo Mori BAE Systems Los Altos, CA, U.S.A. Abstract: rack fusion over a network of sensors requires association of the tracks before the state estimates can be combined. rack association generally involves two steps: evaluating an association metric to score each track-to-track association hypothesis, selecting the best assignment between two sets of tracks. In many applications feature-aided track association can provide better performance than association with only kinematic data (e.g., position velocity when the target density is high. his paper develops a general association metric to support feature-aided track association that considers similarity in both the feature kinematic domains. he association metric is based upon the maximum a posteriori probability (MAP approach can be used for general target sensor models. Special forms of the association metric are given for some common situations. Numerical results illustrate the performance of different feature association metrics. eywords: racking, track association, metrics, featureaided tracking Introduction Multiple sensors can provide better tracking performance than a single sensor because of the diverse viewing geometry phenomenology. he tracking architecture for a network of sensors can be either centralized or distributed. In a centralized architecture, all sensor measurements are processed at a central location to generate tracks. In a distributed architecture, the measurements of a single sensor or a group of sensors are processed to generate local tracks. hese local tracks are then fused with the tracks from another sensor. rack fusion consists of two main operations: track association or track-to-track association to determine tracks that correspond to the same targets, state estimate fusion to update the estimates given the associations. When the target density in the kinematic (e.g., position/velocity space is high, association using only the kinematic states is difficult. In fact, analytic models can be used to predict the probability of correct association [, ]. In this case, other features such as size, radar cross section, high-range resolution (HRR profiles, etc., can improve association performance because the tracks may have lower density in the joint feature/kinematic space. In fact, emitter parameters are always used in ELIN correlation because of the large kinematic uncertainties. References [3 6] describe a taxonomy of features, attributes, categorical features, discuss gating thresholds some association metrics. However, the emphasis is on centralized tracking using sensor measurements not on track association. A key step in track association is computing an association metric that scores the similarity between two tracks. Reference [7] presents feature-aided track association metrics based upon the maximum a posterior probability (MAP approach introduced in [8, 9]. he MAP association metric was derived assuming conditional independence of the individual sensor measurements given the target states. In some situations, this conditional independence condition is not satisfied, the joint posterior probability distribution of the kinematic feature states may not factor into a product of the individual probabilities as assumed in [7]. he exact association metric is then more complicated. his paper examines the relationship between the feature kinematic states develops more general association metrics based upon the MAP approach. Numerical results are presented to illustrate the performance of several metrics for associating features. he rest of this paper is organized as follows. Section provides an overview of the track association problem presents the general track association metric based upon the MAP approach. Section 3 considers different models of the kinematic feature states develops association metrics for each case. Numerical results comparing the performance of several feature-aided association metrics are given in Section 4. Problem ormulation In this section we present the track-to-track association problem review the maximum a posteriori probability (MAP approach to track association. he general association metric from the MAP approach will be used to develop various feature-aided association metrics in Section 3.. rack association architecture he overall system consists of multiple sensors observing a set of targets. A local tracker processes the measurements from a single sensor or a group of sensors to generate target tracks. he tracks from different sensors or sensor groups are then associated to determine the ones that correspond to the same targets. he sensors may

