Combining IMM and JPDA for tracking multiple maneuvering targets in

Combining IMM and JPDA for tracking multiple maneuvering targets in clutter Henk A.P. Blom and Edwin A. Bloem National Aerospace Laboratory NLR P.O. Box 90502, 1006 BM Amsterdam, The Netherlands [email protected] & [email protected] Abstract – The paper combines IMM and JPDA for tracking of multiple possibly maneuvering targets in case of clutter and possibly missed measurements while avoiding sensitivity to track coalescence. The effectiveness of the filter is illustrated through Monte Carlo simulations. Keywords: Bayesian estimation, Multitarget tracking, Sudden maneuvers, Clutter, Missed detections, Hidden Markov model, Descriptor system.

1

Introduction

We consider the problem of tracking multiple maneuvering targets in clutter with a proper combination of two well known approaches in target tracking: IMM and JPDA. Since each of these two solve complementary tracking problems one might expect that it should be useful to combine these two approaches. In literature the problem of combining IMM (Blom & Bar-Shalom, 1988) and JPDA (Bar-Shalom and Fortmann, 1988) has been studied by BarShalom et al. (1992), DeFeo et al. (1997) and Chen and Tugnait (2001). Bar-Shalom et al. (1992) developed an IMMJPDA-Coupled filter for situations where the measurements of two targets are unresolved during periods of close encounter. In Blom & Bloem (2000) it has been shown that these IMMJPDA-Coupled filter equations are rather heuristic. Chen and Tugnait (2001) developed an IMMJPDAUncoupled fixed-lag smoothing algorithm with IMMJPDA uncoupled tracking as a special case. They also showed that the IMMJPDA of De Feo et al. (1997) does not account for ”interactions” between the target modes. All in all, in spite of the significant headway which has been made regarding the combination of IMM and JPDA, there is a lack of insight in the proper choices to be made when combining IMM and JPDA for multiple maneuvering target tracking. In order to improve this situation, the paper studies the problem of combining IMM and JPDA following an approach that is based on recent new insight gained regarding the derivation of a track coalescence avoiding JPDA version

ISIF © 2002

705

(Blom & Bloem, 2000). The basis for this development is to embed the multi target tracking problem with possibly false and missing measurements into one of filtering for a linear descriptor system with random coefficients. In this paper this embedding approach is extended towards the development of various IMMJPDA filters, and it is shown how these compare with known IMMJPDA filters. The paper is organized as follows. Section 2 develops the stochastic model for the tracking problem considered. Section 3 presents exact filter equations. Section 4 develops IMMJPDA filter equations. Section 5 develops the track coalescence avoiding IMMJPDA filter equations. Section 6 shows the effectiveness of the approach through Monte Carlo simulation results.

2

Stochastic modeling

This section describes the target model and the measurement model.

2.1

Target model

Consider M targets and assume that the state of the i-th target is modeled as a jump linear system:

xit+1 = ai (ti+1 )xit + bi (ti+1 )wti ;

i = 1; :::; M; (1)

where xit is the n-vectorial state of the i-th target, ti is the mode of the i-th target and assumes values from f1; ::; N g, ai (ti ) and bi (ti ) are (n n)-matrices and wti is a sequence of i.i.d. standard Gaussian variables of dimension n with wti , wtj independent for all i 6= j and wti ,xi0 ; xj0 inde-

6=

4

4

j . Let xt = Colfx1t ; :::; xM t g, t = 4 1 M 1 1 Colft ; :::; t g, A(t ) = Diagfa (t ); :::; aM (tM )g, 4 4 B (t ) = Diagfb1 (t1 ); :::; bM (tM )g, and wt = Colfwt1 ; :::; wtM g. Then we can model the state of our M

pendent for all i

targets as follows:

xt+1 = A(t+1 )xt + B (t+1 )wt

(2)

2.2 Measurement model A set of measurements consists of measurements originating from targets and measurements originating from clutter. Firstly the measurements originating from targets are treated. Subsequently the clutter measurements are randomly inserted between the target measurements. A Measurements originating from targets We assume that a potential measurement associated with state xit (which we will denote by zti ) is modeled as a jump linear system:

zti = hi (ti )xit + g i (ti )vti

; i = 1; :::; M

(3)

where zti is an m-vector, hi (ti ) is an (m n)-matrix and g i (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 6= j . Moreover vti is in-

