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worse than the probability of correct measurement association. Hence ... approach did lead to novel tracking filters which were referred to as CPDA, CPDA* and. JPDA*, where ...... Interacting Multiple Model Joint Probabilistic Data Association.
Nationaal Lucht- en Ruimtevaartlaboratorium National Aerospace Laboratory NLR

NLR-TP-2005-667

Tracking Multiple Maneuvering Targets From Possibly Unresolved, Missing or False Measurements

H.A.P. Blom and E.A. Bloem

This report has been based on a paper presented at 8th Conference on Information Fusion FUSION 2005, Philadelphia, U.S.A., July 25 - July 29, 2005. This report may be cited on condition that full credit is given to NLR and the authors.

Customer: Working Plan number: Owner: Division: Distribution: Classification title:

National Aerospace Laboratory NLR AT.1.E.3 National Aerospace Laboratory NLR Air Transport Unlimited Unclassified October 2005

Approved by: Author

Reviewer: Anonymous peer reviewers

Managing department

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Summary In Daum (1992) it has been very well explained that sensor resolution modeling is crucial for the tracking of closely spaced targets. For non-maneuvering targets appropriate models and tracking approaches have been developed by Chang & Bar-Shalom (1984) and by Koch & Van Keuk (1997). This paper combines their sensor resolution submodels with a descriptor system approach towards tracking two suddenly maneuvering closely spaced targets from measurements that may be false, missing or unresolved. Using this descriptor system formalism, exact Bayesian and approximate filter equations are derived.

-4NLR-TP-2005-667

Contents

1

Introduction

5

2

The two-target tracking problem

7

3

Descriptor system formulation

10

4

Bayesian filter equations

12

5

Conditional mean and covariance

15

6

Joint IMM Coupled PDA filter with Resolution

17

7

Track-coalescence-avoiding JIMMCPDAR filter

19

8

Discussion of results

21

9

References

22

Appendix A

Acronyms

23

Appendix B

List of Symbols

24

(25 pages in total)

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1

Introduction

In [1] and [2] it has been explained that sensor resolution modeling is very important for the tracking of closely spaced targets. The key issue is that the probability of resolution typically is worse than the probability of correct measurement association. Hence, optimizing the latter requires modeling the former. In literature there are a few papers which develop good resolution models and incorporate them into effective target tracking. [3] introduces a hard measurement distance threshold model regarding yes/no resolution, and incorporates the corresponding Error density function within JPDA. [4] incorporates this Error function modeling with Multiple Hypothesis Tracking (MHT). [5] introduces a Gaussian shaped measure for the probability of resolution, and shows that this combines smoothly with a Gaussian mixture framework of MHT. [6] enhances this MHT approach towards suddenly maneuvering targets by incorporating IMM. [7] utilizes group tracking to determine false and missing measurements and assess targets that merge with or split from a group. The aim of this paper is to combine resolution submodels of [3] and [5] with the descriptor system approach by the authors towards tracking multiple maneuvering targets from missing and false measurements. This approach was introduced in [8] to derive, in a mathematically unambiguous way, exact and approximate Bayesian equations for filtering multiple targets from possibly missing and false measurements. For linear Gaussian targets the descriptor system approach did lead to novel tracking filters which were referred to as CPDA, CPDA* and JPDA*, where the * refers to a particular track coalescence avoiding pruning of permutation hypotheses [8]. Of these filters, CPDA performs comparable to JPDA [9] and JPDA-Coupled [10], whereas CPDA* and JPDA* performs significantly better. In a follow-up series of studies the descriptor system approach is extended to situations of Markov jump linear targets, including the development of several novel approximate Bayesian filters, i.e.: •

IMMJPDA* [11], [12] which prunes particular permutation hypotheses (similar as JPDA*) into a descriptor system version of the IMMJPDA filter by [13];



JIMMCPDA [14] which is in theory the best combination of PDA and IMM for multiple Markov jump linear targets. 1



JIMMCPDA* [15] which prunes particular permutation hypotheses in JIMMCPDA similar as in CPDA* ;



Particle filter implementations of the exact Bayesian filter equations [14], [16].

1

In [17] a version of JIMMCPDA for independently maneuvering targets is developed under the name IMMJPDA-Coupled filter.

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A comparison of these tracking filters through Monte Carlo simulations [14] shows that JIMMCPDA* and IMMJPDA* performs much better than JIMMCPDA and IMMJPDA, and also remarkably well in comparison to a good particle filter implementation of the exact Bayesian filter equations. This paper is organized as follows. Section 2 defines the two-target tracking problem considered. Section 3 formulates this as a problem of filtering for a jump-linear descriptor system with independent identically distributed (i.i.d.) stochastic coefficients. Section 4 develops exact Bayesian filter equations. Section 5 develops equations for the conditional mean and covariance. Sections 6 and 7 specify the JIMMCPDAR and JIMMCPDAR* filters respectively. Section 8 presents a discussion of the results.