2 observe the targets at the same time or at different times, e.g., one sensor before another sensor. In the former case, the objective of track association is to support track state fusion so that more accurate state estimates can be generated. In the latter case, track association provides hover so that the information generated by one sensor, e.g., state or classification information, can be passed on to the other sensor. igure shows the structure of track association for track fusion hover. Each track is characterized by some estimate of the state, which may include kinematic information such as position velocity, feature such as target size or radar cross section, or classification of the target, its confidence on the estimates. A taxonomy of features is provided in [3 6]. ypically, the estimates have to be aligned to the same time before track association can be performed. his is especially true for hover when the downstream fusion system has to extrapolate the tracks from the upstream tracker to the current time. We also assume that the association operation is performed on sensor tracks that have not been associated fused before. hen, the tracks to be associated do not contain common information due to past communication. his assumption allows us to ignore the statistical correlation due to past communication focus on statistical correlation due to other sources. Sensor(s Sensor(s Local racker Local racker rack usion rack Association rack State Estimate usion used racks i Dom( a i j i Dom( a j Im( a ( igure : rack fusion problem where C is a normalizing constant, each track association likelihood function L(, in ( is defined by. Maximum a posteriori probability (MAP association ( ( he traditional approach to track association with only kinematic states uses a chi-squared distance to measure the similarity between two tracks. Once the matrix of all chi-squared distances is formed, an optimal assignment algorithm such as Munkres or Jonker-Volgenant- Castanon (JVC algorithm [] is used to find the best association hypothesis between the two sets of tracks. However, this approach cannot be generalized easily to hle features which may contain both continuous discrete states. he maximum a posteriori probability (MAP association approach was first introduced in [8, 9] to deal with asymmetric conditions with unpaired objects. It will be the basis for feature-aided track association since it can hle arbitrary target states. Suppose there are two sensor systems,, generating two sets of tracks, ( τ,..., τ m ( τ,..., τ n. Note that if the detection probability of each sensor is less than, the two sensors in general have different numbers of tracks because some targets may be detected tracked by one sensor not by the other. Let x be the state of each target extrapolated to a common time for association. he state includes the usual kinematic variables such as position velocity, but it may also include features such as size cross section, target type. Some components of this state, such as position velocity, are continuous rom variables, while target type is usually discrete-valued. eatures can be either continuous or discrete. Let Z i Z j be the measurements associated with the tracks τ i τ j from sensors. hese measurements are usually accumulated over multiple observation times. Let a be an association function defined on a subset Dom( a of {,..., m } taking values in {,..., n}, where Dom( f Im( f denote the domain range of a function f. he set of all (rom association functions is the set D Dom ( α {,..., m} D α {,..., n} #( D # ( Im( α where #( A denotes the number of elements in the set A. his set is basically the set of all one-to-one mappings. It was shown in [8, 9] that the posterior probability of the track association a given all available information represented by Z ( Z,..., Z, Z,..., Z is m n Prob( a Z m n C L( i,a( i L(,0 i L(0, j ( p i x pj x if i m j n qx ( p ( x q ( x Li j i m j qx ( qx ( i (, if 0 q( x pj ( x if i 0 j n his likelihood involves the following probability density functions: q(, the a priori target density in the sense that q ( x is the expected number of objects for any B given region B in psi (, the probability density function of the target state in the state space associated with the i-th track from sensor s, based upon the measurements (3 Z si