4

dependent of xj0 and wtj for all i,j . Next with zt = 4 Colfzt1 ; :::; ztM g, H (t ) = Diagfh1 (t1 ); :::; hM (tM )g,

4

G(t ) = Diagfg 1 (t1 ); :::; g M (tM )g, and vt Colfvt1 ; :::; vtM g, we obtain: zt = H (t )xt + G(t )vt

=4

(4)

We next introduce a model that takes into account that not all targets have to be detected at moment t, which implies that not all potential measurements zti have to be available as true measurements at moment t. To this end, let Pdi be the detection probability of target i and let i,t 2f0,1g be the detection indicator for target i, which assumes the value one with probability Pdi > 0, independently of j;t , j 6= i. This approach yields the following detection indicator vector t of size M : 4 t = Colf1;t ; :::; M;t g:

4 Thus, the number of detected targets is Dt =

PM

i=1 i;t . Furthermore, we assume that ft g is a sequence of i.i.d. vectors. In order to link the detection indicator vector with the measurement model, we introduce the following operator : for an arbitrary (0,1)-valued M 0 -vector 0 we define

4 P 0 0 and the operator producing (0 ) as D(0 ) = M i=1 i a (0; 1)-valued matrix of size D(0 ) M 0 of which the ith row equals the ith non-zero row of Diagf0 g. Next we define, for Dt > 0, a vector that contains all measurements originating from targets at moment t in a fixed order. 4

4 ( ) I ; (t ) = t m with Im a unit-matrix of size m, and the tensor product. z~t = (t )zt ; where

In reality, however, we do not know the order of the targets. Hence, we introduce the stochastic Dt Dt permutation matrix t , which is conditionally independent of ft g. We

706

also assume that ft g is a sequence of independent matrices. Hence, for Dt > 0,

4

4

z~t = t z~t ; where t = t Im ; is a vector that contains all measurements originating from targets at moment t in a random order. B Measurements originating from clutter Let the random variable Ft be the number of false measurements at moment t. We assume that Ft has Poisson distribution:

pFt (F ) =

V )F F!

(

exp

V ,

= 0,

F = 0, 1, 2, . .. else

where is the spatial density of false measurements (i.e. the average number per unit volume) and V is the volume of the validation region. Thus, V is the expected number of false measurements in the validation gate. We assume that the false measurements are uniformly distributed in the validation region, which means that a column-vector vt of Ft i.i.d. false measurements has the following density:

pvt jFt (v jF ) = V

F

where V is the volume of the validation region. Furthermore we assume that the process fvt g is a sequence of independent vectors, which are independent of fxt g; fwtg; fvt g and ft g. C Random insertion of clutter measurements Let the random variable Lt be the total number of measurements at moment t. Thus,

Lt = Dt + Ft

4

With ~ yt = Colfz~t ; vt g, it follows with the above defined variables that 2

3

t (t )zt ~yt = 4 :::::::::::::: 5 ; if Lt > Dt > 0 vt

(5)

whereas the upper and lower subvector parts disappear for Dt = 0 and Lt = Dt respectively. With this equation, the measurements originating from clutter still have to be randomly inserted between the measurements originating from the detected targets. To do so, we first define target indicator and clutter indicator processes, denoted by f t g and f t g, respectively. Let the random variable i,t 2f0,1g be a target indicator at moment t for measurement i, which assumes the value one if measurement i belongs to a detected target and zero if measurement i comes from clutter. This approach yields the following target indicator vector t of size Lt : 4 t = Colf 1;t ; :::; Lt ,t g:

Let the random variable i,t 2f0,1g be a clutter indicator at moment t for measurement i, which assumes the value one if measurement i comes from clutter and zero if measurement i belongs to an aircraft (thus i,t = 1 i,t ). This approach yields the following clutter indicator vector t of size Lt : 4 t = Colf 1;t ; :::; Lt ,t g: In order to link the target and clutter indicator vectors with the measurement model, we make use of the operator introduced before. With this the measurement vector with clutter inserted reads as follows:

yt

. = ( t )T .. ( t )T ~yt

if Lt > Dt > 0

QM

Theorem Let pt jYt 1 () = i=1 pti jYt 1 (i ) and let pxt jt ;Yt 1 (xj) be Gaussian with mean xt () = Colfx1t (1 ); :::; xM (M )g and covariance Pt () = DiagfPt1(1 ); :::;tPtM (M )g, then t (; ~; ) satisfies for 6= f0gM :

t (; ~; ) = (Lt

D())