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2

The two-target tracking problem

We consider two targets and assume that the state of each target is modeled as a jump linear system: xti+1 = a i (θ ti+1 ) xti + b i (θ ti+1 ) wti ,

i = 1, 2

(1)

where xti is the n -vectorial state of the i -th target, θ ti is the Markovian switching mode of the i -th target and assumes values from M

{1,.., N } according to a transition probability matrix

Π , a (θ ) and b (θ ) are ( n × n) - and ( n × n′) -matrices and wti is a sequence of i.i.d. standard i

i

i t

i

i t

Gaussian variables of dimension n′ with wti , wtj independent for all i ≠ j and wti , x0i , x0j independent for all i ≠ j . We assume that a potential measurement originating from target i is also modeled as a jump linear system: zti = h i (θ ti ) xti + g i (θ ti )vti ,

i = 1, 2

(2)

where zti is an m -vector, hi (θ ti ) is an ( m × n )-matrix and g i (θ ti ) is an ( m × m′ )-matrix, and vti is a sequence of i.i.d. standard Gaussian variables of dimension m′ with vti and vtj independent for all i ≠ j . Moreover vti is independent of x0j and wtj for all i , j . ⎡ x1 ⎤ ⎡θ 1 ⎤ ⎡ w1 ⎤ Let xt = ⎢ t2 ⎥ , θt = ⎢ t2 ⎥ , wt = ⎢ t2 ⎥ , and A(θ t ) = Diag{a1 (θ t1 ), a 2 (θ t2 )} , B (θ t ) = Diag{b1 (θ t1 ), b 2 (θ t2 )} , ⎣ xt ⎦ ⎣θ t ⎦ ⎣ wt ⎦

Then we can model the state of our 2 targets as follows: xt +1 = A(θ t +1 ) xt + B (θ t +1 ) wt

(3)

with A and B of size 2n × 2n and 2n × 2n′ respectively, with {θt } assuming values from {1,..., N }2 according to transition probability matrix Π = ⎡⎣⎢ Πη ,θ ⎤⎦⎥ . If the 2 targets switch mode

independently of each other, then: Πη ,θ = ∏ i =1 Πηi i ,θ i 2

for every η and θ ∈ M 2 = {1,..., N }2 .

(4)

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⎡ z1 ⎤

⎡ v1 ⎤

⎣ zt ⎦

⎣vt ⎦

Next with zt = ⎢ t2 ⎥ , vt = ⎢ t2 ⎥ , and H (θ t ) = Diag{h1 (θ t1 ), h 2 (θ t2 )} , G (θ t ) = Diag{g 1 (θ t1 ), g 2 (θ t2 )} , we obtain: zt = H (θ t ) xt + G (θ t )vt

(5)

with H (θt ) and G (θt ) of size 2m × 2n and 2m × 2m′ respectively. In order to incorporate limited sensor resolution with Bayesian filtering, [5] adopts a non-zero probability that two close targets merge. This event of merging or not is represented by a zeroone-valued process κ t , where κ t = 1 refers to merging, and κ t = 0 means non-merging. Following [5] we assume: pκt |xt ,θt (0 | x, θ ) = 1 − pκt |xt ,θt (1 | x, θ ) 1⎛ 1 1 1 2 2 2 ⎞T −1 ⎛ 1 1 1 2 2 2⎞ ⎜ h (θ ) x − h (θ ) x ⎟ R (θ ) ⋅ ⎜ h (θ ) x − h (θ ) x ⎟} = ⎠ ⎝ ⎠ 2⎝ ⎡I ⎤ ⎪⎧ 1 ⎪⎫ = exp ⎨− xT H (θ )T ⎢ ⎥ R(θ ) −1[ I M − I ]H (θ ) x ⎬ ⎣− I ⎦ ⎩⎪ 2 ⎭⎪

(6)

pκt | xt ,θt (1 | x, θ ) = exp{ −

(7)

where R (θ ) is an m × m resolution capability matrix: R(θ ) = ⎛⎜⎝ g 1 (θ 1 ) I r g 1 (θ 1 )T + g 2 (θ 2 ) I r g 2 (θ 2 )T ⎞⎟⎠ = = [ I M I ]G (θ ) Diag{I r , I r }G (θ )T [ I M I ]T

(8)

with I r = Diag{r1 ,.., rm ' } and typically ri > 1. Following [3], [10], for κ t = 1 we assume that with probability Pd0 the merged potential measurement ( zt1 + zt2 ) / 2 is observed at moment t . And for κ t = 0 , we assume that with a nonzero detection probability, Pdi , the potential measurement zti is observed at moment t , independently per target. Let Ft denote the number of false measurements at moment t , we assume Ft to be Poisson distributed: (λV ) F exp(−λV ), F = 0,1, 2,... F! else = 0,

pFt ( F ) =

where λ is the spatial density of false measurements and V is the volume of the observed region. Thus λV is the expected number of false measurements in the observed region. We

(9.a)

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assume that the false measurements are uniformly distributed in the observed region, which means that a column-vector vt∗ of Ft i.i.d. false measurements has the following density: pv∗|F (v∗ | F ) = V − F t

(9.b)

t

Furthermore we assume that the process {vt∗ } is a sequence of independent vectors, which are independent of {xt }, {wt }, {vt } and of the merging and detection. At moment t a vector observation yt is made, the components of which consist of Ft false measurements and Dt detected (merged) potential measurements, in an arbitrary order. The total number Lt of measurements is: Lt = Dt + Ft .

(9.c) Δ

The multi-target tracking problem is to estimate ( xt , θt ) from observations Yt {Ls , ys ;0 ≤ s ≤ t} .

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3

Descriptor system formulation

In order to prepare for a Bayesian evaluation of this tracking problem, in this section we develop an equation which relates the potential measurements zt mathematically to the true measurements yt . Let φi ,t be the detection indicator of target i . For κ t = 0 it assumes the value 1 with probability Pdi > 0 , independently of φ j ,t , j ≠ i , and the value 0 with probability (1 − Pdi ) . For κ t = 1 , with

probability Pd0 > 0 the two potential measurements merge to form one true measurement, i.e. 1 for i = 1, 2 , and with probability 1 − Pd0 the merged potential measurements do not form 2 a true measurement, i.e. φi ,t = 0 for i = 1, 2 . The resulting detection indicator vector

φi ,t =

φt = [φ1,t φ2,t ]T is a sequence {φt } of i.i.d. vectors, and with a κ t -conditional distribution: 2

pφt |κt (φ | κ ) = ∏(1 − Pdi )

1−φi

i =1

2

1 0 2 −φi d

= ∏(1 − P i =1

( Pdi )

φi

) (P )

0 φi d

=0

if κ = 0, φ ∈{0,1}

2

⎧ ⎡ 1 ⎤⎫ ⎪⎪⎡0⎤ ⎢ 2 ⎥ ⎪⎪ if κ = 1, φ ∈ ⎨⎢ ⎥ , ⎢ ⎥ ⎬ ⎪⎣0⎦ ⎢ 1 ⎥ ⎪ ⎢⎣ 2 ⎥⎦ ⎭⎪ ⎪⎩ else.