3 qs (, the density of objects undetected by sensor s, with the same definition of object density. he probability density for undetected targets for each sensor can be computed as q ( x ( P ( x q( x (4 s where PDs ( x is the probability of an object at state x being detected as a track by sensor s. hen Ds D i m (5 L( i,0 ( P ( x p ( x for each i {,..., } L(0, j ( P ( x p ( x for each j {,..., n} (6 D j he integrals in (3, (5 (6 should be interpreted as summation of the probabilities distributions for those components of x that are discrete rom variables. Certain conditions have to be satisfied for ( (3 to be valid. irst, the number of targets has to be Poisson distributed. his condition has been shown [] to be both necessary sufficient. Second, the measurements for two tracks have to be conditionally independent given the target state, i.e.,, Z x xpz ( x i j i j Note that the likelihoods of tracks from one sensor being undetected by another sensor depend on the detection probabilities, PDs ( x. Just as track state estimates (or means covariances for Gaussian rom variables are used for evaluating the likelihood of two tracks to have originated from the same target, detection probabilities are used to evaluate the likelihood of a track being undetected by another sensor. max Let us assume that 0 PDs ( x PDs <. hen max Li (,0 P D > 0 for every i {,..., m} max L(0, j P D > 0 for every j {,..., n}. Hence, we m n can divide (3 by Li (,0 L(0, j > 0 to obtain i j Prob( a Z C' ( i, j i Dom( a where C ' is a normalizing constant (, i j p i( x p j( x q( x ( P ( x p ( x ( P ( x p ( x D i D j he best association of the tracks from the two sensors is then obtained by selecting the association function that maximizes ( or (8. (7 (8 (9 3 Special Cases of Association Metrics he association likelihoods Li (, j or (, i j in Section were derived for a general target state x assuming that the conditional independence condition (7 is satisfied. In general, this target state is given by x ( x, x, x, where x is the (usually continuous dynamic kinematic state, x is the (continuous or discrete, static or dynamic feature state, x is the (discrete static target type. Special forms of these association metrics can be obtained by exploiting the relationship between the component states. or example, two sensors of the same type may observe the same feature, e.g., size, but sensors of different types may observe different features, e.g., radar observing radar cross section (RCS infrared sensor observing temperature. While commensurate or matching features can be associated directly, noncommensurate or non-matching features cannot. In these cases, we often have to relate the non-commensurate features to the target type. inematic states features may also depend on each other. or example, the acoustic signature of a moving target may depend on its speed. he relationship between the measurements the states should also be exploited. In some cases, the measurements on the kinematic feature states are independent. In other situations, the feature measurement may depend on the kinematic state. requently, the feature measurements (e.g., projected length measurements are coupled to the static features (e.g., length through some dynamic features such as aspect angle. 3. inematic state only Gaussian models We present this case to relate to the stard kinematic track association problem also to illustrate the importance of the conditional independence assumption. Suppose the target state is kinematic only (i.e., is Euclidean space the state distribution associated with track τ is Gaussian with state vector estimate si estimation error covariance matrix. If we assume also that the a priori target state distribution q is flat enough with respect to the state distribution associated with any track, we can approximate (9 by with V si xˆsi ( ( ( ( ( ( i j D D π i j (, i j q P P det ( V + V exp χij for each χ xˆ xˆ ij i j ( Vi+ V j ( xˆ xˆ ( V + V ( xˆ xˆ i j i j i j ( i, j {,..., m} {,..., n}, (0 (

4 ( ( for {,..., m} P P x p x i ( i D D i } ( ( j P P ( x p ( x for j,..., n (3 D D j he detection probability ( i D P is the average probability of the object in the i-th track { τ i from sensor being ( j detected by sensor, while the detection probability P D is the average probability of the object in the j-th track τ from sensor being detected by sensor. Both of j these detection probabilities depend on the specific tracks. If we apply ln( to the a posteriori probability (8 for each assignment a, the maximization of (8 becomes the minimization of J ( a C( i, a( i (4 i Dom( a with cost matrix defined by Ci (, j ln (, i j χij Aij where ( i ( j (( π A ln q( P ( P det( ( V + V ij D D i j (5 (6 he cost coefficient in (4 is different from the stard chi-squared distance by having the additional term A ij to represent the effect of the unpaired targets in association. Basically, the assignment cost depends not only on the distance between the (kinematic states of the two tracks the tracking accuracy (as represented by ( i Vi V j but also on the detection performance P Ds a priori target density q. In particular, the adaptive threshold A ij depends on the track-dependent detection ( j ( i probabilities P. Low detection probabilities P D D high target density will increase the assignment cost. A similar expression that also includes false tracks is presented in [0] without derivation. As discussed before, the validity of the MAP association