M h Y i=1

Pdi )(1

fti (; ~; i )(1

i )

i

(Pdi )i ptijYt 1 (i ) =ct (11)

with: (6)

This, together with equation (2), (4) and (5), forms a complete characterization of our tracking problem in terms of stochastic difference equations.

fti (; ~; i ) = [(2 )m DetfQit (i )g]

expf 12 k

Lt X

1 2 i

[()]Ti ~k ikt (i )T [Qit (i )]

1

i ik t ( )

g

=1

(12.a) where:

3

4 hi (i )xit (i ) (12.b) 4 Qit (i ) = hi (i )Pti (i )hi (i )T + g i (i )g i (i )T (12.c) k i ik t ( ) = yt

Exact filter equations

Next we introduce an auxiliary indicator process ~t as follows: 4 ~t = Tt ( t ) if Dt > 0: Following the approach of Blom & Bloem (2000) equations (4), (5) and (6) can be transformed to:

~t yt = (t )H (t )xt + (t )G(t )vt if Dt > 0 (7) Notice that (7) is a linear Gaussian descriptor system (Dai, 1989) with stochastic i.i.d. coefficients (t ) and ~t . From (7), it follows that for Dt > 0 all relevant associations and permutations can be covered by (t ; ~t )-hypotheses. We extend this to Dt = 0 by adding the combination ~t = fgLt M and t = f0g . Hence, through defining the weights

4

t (; ~; ) = Probft = ; ~t = ~; t = j Yt g; where Yt denotes the -algebra generated by measurements yt up to and including moment t, then the law of total probability yields:

pt jYt () = pxt ;t jYt (x; ) =

X ; ~

X ; ~

t (; ~; )

(8)

t (; ~; )pxt jt ;t ;~t ;Yt (x j ; ; ~) (9)

Since

pxt jt ;Yt (x j ) = pxt ;t jYt (x; )=pjYt ()

(10)

our problem is to characterize the right-hand terms in (9). This has been accomplished in the following Theorem.

707

~k are the i-th and k-th columns of whereas [()]i and () and ~, respectively. Moreover, pxit jti;Yt (xi ji ), i 2 f1; :::; M g, is a Gaussian mixture, while its overall mean xît (i ) and its overall covariance P^ti (i ) satisfy: X

pti jYt (i ) =

;; ~ i =i

^ ( )= ( )+

xit

i

xit

i

t (; ~; ) Wti

^ ( ) = ( )

Pti

i

Pti

i

+ ( )

Wti

i

Wti

( )

Lt X k0 =1

i

0 tik

Lt X k=1 Lt X k=1

i

k=1

tik

( ) ( ) i

ik t

i

i

Pti

i

k=1

( ) ( ) ( )

tik

( ) ( )

i

ik t

i

ik t

i T

!

i

Lt X

( )h ( ) ( ) i

tik

!

(13.b) !

tik

( ) + i

Wti (i )T

+

!

0 ik t

( ) i

Lt X

( )

Wti

(13.a)

i

ik t

!T

( ) i

i

Wti (i )T

(13.c)

with:

Wti (i ) = Pti (i )hi (i )T [Qit (i )] 1 (13.d) X tik (i ) = ()Ti ~k t (; ~; )]=pti jYt (i ) (13.e) ;; ~ 6=0

i =i

Proof: See Blom & Bloem (2002)

4

IMMJPDA filter

In this section the IMMJPDA filter algorithm is specified. To do so use is made of the IMM filter algorithm and of the Theorem. One cycle of the IMMJPDA filter algorithm consists of the following six steps. IMMJPDA Step 1: For each target this comes down to the mixing/interaction step of the IMM algorithm (Blom & Bar-Shalom, 1988) for all i 2 f1, . . . ,M g: Starting with the weights