(10)

Summation over all components of φt yields Dt = ∑ i =1 φi ,t . M

(11)

We want to use this target indicator process {φt } within an equation that relates yt to zt . To accomplish this, we first introduce the following operator Φ : for an M ′ -vector φ ′ with (0,1)valued components φ ′i, i ∈ {1, M ′}, we define D(φ ′)

Δ



M′ i =1

φ ′i and the operator Φ producing

Φ (φ ′) as a (0,1) -valued matrix of size D (φ ′) × M ′ of which the i th row equals the i th non-zero

row of Diag {φ ′} . And, if φ ′ = [ Δ

1 1T 1 1 ] , then Φ(φ ′) = [ ] . We also define an underlining 2 2 2 2

notation: Φ (φ ′) Φ (φ ′) ⊗ I m , with I m a unit-matrix of size m and ⊗ Kronecker product, i.e.

⎡ aI Δ⎢ m ⎡a b ⎤ ⊗ I m ⎢c d ⎥ ⎢ ⎣ ⎦ ⎢⎣ cI m

M bI m ⎤ ⎥ ...... ⎥ M dI m ⎥⎦

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With this the vector of detected measurements z% t satisfies: z% t = Φ (φt ) zt

if Dt > 0

(12)

We next introduce a stochastic Dt × Lt -matrix process {χ% t} such that χ% t yt = z% t

if Dt > 0

(13)

Δ

where χ% t χ% t ⊗ I m . Substitution of (5) into (12) and this into (13) yield: χ% t yt = Φ(φt ) H (θt ) xt + Φ(φt )G(θ t )vt

if Dt > 0

(14)

Notice that the size of χ% t is Dt m × Lt m and the size of Φ (φt ) is Dt m × Mm . Equation (14) is a jump-linear Gaussian descriptor system [18] with stochastic i.i.d. coefficients χ% t and Φ (φt ) . Together with equations (3), (4) and (6) through (11), equation (14) captures the filtering problem to be solved in a mathematically well defined system of equations. Following (14), all target to measurement relevant associations and permutations are covered by (φt , χ% t ) -hypotheses with Dt > 0 . To this set of hypotheses we add one for the situation Dt = 0 through the hypothesis φt = {0}2 and χ% t = {}L . Hence, through defining the weights t

Δ

βt (φ , κ , χ% ,θ ) Prob{φt = φ , κ t = κ , χ% t = χ% ,θt = θ | Yt }

(15)

the law of total probability yields: pxt ,θt |Yt ( x,θ ) =

∑ β (φ, κ , χ% ,θ ) p θ φ κ χ

χ% ,φ ,κ

t

xt | t , t , t , % t ,Yt

( x | θ , φ, κ , χ% )

(16)

With this the problem is to characterize the terms in the last summation, which is done in the next section.

-12NLR-TP-2005-667

4

Bayesian filter equations

In this section a Bayesian characterization of the conditional density px ,θ |Y ( x, θ ) is given where t

t t

Yt denotes the σ -algebra generated by measurements up to and including moment t .

Proposition 1

{

For any φ ∈ {0,1}2 ∪ [

}

Δ 2 1 1T ] , such that D(φ ) ∑ φi ≤ Lt , and any χ% t matrix realization χ% of 2 2 i =1

size D(φ ) × Lt , the following holds true: pxt |θt ,κt ,φt , χ% t ,Yt ( x | θ , κ , φ , χ% ) =

pz% t | xt ,θt ,φt ( χ% yt | x, θ , φ ) ⋅ pxt |θt ,κt ,Yt −1 ( x | θ , κ )

(17)

Ft (φ , κ , χ% , θ )

βt (φ , κ , χ% ,θ ) = Ft (φ , κ , χ% ,θ )λ ( L − D (φ ) ) ⋅ pφ |κ (φ | κ ) ⋅ pκ |θ ,Y (κ | θ ) ⋅ pθ |Y (θ ) /ct t

t

t

t

t

t −1

t

t −1

(18)

where Ft (φ , κ , χ% , θ ) and ct are such that they normalize px |θ ,κ ,φ ,χ% ,Y ( x | θ , κ , φ , χ% ) and β t (φ , κ , χ% , θ ) t

t

t

t

t

t

respectively. Proof: Omitted due to space limitation. It largely follows the proof of Theorem 1 in [16] Proposition 2 The conditional density px |θ ,κ ,Y ( x | θ , κ ) in Proposition 1 satisfies for κ t = 0 and κ t = 1 t

respectively: pxt |θt ,κt ,Yt −1 ( x | θ , 0) =

t

t

t −1

c (θ ) 1 pxt |θt ,Yt −1 ( x | θ ) − t px |θ ,κ ,Y ( x | θ ,1) 1 − ct (θ ) 1 − ct (θ ) t t t t −1

(19)

pxt |θt ,κt ,Yt −1 ( x | θ ,1) = pκt |xt ,θt (1| x, θ ) pxt |θt ,Yt −1 ( x | θ ) / ct (θ )

(20)

ct (θ ) = pκ t |θt ,Yt −1 (1| θ )

(21)

Proof: Follows from Bayes and subsequent evaluation. The next step is to combine Propositions 1 and 2 for the derivation of a characterization of the exact Bayesian measurement update equations in the following Theorem.

