A Branching Particle-based Nonlinear Filter for Multi-target Tracking

A Branching Particle-based Nonlinear Filter for Multi-target Tracking David J. Ballantyne University of Alberta Edmonton, AB, Canada [email protected]

Hubert Y. Chan University of Alberta Edmonton, AB, Canada [email protected]

Abstract – A branching particle-based filter is used to detect and track multiple simulated maneuvering ships in a region of water. The ship trajectories exhibit nonlinear dynamics and interact in a nonlinear manner so that the ships do not collide. There is no a priori knowledge of the number of ships in the region. Observations model a sensor tracking the ships from above the region as in a low observable problem. The branching filter simulates particles, each of which is a sample from the domain of possible combinations of ship number and the state of those ships, and each of which is evolved independently using the stochastic law of the signal between observations. The branching filter employs these particles to provide the approximated conditional distribution of the signal in the combined domain, given all observations. Quantitative results recording the capacity of the branching filter to determine the number of ships in the region and the location of each ship are presented.

Michael A. Kouritzin University of Alberta Edmonton, AB, Canada [email protected]

of targets is unknown and that number may change over time. It can be critical in applications to determine how many, if any, targets are located in a region of space based upon a noisy, corrupted sequence of observations. In this paper, we address this problem for multiple interacting targets diffusing in a plane and spatial observations with high noise intensity. For example, Figure 1 depicts our particle system filter trying to detect and track possible targets (in the top left frame) based upon observations corrupted by extreme noise (as shown in the top middle frame). The estimated locations of the targets given that there are one, two, or three targets to track are displayed in the lower left, middle, and right frames, and a greater length of a bar along the bottom of each of these frames indicates a greater estimated probability that the signal contains that number of targets. Any or all of the target locations may not closely represent an actual target, but rather be an artifact of the observation noise.

Keywords: Tracking, particle-based filtering, multiple target, branching filter.

1 Introduction Properly designed adaptive particle systems provide versatile yet under-exploited methods for asymptotic solutions to filtering problems. Hitherto, most practical models have concentrated on the single target scenario. However, the merits of particle systems may be best illustrated in the perennially problematic multiple-target problems. Del Moral and Salut[3] describe the adaptive particle system approach to filtering; our branching method was introduced by Kouritzin and first discussed in Ballantyne, Chan, and Kouritzin [1]; and Salmond and Gordon[8] and Hue, Le Cadre, and P´erez[5] have used particle systems in multiple target problems. Our approach is different than that of Hue et. al. in that the filter is constructed directly by expanding the domain of the signal, rather than introducing Gibbs sampling, and can thus readily handle cases in which the initial number

Figure 1: Example multi-target filter How does one best decide how many targets really exist in the viewing area? In the case that the targets act independently or interact weakly, this question can be answered by imbedding the targets into a measure-valued Markov process. Since we are filtering measures rather

\

than a single target we quickly review the theory of filtering for signals in Polish spaces.

1.1 Filtering continuous Markov signals 
 

Suppose is a probability space, is a com

plete, separable metric space, and is the set of bounded, measurable functions on . Then,  is a cad

lag homogeneous -Markov process on if:  

is right continuous with left hand limits for each ,   

!"#

$% for all (the Borel 

& -field   1032

!6 

of ), ('*)

+,&.-/ 54 ('*)

7+8 $%

for sets , and 9;: <  

>=?

9@:
=EGF )

(1)

IHJK ML

<



for all bounded measurable functions ,  the (weak) domain of . Suppose further that this signal cannot be measured directly, but rather only through an imperfect sensor. In particular, it is common to assume that the observations occur at times 1N and take the form O R

N

6P

 N

 R

KQ

S.TVUWYXZ5[B]\^\^\_ N

(2)

`UYX# [a]\^\_\

where - N 4 is an independent vector-valued noise sequence. Then, the conditional distribution bdc of  given  2 O  N  is solved by Bayes’ rule N b c

< ! 9


³ ½¸¾

®

ª ) ¯ ° +

-

° Œ

yi –

¹

F

–

F –

+ ½v¿

nFª –µ´



(7)

(8)

º Fªf

·  F ª



(9)

–

XZ5[B { A

nFª

¶ F ª

·  5Ffª

.i F

–

®

4

, where )1e,°>³ ½v¿

®

) ¯ ° +

ª 

(10)

Ž

and · ¶ , · º denote projection onto the  and components. The effect of this field, when incorporated into the stochastic Itˆo equations that describe the motion of the individual targets, is to draw a target slightly towards the pack if it is distant and to repel it from any oneÁX other target { to which it becomes too close. The value À represents the closest distance that two targets should come to each other. We define the following variables to enact this attraction-repulsion: tdÂ