^ti

1

(i ) =4 pti 1 jYt 1 (i );

i 2 f1; :::; N g

xît

1

t

1

jti = i ; Yt g;

i 2 f1; :::; N g

1

1

and the associated covariances

P^ti

1

4 E f[xi (i ) = t

1

j ti

xît 1

(i )][xit xît (i )]T j = i ; Yt g; i 2 f1; :::; N g 1

1

1

1

one evaluates the mixed initial condition for the filter matched to ti = i as follows:

ti (i ) =

N X i =1

xît 1jti (i ) = P^ti 1jti (i ) =

i ;i ^ti (i ) 1

N X i =1 N X i =1

Qit (i ) = hi (i )Pti (i )hi (i )T

Subsequently identify for each target the mode for which Det Qit (i ) is largest:

ti = Argmax fDet Qit (i )g i

4

with the gate size. If the j -th measurement yjt falls = Git , then the j -th component outside gate Git ; i.e. yjt 2 T of the i-th row of [() ~t ] is assumed to equal zero. This reduces the set of possible detection/permutation hypotheses to be evaluated at moment t for various to X~t (). IMMJPDA Step 4: Evaluation of the detection / association / mode hypotheses is based on the Theorem. For all 2 f0; 1gM , ~ 2 f0; 1gD()D(), 2 f1; :::; N gM :

i ;i ^ti (i ) xît (i )= ti (i ) 1

i ;i ^ti (i )

P^ti (i ) + [^xit (i ) xît jti (i )] [^xit (i ) xît jti (i )]T = ti (i ) 1

1

1

= Lt (1 (

D()) QM f i ; ; i i=1 t Pdi (1 i ) Pdi i ti i

1

1

)

=0 with fti (f0gM ; fgLt ; i )

1

2 IRm as

Git = fz i 2 IRm ; [z i hi (ti )xit (ti )]T Qit (ti ) 1 [z i hi (ti )xit (ti )] g

t (; ~; )

1

+ gi (i )gi (i )T

and use this to define for each target i a gate Git follows:

the means

(i ) =4 E fxi

IMMJPDA Step 3: Gating, which is based on Bar-Shalom & Li (1995). Evaluate for each i and i the crosscovariance as follows:

[ ( ~ ) ( ) ( )=ct for ~ 2 X~t (); else

= 1 and for 6= f0gM :

[(2)m DetfQi (i )g] fti (; ~; i ) = t

(15.a)

1 2 i

Lt X expf 12 [()Ti ~k ikt (i )T [Qit (i )] k

1

i ik t ( )]g

=1

(15.b)

with

4 P r f i = i j i = i g i ;i = t t 4 i i i

t ( ) = pti jYt 1 ( ) 4 xît jti (i ) = E fxit j ti = i ; Yt g 4 P^ti jti (i ) = E f[xit xît (i )] [xit xît (i )]T jti = i ; Yt g IMMJPDA Step 2: Prediction for all i 2 f1, . . .,M g, i 2 f1, . . . ,N g : xit (i ) = ai (i )^xit jti (i ) (14.a) Pti (i ) = ai (i )P^ti jti (i )ai (i )T + bi (i )bi (i )T (14.b) 1

1

1

1

1

1

1

1

1

1

1

1

708

where

i k ik t ( ) = yt

hi (i )xit (i )

(15.c)

IMMJPDA Step 5: Measurement update equations are based on the Theorem. For all i 2 f1; :::; M g, i 2 f1; :::; N g :

^ti (i ) =

X ;; ~ i =i

t (; ~; )

(16)

and using (13b, c, d, e) as approximate equations to evaluate x ît (i ) and P^ti (i ).

One cycle of the IMMJPDA* filter algorithm consists of the following 7 steps:

IMMJPDA Step 6: Output equations:

xît = P^ti =

N X i =1 N X

^ti (i ) xît (i )

i =1

^ti (i ) P^ti (i ) + [^xit (i )

IMMJPDA* Step 1: Mixing for all i 2 f1; :::; M g, i 2 f1; :::; N g: Equivalent to IMMJPDA step 1, section 5.

xît ]

IMMJPDA* Step 2: Prediction for all i 2 f1; :::; M g, i 2 f1; :::; N g: Equivalent to IMMJPDA step 2, section 5.

xi T

[^xit (i ) ^t ]

Remark: It can be verified that the above IMMJPDA filter algorithm is similar to the IMMJPDA filter algorithm of Chen & Tugnait (2001). The main new element is that the above specification of IMMJPDA Steps 4 and 5 explicitly show the relation to the processes f ~tg and ft g. In the sequel this relation is exploited for the development of a track coalescence avoiding IMMJPDA filter, for short IMMJPDA* filter.