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Theorem 1 The measurement updating of px ,θ |Y ( x,θ ) to px ,θ |Y ( x,θ ) satisfies, for θ ∈ {1,.., N }2 : t

t

t −1

t

t t

pxt ,θt |Yt ( x,θ ) = ∑ βt0 (θ , χ% , φ ) pxrt=|θ0t , χ%t ,φt ,Yt ( x | θ , χ% , φ ) + ∑ βt1 (θ , χ% ,φ ) pxt |θt ,κt , χ%t ,φt ,Yt ( x | θ ,1, χ% ,φ ) φ , χ%

β tκ (θ , χ% , φ ) =

1 κ 2 − D (φ ) ) ⋅ ⎡⎣ pφt |κt (φ | κ ) − κ pφt |κt (φ | 0) ⎤⎦ pθt |Yt −1 (θ ) Ft (θ , χ% , φ )ct (θ )κ λ ( ct

pxrt=|θ0t , χ%t ,φt ,Yt ( x | θ , χ% , φ ) =

(22)

φ , χ%

pz%t | xt ,θt ,φt ( χ% yt | x, θ , φ ) pxt |θt ,Yt −1 ( x | θ )

(24)

Ft 0 (θ , χ% , φ )

pxt |θt ,κt , χ%t ,φt ,Yt ( x | θ ,1, χ% , φ ) =

pz%t | xt ,θt ,φt ( χ% yt | x, θ , φ ) pxt |θt ,κt ,Yt −1 ( x | θ ,1)

(25)

Ft1 (θ , χ% , φ )

{

pz%t | xt ,θt ,φt ( χ% yt | x, θ , φ ) = N χ% yt ; Φ (φ ) H (θ ) x, Φ (φ )G (θ )G (θ )T Φ (φ )T

(23)

}

(26)

with px |θ ,κ ,Y ( x | θ ,1) and ct (θ ) satisfying (20) and (21) respectively, Ft 0 (θ , χ% , φ ) and Ft1 (θ , χ% , φ ) t

t

t

t −1

normalization functions, and ct such that



θ ,κ ,φ , χ%

β tκ (θ , χ , φ ) = 1 .

Proof of Theorem 1 When r = 0 there is always resolution, which implies eq. (24). Under the condition r > 0 we substitute (20) into (17) for κ = 1 to get equation (25) with Ft1 (θ , χ% , φ ) = Ft (θ ,1, χ% , φ ) . To get (22) and (23), we first substitute (19) into (17) for κ = 0 : pxt |θt ,Yt −1 ( x | θ ) ct (θ ) pxt |θt ,κt ,Yt −1 ( x | θ ,1) ⎤ ⎡ pxt |θt ,κt , χ%t ,φt ,Yt ( x | θ , 0, χ% , φ ) = pz%t | xt ,θt ,φt ( χ% yt | x,θ , φ ) ⋅ ⎢ − ⎥ (27) ⎣⎢ [1 − ct (θ )] Ft (θ , 0, χ% , φ ) [1 − ct (θ ) ] Ft (θ , 0, χ% , φ ) ⎦⎥

Next, substituting (24) and (25) into (27) and subsequent evaluation yields: pxt |θt ,κt , χ%t ,φt ,Yt ( x | θ ,0, χ% , φ ) = =

=

Ft 0 (θ , χ% , φ ) r =0 F 1 (θ , χ% ,φ ) pxt |θt , χ%t ,φt ,Yt ( x | θ , χ% , φ ) / (1 − ct (θ ) ) − ct (θ ) t px |θ ,κ , χ% ,φ ,Y ( x | θ ,1, χ% , φ ) / (1 − ct (θ ) ) = Ft (θ ,0, χ% , φ ) Ft (θ ,0, χ% , φ ) t t t t t t

1 1 − ct1 (θ , φ , χ% )

pxrt=|θ0t , χ%t ,φt ,Yt ( x | θ , χ% , φ ) −

ct1 (θ , φ , χ% ) px |θ ,κ , χ% ,φ ,Y ( x | θ ,1, χ% , φ ) 1 − ct1 (θ , φ , χ% ) t t t t t t

(H1)

with: ct1 (θ , χ% , φ ) =

Ft1 (θ , χ% , φ ) ct (θ ) , and Ft 0 (θ , χ% , φ )

Ft (θ , 0, χ% , φ ) =

Ft 0 (θ , χ% , φ ) F 1 (θ , χ% , φ ) − ct (θ ) t (1 − ct (θ ) ) (1 − ct (θ ) )

(H2) (H3)

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Substitution of equation (H1) into equation (16) yields: β t (θ , 0, χ% , φ ) r = 0 ( x | θ , χ% , φ ) + p (1 − ct1 (θ , φ , χ% ) ) x |θ , χ% ,φ ,Y

pxt ,θt |Yt ( x | θ ) = ∑ β t (θ ,1, χ% , φ ) pxt |θt ,κt , χ%t ,φt ,Yt ( x | θ ,1, χ% , φ ) + ∑ φ , χ%

t

φ , χ%

−∑ φ , χ%

t

t

t

t

β t (θ , 0, χ% , φ )ct1 (θ , φ , χ% ) px |θ ,κ , χ% ,φ ,Y ( x | θ ,1, χ% , φ ) (1 − ct1 (θ ,φ , χ% ) ) t

t

t

t

t

t

which implies (22) with: β t0 (θ , χ% , φ ) =

β t (θ , 0, χ% , φ ) , and (1 − ct1 (θ , φ , χ% ) )