metric hinges on the conditional independence of the sensor measurements given the target state x. If the target kinematic state is that of a deterministic dynamic system with no process noise, then the conditional independence condition (7 is satisfied, the assignment cost (4 or (5 is exact. If the dynamics is not deterministic but the process noise is low, then this association metric or assignment cost is still a good approximation. However, if the process noise is high, then the association metric needs to be modified. One approach modifies the assignment cost (as in [3] by adding cross-covariance terms to the chi-squared distance. Another approach, first discussed in [4] further developed in [5], uses the kinematic state trajectory as the state to be used in association. Let x Z be the vector of kinematic states at the observation times in the measurement set Z. hen the conditional independence condition of the measurements is satisfied if x Z is the state, i.e.,, Z x x x (7 i j Z i Z j Z he association metric in Section 3. is applicable with x replaced by xz. In particular, the chi-squared distance compares the state estimates at the measurement instants. his makes sense since track association should depend on the entire track history in a general case. However, there is additional cost to compute this association metric. 3. Independent kinematic feature measurements Suppose each sensor produces independent measurements on the kinematic feature states the states are independent, i.e., Z i ( Z i, Z i, Z j ( Z j, Z j p( Z, Z x, x p( Z x p( Z x (8 si si si si x Z igure : Independent kinematic feature measurements igure shows the target sensor model as a Bayesian network. urthermore, let the detection probability for each sensor be constant given by P Ds on the kinematic state, qx ( qx ( ν where ν is the expected number of targets, p( x p( x are the prior probability densities of the kinematic features states. hen p ( x p ( x, x p ( x p ( x (9 x Z si si si si ( q ( x P ( x q( x (0 s Ds he association likelihood decomposes into a product of kinematic feature association likelihoods, i.e., with (, i j (, i j (, i j (, i j ( p i( x p j( x q( x ( P ( x p ( x ( P ( x p ( x D i D j ( p i( x p j( x (, i j (3

5 When the posterior state distributions for the kinematic states are Gaussian, the same development used in Section 3. gives ( ( ( ( ( ( i j π (, i j q P P det ( V + V D D i j exp χij (4 When the feature states are also continuous Gaussian, we obtain ( ( π (, i j det ( V i+ V j exp χij (5 where χ ij χ ij are given by the equation ( with appropriately defined means covariances, i.e., χ χ xˆ xˆ ij i j xˆ xˆ ij i j ( Vi+ V j ( Vi+ V j When the feature states are discrete, (3 becomes a summation p i( x p j( x (, i j (6 x When the logarithm is taken the association metric is converted to an assignment cost, the net effect of using independent features is to add χ to the kinematic assignment cost, with additional determinant terms that represent the uncertainty in the feature estimates. Effectively, the independent feature introduces an additional dimension to the assignment problem. hen ij ( i ( j / ( D ( D π i j / ( V V Ci (, j ln q( P ( P det ( V + V + χ + ln det π( + + χ ij i j ij (7 his association metric of (4 with (6 is used in [7], which assumes that the prior posterior distributions of the feature/type states are independent of those of the kinematic state. 3.3 Dependent kinematic feature (type measurements In general, the posterior distributions of kinematic feature states are not independent even though the prior distributions may be independent. hus, the association metrics cannot be factored into a product of kinematic feature association metrics. his dependence may result from measurements that depend on both kinematic feature states. An example is when the feature is the length of the target. he feature measurement is the projected length, which depends on the actual length the aspect angle. If the target is moving on a road that is known, the aspect angle is determined by the kinematic state. x Z x Z igure 3: Dependent kinematic feature (type measurements Assume that the kinematic measurement depends only on the kinematic state the feature measurement depends on both the kinematic the feature states. hen the state-to-measurement transition probability is given by (left side of igure 3 p( Z, Z x, x x, x x (8 x Z s s s s he posterior state distribution for each track is, Z x, x, x, x Z, Z, Z x, x x, x, Z x, x x x, Z Z, x x Z, Z (9 When this posterior distribution is substituted into (3, we obtain Li (, j C p i( x Z i p j( x Z j i x j x q( x p i( x Z i, x p j( x Z j, x where psi( x Zsi, x is the probability distribution of the feature state for track i of sensor s given the measurements x. hus Li (, j is of the form p i( x Z i p j( x Z j Li (, j C q( x i x j x L ( i, jx ; (30 where L (, i j; x is the association likelihood using only feature given x p i( x Z i, x p j( x Z j, x L (, i j; x (3 x Z