5

A shortcoming of JPDA is its sensitivity to track coalescence. With their JPDA* approach, Blom & Bloem (2000) have shown that hypothesis pruning can provide an effective track-coalescence avoidance. The JPDA* filter equations can be obtained from the JPDA algorithm by pruning per (t ; t )-hypothesis all less likely t -hypotheses prior to measurement updating. In order to apply this approach to IMMJPDA the JPDA* hypothesis pruning strategy is now extended: evaluate all (t ; t ,t ) hypotheses and prune per (t ; t ,t )-hypothesis all less-likely t -hypotheses. To do so, define for every , and , satisfying D( ) = D() MinfM; Lt g, a mapping ^t (; ,): Argmax t (

; T

( ); )

where the maximization is over all permutation matrices of size D() D(). The pruning strategy of evaluating all (; ; )-hypotheses and only one -hypothesis per (; ; )-hypothesis implies that for D() > 0 we adopt the following pruned hypothesis weights ^t (; ; ):

^t (; ; ) = t (; ^(; ; )T ( ); )=c^t if D() = D( ) MinfM; Lt g

=0

IMMJPDA* Step 5: Track-coalescence hypothesis pruning: First evaluate for every (; ,) such that D( ) = D() MinfM; Lt g:

Next evaluate all ^t (; ,) hypothesis weights:

^t (; ; ) = t (; ^Tt (; ; )( ); )=c^t if 0 < D( ) = D() MinfM; Lt g = t (f0gM ; fgLt ; )=c^t if D( ) = D() = 0

=0

IMMJPDA* Step 6: Measurement update equations for all i 2 f1; :::; M g, i 2 f1; :::; N g :

^ti (i ) =

X

; ; i =i

^t (; ; )

^ ( ) = xi (i ) + W i (i )

xit

i

t

t

P i (i ) P^ti (i ) = t

+Wti (i ) Wti (i )

; ;

D()

else

where c^t is a normalizing constant for ^t .

else

with c^t a normalization constant for ^t ; i.e. such that X ^t (; ; ) = 1 D(

IMMJPDA* Step 4: Evaluation of the detection/evaluation hypotheses: Equivalent to IMMJPDA step 4, section 5.

^t (; ; ) = Argmax t (; T ( ); )

IMMJPDA* filter

4 ^t (; ; ) =

IMMJPDA* Step 3: Gating: Equivalent to IMMJPDA step 3, section 5.

Lt X k0 =1

Wti

Lt X k=1 Lt X k=1

(17.a) Lt X k=1

i ^tik (i )ik t ( )

( )h ( )Pti (i ) i

i

i

Lt X k=1

!

!

(17.b) !

^tik (i ) +

i ik i T ^tik (i )ik Wti (i )T t ( )t ( ) i ^tik (i )ik t ( )

!T

0 0 i ^tik (i )ik t ( )

!

+

Wti (i )T

(17.c)

)=

with:

By inserting these particular weights within IMMJPDA, we get IMMJPDA*.

709

4

Wti (i ) = Pti (i )hi (i )T [Qit (i )]

1

(17.d)

; ; ; 6=0 i =i

[()Ti [^t (;

)]k

; )T (

^t (; where [:]k is the k -th column of [:] .

Trajectories for d > 0

; )]= ^ti (i ) (17.e)

500

IMMJPDA* Step 7: Output Equations: Equivalent to IMMJPDA step 6, section 5.

0

In this section some Monte Carlo simulation results are given for the IMMPDA, IMMJPDA and IMMJPDA* filter algorithms. The simulations primarily aim at gaining insight into the behaviour and performance of the filters when objects move in and out close approach situations, while giving the filters enough time to converge after a manoeuvre has taken place. In the example scenarios there are two targets, each modeled with two possible modes. The first mode represents a constant velocity model and the second mode represents a constant acceleration model. Both objects start moving towards eachother, each with constant initial velocity Vinitial (i.e. the initial relative velocity Vrel, initial = 2V ). At a certain moment in time both objects start decelerating with -0.5 m/s2 until they both have zero velocity. The moment at which the deceleration starts is such that when the objects both have zero velocity, the distance between the two objects equals d. After spending a significant number of scans with zero velocity, both objects start accelerating with 0.5 m/s2 away from each other without crossing until their velocity equals the opposite of their initial velocity. From that moment on the velocity of both objects remains constant again (thus the final relative velocity Vrel, final = Vrel, initial ). Note that d < 0 implies that the objects have crossed each other before they have reached zero velocity. Each simulation the filters start with perfect estimates and run for 40 scans. Examples of the trajectories for d > 0 and d < 0 are depicted in figures 1a and 1b respectively. Table 1: Scenario parameter values. IMMPDA’s = 0:00001 for scenarios 1 and 3 Scenario 1 2 3 4