βt1 (θ , χ% , φ ) = β t (θ ,1, χ% , φ ) − β t (θ , 0, χ% , φ )

ct1 (θ , φ , χ% ) (1 − ct1 (θ , φ , χ% ) )

Substitution of (18) and subsequent evaluation, using (H2) and (H3), yield: βt0 (θ , χ% , φ ) =

βt1 (θ , χ% , φ ) =

1 0 Ft (θ , χ% , φ )λ ( 2−D (φ ) ) pφt |κt (φ | 0) pθt |Yt −1 (θ ) ct

1 1 Ft (θ , χ% ,φ )ct (θ )λ ( 2− D (φ )) pφt |κt (φ |1) pθt |Yt −1 (θ ) − βt0 (θ , χ% ,φ )ct1 (θ , χ% , φ ) = ct

1 1 1 Ft (θ , χ% ,φ )ct (θ )λ ( 2− D(φ )) pφt |κt (φ |1) pθt |Yt−1 (θ ) − Ft1 (θ , χ% ,φ )ct (θ )λ ( 2− D(φ )) pφt |κt (φ | 0) pθt |Yt−1 (θ ) = ct ct 1 = Ft1 (θ , χ% ,φ )ct (θ )λ( 2−D(φ )) ⎡⎣ pφt |κt (φ |1) − pφt |κt (φ | 0)⎤⎦ pθt |Yt −1 (θ ) ct =

1

which implies eq. (23). From the above also follows

1

β κ (θ , χ% , φ ) = ∑ β (θ , κ , χ% , φ ) . Hence c ∑ κ κ =0

normalizes not only β t (θ , κ , χ% , φ ) but also β tκ (θ , χ% , φ ) .

t

=0

t

t

Q.E.D.

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5

Conditional mean and covariance

Having characterized a set of equations for the measurement update of the conditional density, our next step is to assume that the joint-mode-conditionally predicted joint target state has a density px |θ ,Y ( x | θ ) which is Gaussian for each joint θ . t

t −1

t

Theorem 2 For each θ ∈ {1,..., N }2 , let px |θ ,Y ( x | θ ) be Gaussian with mean xt (θ ) , covariance Pt (θ ) and t

t

t −1

let pθ |Y (θ ) > 0 . Then px |θ ,Y ( x | θ ) is a Gaussian mixture, with overall weight pθ |Y (θ ) , overall t

t −1

t

t

t

t t

mean xˆt (θ ) and overall covariance Pˆt (θ ) , satisfying: pθt |Yt (θ ) =

∑ β κ (φ , χ% ,θ )

κ ,φ , χ%

(28)

t

where β tκ (φ , χ% ,θ ) satisfies (25), and: xˆt (θ ) = ∑ β t1|θ ( χ% , φ ) xˆt1 (θ , χ% , φ ) + ∑ β t0|θ ( χ% , φ ) xˆt0 (θ , χ% , φ ) φ , χ%

(29)

φ , χ%

T 1 ⎤ Pˆt (θ) = ∑βt1|θ (χ%,φ) ⎡⎣Pˆt1(θ,φ) + ⎡⎣xˆt1(θ, χ%,φ) − xˆt (θ)⎤⎡ ⎦⎣xˆt (θ, χ%,φ) − xˆt (θ)⎤⎦ ⎦ +

φ,χ%

T 0 ⎤ +∑βt0|θ (χ%,φ) ⎡⎣Pˆt0 (θ,φ) + ⎡⎣xˆt0 (θ, χ%,φ) − xˆt (θ)⎤⎡ ⎦⎣xˆt (θ, χ%,φ) − xˆt (θ)⎤⎦ ⎦

(30)

φ,χ%

with: β tκ|θ (φ , χ% ) = β tκ (φ , χ% ,θ ) /pθ |Y (θ )

(31)

xˆtκ (θ , φ , χ% ) = xtκ (θ ) + K tκ (θ , φ ) μ tκ (φ , χ% ,θ )

(32.a)

Pˆtκ (θ , φ ) = Pt κ (θ ) − Ktκ (θ , φ )Φ(φ ) H (θ ) Pt κ (θ )

(32.b)

t t

where: μtκ (φ , χ% ,θ ) = χ% yt − Φ(φ ) H (θ ) xtκ (θ )

(33.a)

⎡0 ⎤ Ktκ (φ ,θ ) = Pt κ (θ ) H (θ )T Φ(φ )T Qtκ (φ ,θ )−1 if φ ≠ ⎢ ⎥ , ⎣0 ⎦ ⎡0⎤ if φ = ⎢ ⎥ =0 ⎣0⎦

(33.b)

xtκ (θ ) = xt (θ ) − κ Kt (θ ) [ I M − I ] H (θ ) xt (θ )

(34.a)

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Pt κ (θ ) = Pt (θ ) − κ Kt (θ ) [ I M − I ] H (θ ) Pt (θ )

(34.b)

⎡I ⎤ K t (θ ) = Pt (θ ) H (θ )T ⎢ ⎥ Qt (θ ) −1 ⎣− I ⎦

(34.c)

⎡I⎤ Qt (θ ) = [ I M − I ] ⎡⎣ H (θ )Pt (θ )H (θ )T ⎤⎦ ⎢ ⎥ + R(θ ) ⎣−I ⎦

(34.d)

1 1 Ftκ (φ, χ%,θ ) = exp{− μtκ (φ, χ% ,θ )T Qtκ (φ,θ )−1 μtκ (φ, χ% ,θ )}/[(2π )mD(φ ) Det{Qtκ (φ,θ )}]2 , if φ ≠ {0}2 2 , if φ = {0}2 =1

(35.a)

Qtκ (φ,θ ) = Φ(φ )(H (θ ) Pt κ (θ ) H (θ )T + G(θ )G(θ )T )Φ(φ )T

(35.b)

ct (θ ) =

R(θ )

1 2

⎧⎪ 1 ⎫⎪ ⎡I⎤ ⋅ exp ⎨− xt (θ )T H (θ )T ⎢ ⎥ Qt (θ )−1 [ I M − I ] H (θ ) xt (θ )⎬ ⎪⎩ 2 ⎪⎭ ⎣−I ⎦ Qt (θ ) 1 2

(36)

Proof outline: Eqs. (28) through (31) follow from integrations over (22). Completing the square in equation (20) yields (34.a,b,c,d) and (36). Completing the square in equation (24) and (25) yield (32.a,b), (33.a,b) and (35.a,b).