6 In other words, the association metric cannot be decomposed into a product of the feature kinematic association metrics. or the case when the target type is observed affects the measurements on the kinematic state (right side of igure 3, the association likelihood is given by Since p ( x, x p ( x, x i j, x p i( x x p i( x p j( x x p j( x x (35 Li (, j p i( x p j( x p i( x x p j( x x qx ( x x x p i( x p j( x L x (, i j; x (3 where p( x is the prior probability distribution of the discrete target type, L (, i j; x is the kinematic x association metric given x, i.e., p ( x x p ( x x L i j x (33 i j x (, ; q( x x Note that this equation again does not decompose into a product as assumed in [7]. 3.4 eatures dependent on target type Suppose the two sensors observe common features that depend on the target type (igure 4. urthermore, suppose the features are independent of the kinematic states. hen the conditional independence condition is satisfied for the feature state x, the association likelihood is a product of the kinematic association feature association likelihoods according to Section 3.. In particular, the association likelihood (, i j is given by (3. However, psi( x, the posterior distributions of the feature, is not necessarily Gaussian because of its dependence on x even though all other variables may be Gaussian. hus, (5 cannot be used to compute the association likelihood we need to start with (3. Z x x igure 4: eature dependent on target type Z It is fairly straightforward to show that equation (34 becomes where p ( x p ( x (, (, ; (36 i j i j i j x x p ( x x p ( x x i j (, i j; x x (37 If psi( x x is a Gaussian distribution given x with prior mean x( x covariance V( x, ˆ x, Z i, Z j has posterior mean xij ( x covariance V ( x, then (37 can be expressed as ij (, i j; x ( Vˆ ij x V x det ( ( ( V i x V j x det ( ( ˆ x x x x exp + ( ( ˆ ij ( i ( Vi xˆ ˆ ij x x j x V j xˆ ij ( x x ( x V Vˆ ( x xˆ ( x V ( x xˆ ( x ij ij i i ˆ j j + V ( x x ( x V ( x x ( x Vˆ ( x V ( x + V ( x V ( x ij i j (38 (39 hus, when the feature x depends on the target type x, the exact association likelihood (36 is considerably more complicated than (5 when the feature is assumed to be Gaussian. In particular, the likelihood is no longer dependent on a single chi-squared term. 4 Numerical Example We present a simple example to illustrate how association performance depends on the choice of the feature association metric. he target observation model consists of the following components: (, i j p i( x p j( x p i( x, x p j( x, x, x x (34 inematic model: arget kinematic state x is uniformly distributed over a two dimensional disk U x R x R { }

7 Number of targets: Poisson with mean ν 0 ν inematic target density: q π R inematic track accuracy: stard deviation σ in both directions, as determined by a given relative target density defined by γ ν R σ arget type feature model: arget type x : {0,} with prior probability p 0 p rue feature state x : x 0 is a zero mean, unit variance Gaussian rom variable x is a unit variance, Gaussian rom variable with mean > 0. he parameter specifies the separation of the two features is the - factor sometimes used to characterize the difficulty of target classification problems. eature observation Z : Z x + w where w is a zero mean Gaussian rom variable with stard deviation σ 0. our different association metrics are used: Gaussian approximation of the feature distribution (d, then using type only (b. Association with features always produces better performance than association with kinematic state alone. he performance improvement is especially significant for high target density. As expected, the performance also increases with a bigger, i.e., with more separation in the features. Probability of Correct Association inematic + ype Optimal inematic Only Normalized Object Density actor Gaussian Approximation of eatures igure 5: Performance comparison for different association metrics ( (a. inematic only with the association cost C (, i j given by (5 (6 (b. inematic plus target type probability with association cost given by p i( x p j( x C (, i j + ln (40 x (c. inematic plus exact combination of target type feature given by (36 to (39 Probability of Correct Association inematic + ype actor Optimal inematic Only Gaussian Approximation of eatures p i( x p j( x C (, i j + ln (, i j; x (4 x where (, i j; x is given by (37 to (39 (d. inematic plus Gaussian approximations for the prior feature distribution x as well x as the posterior feature distribution p ( x p( x Z p( x Z, x p( x Z si si si si x obtained by equating the first second moments. he association metric (b is sub-optimal since the measurements from the two sensors are not conditionally independent given the target type. he association metric (d is suboptimal since the exact feature state distribution consists of a sum of Gaussian distributions. igure 5 to 7 shows the performance of the four association metrics as a function of the normalized target density the factor. As expected, the optimal association metric (c using the kinematic state the feature state distributions has the best performance, followed by Normalized Object Density igure 6: Performance comparison for different association metrics ( Probability of Correct Association inematic + ype actor 3 Optimal Gaussian Approximation of eatures inematic Only Normalized Object Density igure 7: Performance comparison for different association metrics ( 3

8 5 Conclusions rack association is an important function in networkcentric tracking since it supports fusion of track states seen by multiple sensors, hover of target data from one sensor to another sensor, preserving track continuity when there is a sensing gap. When the target density is high, association using kinematic data alone may not provide adequate performance. In this case, using features in addition to kinematic data may improve association performance since the target density may be lower in the augmented feature/kinematic state space. An association metric is used to score the similarity between tracks before assignments can be made. In this paper, we have developed association metrics that incorporate both feature kinematic data. hese metrics are based upon the maximum a posteriori probability (MAP association approach [3, 4] that has been shown to perform better than the traditional chi-squared distance when unpaired objects are involved. In order to satisfy the conditional independence condition required for the MAP approach, different target/sensor models will produce different association metrics. We have used a simulated example to show that association performance is sensitive to the choice of the association metric, especially when the normalized target density is high. eature-aided track association is a relatively new area in tracking. We have developed some possible metrics based on the MAP approach demonstrated their performance with a simple example. urther research is needed to address computational issues as well as evaluate performance in more realistic scenarios. References. S. Mori,. C. Chang, C. Y. Chong, Performance Analysis of Optimal Data Association with Applications to Multiple arget racking, Chapter 7 in Multitarget-Multisensor racking: Applications Advances, Vol. II, Y. Bar-Shalom, ed., pp , Artech House, 99.. S. Mori,.C. Chang, C.Y. Chong,.P. Dunn, Prediction of rack Purity rack Accuracy in Dense arget Environments, IEEE ransactions on Automatic Control, Vol. AC-40, No. 5, pp , May O. E. Drummond, On eatures Attributes in Multisensor Multitarget racking, Proc. nd Inter. Conf. on Information usion (USION 99, July 999, Sunnyvale, CA, pp O. E. Drummond, Integration of eatures Attributes into arget racking, Signal Data Processing of Small argets 000, Proc. SPIE, Vol. 4048, pp. 60-6, O. E. Drummond, eature, Attribute, Classification Aided arget racking, Signal Data Processing of Small argets 00, Proc. SPIE, Vol. 4473, pp , O. E. Drummond, On Categorical eature Aided arget racking, Signal Data Processing of Small argets 003, Proc. SPIE, Vol. 504, L. D. Stone, M. L. Williams,. M. ran, rackto-rack Association Bias Removal, Signal Data Processing of Small argets 00, Proc. SPIE, Vol. 478, pp , S. Mori C. Y. Chong, Effects of Unpaired Objects Sensor Biases on rack-to-rack Association Problems Solutions, Proc. 000 MSS National Symposium on Sensor Data usion, San Antonio,, June S. Mori C. Y. Chong, BMD Mid-Course Object racking: rack usion under Asymmetric Conditions, Proc. 00 MSS National Symposium on Sensor Data usion, San Diego, CA, June 6-8, S. Blackman R.Popoli, Design Analysis of Modern racking Systems, Artech House, Norwood, D. A. Canstañon, New Assignment Algorithms for Data Association Proc SPIE, Signal Data Processing of Small argets, Vol. 698, 99.. S. Mori C. Y. Chong, rack-to-rack Association Metric - I.I.D Non-Poisson Cases, Proc. 6th Intern. Conf. on Information usion (USION 003, pp , Cairns, Australia, July 8-0, Y. Bar-Shalom, On the rack-to-rack Correlation Problem, IEEE rans. Automatic Control, Vol. AC- 6, pp , April C.-Y. Chong, S. Mori,.-C. Chang, Distributed Multitarget Multisensor racking, Chapter 8 in Multitarget-Multisensor racking: Advanced Applications, edited by Y. Bar-Shalom, Artech House, MA, S. Mori, W. H. Barker, C. Y. Chong,. C. Chang, rack Association rack usion with Nondeterministic arget Dynamics, IEEE rans. Aerospace Electronic Systems, Vol. 38, No.. pp , April 00.