Pd

d

Vinitial

Ts

1 1 0.9 0.9

0 0.001 0 0.001

Variable Variable Variable Variable

7.5 7.5 7.5 7.5

10 10 10 10

=

ai (ti+1 )xit + bi (ti+1 )wti

−1000

0

100

200 time

0

Trajectories for d < 0

500

d

2

For each target, the underlying model of the potential target measurements is given by (1) and (3)

xit+1

d>0

−500

Monte Carlo simulations

position

6

1000

position

X

^tik (i ) =

1

2

where represents the standard deviation of acceleration i represents the standard deviation of the meanoise and m surement error. For simplicity we consider the situation of i = , Pi = P . similar targets only; i.e. ai = a , m m d d With this, the scenario parameters are Pd , , d, Vinitial , Ts , m , a , 1 , 2 , and the gate size . We used fixed parameters m = 30, a = 0:5, 1 = 500, 2 = 50, and = 25.

Both Tracks O.K. 100

IMMJPDA*

Both Tracks O.K. 100

IMMJPDA

IMMJPDA*

80 percentage

percentage

80 60 40 20

60 IMMJPDA

40 20 IMMPDA

IMMPDA

0

−10

−5

0 d/σm

5

0

10

−10

2a. ”Both tracks O.K.” percentage

mean time in coalescence

5

10

mean time in coalescence 30

25

25

IMMPDA

number of scans

number of scans

0 d/σm


30

20 15 10 IMMJPDA

5 0

−5

−5

0 d/σm

20 15 10 IMMJPDA

5

IMMJPDA*

−10

IMMPDA

5

0

10

IMMJPDA*

−10

−5

0 d/σm

5

10

2b. Average number of ”coalescing” scans


Figure 2: Simulation results for scenario 1


Table 1 gives the other scenario parameter values that are being used for the Monte Carlo simulations. During our simulations we counted track i ”O.K.”, if

the three scenarios are depicted as function of the distance relative to m in two types of figures, showing respectively:

j hi xîT hi xiT j 9m and we counted track i 6= j ”Swapped”, if j hi xîT hj xjT j 9m Furthermore, two tracks i 6= j are counted “Coalescing” at

scan t, if

j hi xît

hj x^jt

j m ^ j hi xit

hj xjt j> m

The percentage of Both tracks “O.K. ” (figures 2a, 3a, 4a, 5a) The average number of “coalescing” scans (figure 2b, 3b, 4b, 5b).

For the examples considered, the simulation results show that both IMMJPDA* and IMMJPDA perform much better than IMMPDA. Moreover the results show that IMMJPDA* avoids track coalescence.

References

For each of the scenarios Monte Carlo simulations containing 100 runs have been performed for each of the tracking filters. To make the comparisons more meaningful, for all tracking mechanisms the same random number streams were used. The results of the Monte Carlo simulations for

711

[1] Y. Bar-Shalom and T.E. Fortmann, Tracking and data association, Academic Press, 1988. [2] H.A.P. Blom and Y. Bar-Shalom, The Interacting Multiple Model algorithm for systems with Markovian

Both Tracks O.K.

Both Tracks O.K.

100

100 IMMJPDA*

80 percentage

percentage

80 60 40

IMMJPDA

20

IMMJPDA* IMMJPDA

60 40 20 IMMPDA

IMMPDA

0

−10

−5

0 d/σm

5

0

10

−10


mean time in coalescence

5

10

mean time in coalescence 30

25

25

IMMPDA

IMMPDA

number of scans

number of scans

0 d/σm


30

20 15 10 IMMJPDA

5 0

−5

−5

0 d/σm

15 10 IMMJPDA

5

IMMJPDA*

−10

20

5

0

10

IMMJPDA*

−10

−5

0 d/σm

5

10





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