Q.E.D.

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6

Joint IMM Coupled PDA filter with Resolution

Proposition 4 and Theorem 2 provide conditional characterizations for the joint targets modes and states. Here we use these equations to specify the JIMMCPDAR filter algorithm (this acronym stands for Joint IMM Coupled PDA Resolution). A filter cycle starts with, for each θ ∈ {1,..., N }2 , conditional mode probability pθ |Y (θ ) and conditional mean and covariance : t −1 t −1

Δ xˆt −1 (θ ) E{xt −1 | θ t −1 = θ , Yt −1} , and Δ Pˆt −1 (θ ) E{[ xt −1 − xˆt −1 (θ )][ xt −1 − xˆt −1 (θ )]T | θ t −1 = θ , Yt −1}

One filter cycle consists of the following seven steps. JIMMCPDAR Step 1: Interaction Equal to step 1 of JIMMCPDA [14]. JIMMCPDAR Step 2: Prediction Equal to step 2 of JIMMCPDA [14]. JIMMCPDAR Step 3: Merging prediction: For all θ ∈ {1 ,..., N }2 , evaluate equations (34.a,b,c,d). JIMMCPDAR Step 4: Gating: Following [10], now per κ value: Let Qtκ ,i (θ ) be the i-th m × m diagonal block matrix of the κ -conditional predicted Qtκ (θ ) , with Qtκ (θ ) = H (θ ) Pt κ (θ ) H (θ )T + G (θ )G (θ )T

(37)

Identify for each target i and κ -value the mode θ t κ ,i for which Det Qtκ ,i (θ ) is largest: θ t κ ,i = Argmax{DetQtκ ,i (θ )} θ

(38)

and identify for each target i a κ -dependent gate Gtκ ,i ∈ R m as follows: Gtκ ,i = { z i ∈ R m ;[ z i − hi (θ t κ ,i ) xtκ ,i (θ t κ ,i )]T ⋅ Qtκ ,i (θ t κ ,i ) −1 ⋅ [ z i − hi (θ t κ ,i ) xtκ ,i (θ t κ ,i )] ≤ ν }

(39)

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with ν the gate size. If the j -th measurement y tj falls outside gate Gtκ ,i ; i.e. ytj ∉ Gtκ ,i , then under κ t = κ the j -th component of the i -th row of [Φ (φ )T χ% ] is assumed to equal zero at moment t . Per κ -value this reduces the set of possible detection/permutation hypotheses to be evaluated for χ% t given φt = φ to χ% ∈ X%tκ (φ ) . JIMMCPDAR Step 5: Hypothesis evaluation. Using (23) as approximation and adapting the Pdi and Pd0 in (10) for reduced detection probability due to limited gate size ν yields: βtκ (φ, χ% ,θ ) ≅ Ftκ (φ, χ% ,θ )ct (θ )κ λ ( 2− D(φ )) ⋅ ⎡ pφ |κ (φ | κ ) − κ pφ |κ (φ | 0)⎤ pθ |Y (θ )/ct for χ% ∈ X%tκ (φ ) ⎣

t

t

t

t



t

t −1

for χ% ∉ X%tκ (φ )

=0 pφ |κ (φ | κ ) = [∏i =1 (1− Pdi ⋅ Chi 2m(ν )) 2

t

= (1− P ⋅ Chi (ν )) 0 d

2 m

( P ⋅ Chi (ν )) ]

(1−φi )

t

(1−D(φ ))

i d

2 m

( P ⋅ Chi (ν )) 0 d

2 m

φi

D(φ )

=0

(40)

if κ =0, φ ∈{0,1}2 ⎧ ⎡ 1 ⎤⎫ ⎪⎪⎡0⎤ ⎢ 2 ⎥⎪⎪ if κ =1, φ ∈ ⎨⎢ ⎥ , ⎢ ⎥⎬ ⎪⎣0⎦ ⎢ 1 ⎥⎪ ⎢⎣ 2 ⎥⎦⎭⎪ ⎩⎪

(41)

else

with μ tκ (φ , χ% ,θ ) , Ftκ (φ , χ% ,θ ) , Qtκ (φ ,θ ) and ct (θ ) to be evaluated by (33.a), (35.a,b) and (36) respectively, with ct normalizing β tκ (φ , χ% , θ ) , and Chi 2m (⋅) the Chi-squared cumulative distribution function with m degrees of freedom. JIMMCPDAR Step 6: Measurement-based update, by evaluating equations (28) - (32) and (33.b). JIMMCPDAR Step 7: Output equations: xˆt =

Pˆt =



θ ∈{1,..., N }2



θ ∈{1,..., N }

2

pθt |Yt (θ ) ⋅ xˆt (θ )

(42)

pθt |Yt (θ )( Pˆt (θ ) + [ xˆt (θ ) − xˆt ] ⋅ [ xˆt (θ ) − xˆt ]T )

(43)

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7

Track-coalescence-avoiding JIMMCPDAR filter

A shortcoming of JIMMCPDA is its sensitivity to track coalescence. With the JIMMCPDA* approach, [15] has shown that this is due to JIMMCPDA’s merging over permutation hypotheses, and that a suitable hypothesis pruning may provide an effective countermeasure. In order to develop such a pruning for JIMMCPDAR we need to introduce some additional processes and notation. Following [8], χ% t can be written as: χ% t = χ t Φ (ψ t )

if

Dt > 0.

(44)

where χ t is a Dt × Dt permutation matrix, which is conditionally independent of φt given Dt , and where ψ t = [ψ 1,t ... ψ L ,t ]T , with ψ i ,t ∈ {0,1} the target indicator at moment t for measurement t

i , which assumes the value one if measurement i belongs to a detected target and zero if measurement i comes from clutter. ψ t is conditionally independent of φt and χ t given Dt and Lt . Moreover, {χ t } and {ψ t } are i.i.d. sequences.

The JIMMCPDA* filter equations are obtained from the JIMMCPDA algorithm by pruning per (φt ,ψ t , θ t ) -hypothesis all except the most likely χ t -hypothesis prior to measurement updating. The physical explanation why this is working for two targets has been given by [5]. If targets move closely spaced for a longer period of time, then the pdf of the individual targets tend to become of a symmetric form invariant against a permutation of the objects. Because for two targets permutations are possible for κ t = 0 and φt = [1 1]T only, the JIMMCPDA* hypothesis pruning strategy needs to be extended for that case only. For κ = 0 and φ = [1 1]T evaluate all (ψ , θ ) hypotheses and prune per such (ψ , θ ) -hypothesis all except the most likely χ -

hypothesis. To do so, define for every ψ and θ a mapping χˆ t (ψ , θ ) : Δ

χˆ t (θ ,ψ ) Argmax β t0 (θ , χΦ (ψ ), [1 1]T )

(45.a)

χ

with maximization over all permutation matrices χ of size 2 × 2 . The strategy of evaluating for κ = 0 and φ = [1 1]T all (ψ ,θ ) -hypotheses and only one χ -hypothesis implies that we adopt per (φ , χ% , θ ) -hypothesis the following hypothesis weights βˆ κ (θ , χ% ,φ ) : t

βˆtκ (θ , χΦ(ψ ),φ ) = βtκ (θ , χΦ(ψ ),φ ) / cˆt =0

if D(φ ) ≤ 1 or if D(φ ) = 2 and χ = χˆ (θ ,ψ ) else

with cˆ t a normalization constant for βˆtκ ; i.e. such that



κ ,φ , χ% ,θ

βˆtκ (θ , χ% , φ ) = 1

(45.b)

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Inserting these particular weights within JIMMCPDAR, yields JIMMCPDAR*, consisting of the following cycle of of 8 steps (the first five are equivalent to the first five JIMMCPDAR steps): JIMMCPDAR* Steps 1-5: Equivalent to JIMMCPDAR Steps 1-5. JIMMCPDAR* Step 6: Hypothesis pruning. Evaluate the new hypothesis weights using eqs. (45.a,b). JIMMCPDAR* Step 7: Measurement-based update: Equivalent to JIMMCPDAR Step 6, but with β tκ replaced by βˆtκ . JIMMCPDAR* Step 8: Output equations: Equivalent to JIMMCPDAR Step 7.

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8

Discussion of results

This paper incorporated resolution submodels of [3] and [5] within the descriptor system approach of [8], [14] and [15] towards filtering two Markov jump linear targets from possibly unresolved, missing and false observations. For this descriptor system we developed the exact Bayesian measurement update equations (Theorem 1), and novel filter algorithms, JIMMCPDAR and JIMMCPDAR*. For closely spaced targets these filter algorithms differ significantly from the JIMMCPDA and JIMMCPDA* filter algorithms. In follow-up work the effectivity of these new filters will be shown through Monte Carlo simulations relative to JIMMCPDA and JIMMCPDA*, and relative to a good particle filter. For such particle filter the exact Bayesian filter equations of Theorem 1 will be used.

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9

References

[1] F.E. Daum, A system approach to multiple target tracking, Ed: Y. Bar-Shalom, MultitargetMultisensor Tracking: applications and advances, Volume II, Artech House, 1992, pp. 149-181. [2] F.E. Daum, R.J. Fitzgerald, The importance of resolution in multiple target tracking, SPIE Signal & Data Processing of Small Targets, 1994, Vol. 2238, pp. 329-338. [3] K.C. Chang, Y. Bar-Shalom, Joint Probabilistic Data Association for multitarget tracking with possibly unresolved measurements and maneuvers, IEEE Tr. Automatic Control, Vol. 29 (1984), pp. 585-594. [4] S. Mori, K.C. Chang, C.Y. Chong, Tracking aircraft by acoustic sensors – multiple hypothesis approach applied to possibly unresolved measurements, Proc. ACC, 1987, pp. 1099-1105. [5] W. Koch, G. Van Keuk, Multiple hypothesis track maintenance with possibly unresolved measurements, IEEE Tr. AES, Vol. 33 (1997), pp. 883-892. [6] W. Koch, Fixed-Interval Retrodiction Approach to Bayesian IMM-MHT for Maneuvering Multiple Targets, IEEE Trans. Aerospace and Electronic Systems, Vol. 36, No. 1, January 2000, pp. 2-14. [7] E.P. Blasch, T. Connare, Improving track maintenance through group tracking, Proc. Workshop on Estimation, Tracking and Fusion: A tribute to Yaakov Bar-Shalom, Monterey, CA, May 2001, pp. 360-371. [8] H.A.P. Blom, E.A. Bloem, Probabilistic Data Association avoiding track coalescence, IEEE Tr. Automatic Control, Vol. 45 (2000), pp. 247-259. [9] Y. Bar-Shalom, T. E. Fortmann, Tracking and data association, Academic Press, 1988. [10] Y. Bar-Shalom, X. R. Li, Multitarget-Multisensor Tracking: Principles and Techniques, YBS Publishing, Storrs, CT, 1995. [11] H. A. P. Blom, E. A. Bloem, Combining IMM and JPDA for tracking multiple maneuvering targets in clutter, Proc. 5th Int. Conf. On Information Fusion, July 8-11, 2002, Annapolis, MD, USA, Vol. 1, pp. 705-712. [12] H. A. P. Blom and E. A. Bloem, Interacting Multiple Model Joint Probabilistic Data Asscoiation avoiding track coalescence, Proc. IEEE Conf. On Decision and Control, 2002, pp. 3408-3415. [13] B. Chen, J. K. Tugnait, Tracking of multiple maneuvering targets in clutter using IMM/JPDA filtering and fixed-lag smoothing, Automatica, vol. 37, pp. 239-249, Feb. 2001. [14] H.A.P. Blom, E.A. Bloem, Tracking multiple maneuvering targets by Joint combinations of IMM and PDA, Proc. 42nd IEEE Conf. on Decision and Control, 2003, pp. 2965-2970. [15] H.A.P. Blom, E.A. Bloem, Joint IMM and Coupled PDA to track closely spaced targets and to avoid track coalescence, Proc. 7th Int. Conf. on Information Fusion, July 2004, Stockholm, pp. 130-137. [16] H.A.P. Blom, E.A. Bloem, Joint IMMPDA Particle filters, Proc. 6th Int. Conf. on Information Fusion, 2003, Vol. 1, pp. 785-792. [17] J.K. Tugnait, Tracking of multiple maneuvering targets in clutter using multiple sensors, IMM and JPDA Coupled Filtering, IEEE Tr. AES, Vol. 40, 2004, pp. 320-330. [18] L. Dai, “Singular control systems”, Lecture notes in Control and information sciences, Vol. 118, Springer, 1989.

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Appendix A Acronyms CPDA

Coupled PDA

CPDA*

Track-coalescence-avoiding CPDA

IMM

Interacting Multiple Model

IMMJPDA

Interacting Multiple Model Joint Probabilistic Data Association

IMMJPDA*

Track-coalescence-avoiding IMMJPDA

IMMPDA

Interacting Multiple Model Probabilistic Data Association

JIMMCPDA

Joint Interacting Multiple Model Coupled Probabilistic Data Association

JIMMCPDA*

Track-coalescence-avoiding JIMMCPDA

JIMMCPDAR

JIMMCPDA with Resolution

JIMMCPDAR*

Track-coalescence-avoiding JIMMCPDAR

JPDA

Joint PDA

JPDA*

Track-coalescence-avoiding JPDA

MHT

Multiple Hypotheses Tracking

PDA

Probabilistic Data Association

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Appendix B List of Symbols a i (θ i )

Target i’s state transition matrix of size n × n as a function of mode θ i

A(θ ) b (θ )

Joint targets state transition matrix as a function of joint mode θ Target i’s state noise gain matrix of size n × n′ as a function of mode θ i

B(θ )

Joint targets state noise gain matrix as a function of joint mode θ

Dt

Total number of detected targets at moment t

Ft φi ,t

Total number of false measurements at moment t

i

i

Detection indicator for target i at moment t

φt

Detection indicator vector at moment t, containing the detection indicators for all targets at moment t

Φ g i (θ i )

Matrix operator to link the detection indicator vector with the measurement model Target i’s measurement noise gain matrix of size m × m′ as a function of mode θ i

G (θ ) χ% t

Joint targets measurement noise gain matrix as a function of joint mode θ

χ% t

(0,1)-matrix that is used to randomly select target measurements from the measurement vector yt “Inflated” χ% t matrix of proper size such that it randomly selects target measurements from the measurement vector yt by means of matrix multiplication.

h (θ )

Target i’s state-to-measurement transition matrix of size m × n as a function of mode θ i

H (θ )

Joint targets state-to-measurement transition matrix as a function of joint mode θ

I Ir

Unit-matrix

i

i

Diagonal matrix with its i-th diagonal equal to the i-th element of the vector r

κt

Merging indicator at moment t

Lt

The number of measurements at moment t

λ

Spatial density of false measurements

M

Total number of targets

M N Pdi Πηθ

Set of possible modes of target

Π

ψ i.t ψt

Total number of modes of a target Detection probability of target i Transition probability of a target switching from mode η to mode θ Transition probability matrix Target indicator for measurement i at moment t Target indicator vector at moment t, containing the target indicators for all measurements at moment t

ri

Resolution capability factor for the i-th noise component of a potential target

R(θ )

measurement Resolution capability matrix of size m × m

θti

Mode of target i at moment t

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θt i t

v

Joint targets mode at moment t Sequence of i.i.d. standard Gaussian variables of dimension m′ representing the measurement noise for target i

vt v

Joint targets measurement noise vector Column vector of Ft i.i.d. false measurements

V wti

Volume of the observed region Sequence of i.i.d. standard Gaussian variables of dimension n′ representing the system

* t

noise for target i

wt i t

x

xt k t

Joint targets system noise vector

n-vectorial state of target i at moment t Joint targets state vector at moment t

y

k-th measurement at moment t

yt Yt

Measurement vector at moment t, containing all measurements at moment t σ -algebra generated by measurements up to and including moment t

zti

m-vectorial potential measurement of target i at moment t

zt

Joint measurements vector at moment t, containing the potential measurements of all targets at moment t

z%t

Joint measurements vector at moment t, containing the potential measurements of all detected targets at moment t in a fixed order