CÅÄ © Æ» Ÿ Ä

¥ s YÃ

ª

s Ä  Ä

E#Æk S



(11)

sK¥ ¦

where  is a three-dimensional standard Brownian motion mentioned previously. To make the simulation more realistic, friction | is included in the Õ equation. The calculation of this friction is similar for each motion type and is described below. 2. Rowing (ž

©»º Æ» Ÿ

s



C

©`¶ Æ» Ÿ 

s



E#Æk 1 Ì 

Ú < ¡s  ¡s  Û

Kâi

ÞSá

sK¥ §

where motion. 

is a two-dimensional standard Brownian



s "{

The target state variable is a Markov chain with state X# [a{ space 4 , representing adrift, rowing, and motoring respectively. The rates are given by: ± Ÿ

s  

ª



Í  – Î sK¥ 

! 

Ÿ 

s 



Í

±

Î sK¥ 



Ÿ

s  

ª



Í

Ÿ

ε– s(¥ 

! X 



s 

{

!



¥s 

†Z†Gt Â

X

(13)



(14)

†

and Í ± µ Î – s(¥  Ÿ 

s  



!

Í ± Î s(¥ 

ª Ÿ



s  



X

! [

\

†#†#tdÂ

¥s

(17)

(18)

3. Motorized (ž

ž s 



k¡ ‹Ž ¡ s   s 1

< ¡s A ß  s E    a¡ s V  Ž Õ ¡ s ×  ¡ s   s

à       Õ Ç× < ¡ s Ì < ¡ s iã{G

E sK¥ §  †  

 \â à † †

(12)

2.2 Maneuver type

Í  – Î sK¥ 

Ú < ¡ ] s Ê Ü#Ý  s   Ý ^‚  s < ¡s    Û

{a\qÙi

’Þ

A

which represent the strength and orientation of the deflecting force, and both of which decompose into sums over the finite set of targets.

ª

In this case,

< ¡s

KEGÆG Ì



)

where  represents scalar velocity in the forward direction, with E

¥ s nȉÉ5ÊSË5ȉ‚V C

s "[

Ú ¡ s   Ž¡ s Û

and td Ç



)

In this case,

 Û

Ú < ¡ ] s Ê Ü#Ý  s   ¡ Ý ^‚  s < s    Û

Ú ¡ s   Ž¡ s



(19)

< ¡s

where  represents scalar velocity in the forward direction, with E

Ú < ¡ s  ¡s  Û Þ

AåÞ á

< ¡s A ß  ¡ ¥s i  s

¥s A Õ × t? Ç  tdÂ Õ ,X i < ¡ s Ì < ¡ s iãäG

†   Käa\qÙi

(15) †

k¡ s ‹Ž ¡ s   s 1

E     B¡ s *Ž ¡ s   ¡ s   s à       ×Ç † †

\â à

E



sK¥ ¨ 

(20) Since the attraction-repulsion field only affects the motion of a target if it is motorized, a strong field increases the likelihood that the target will switch into, or stay in, the motorized maneuver. The differential equations describing the motion of tars get › depend on the maneuver type ž  , as follows: 1. Adrift

YX ( ž s

¡ s   E ÏÑ Ž ¡ s Ð  ¡s ÒÓ

)

A

s   s

  s   s

 ¡   s   s

  ÓÒ X † ÏÑ X † †Ø†Ø†



sK¥ ¨

is a two-dimensional standard Brownian

Friction is calculated according to the following model: ¶væ ¶ ¯ º ¯ ' × è]® ¶ × º × ç# ¥ ¥ ° × × Ç i

¡   !¬« ¶ B¡ VŽ?  Õ ×

In this case,

¶ B¡ s *Ž ¡    Ž¡ ÔÏÑÖÕº × a¡ s *   ÕB¡× s *Ž ¡ s     Õ Ç×

where motion.

†

|

¥

we define +^+ – +^+ where for two ships – ¹ F 5Ffª

– as in Equation (10), that is, the distance between two ships is taken as the® distance in the plane of 1Ža

Ÿ  s_°>¥ Kœ their  locations. Here, KQ ü is the › target under Kœ the permutation & of the particle at time  N . This formula calculates the average distance between each ship in the signal and its associated ship in the particle under the best possible association. The value used to represent the distance from the signal of the filter measure is then calculated as X

Results

40

5 Evaluation



6

20 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 3: Error in position for one target 100 80 60 40 20 0

(30)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 4: Error in position for two targets

“

where is the set defined above of particles with the cor rect number X of targets, and the norm +^+  Q Mü Q +_+ above is taken to be for zero-target particles. This introduces a bias to penalize filter estimates that indicate the incorrect number of targets. The resulting distances are averaged at each time over ⠆#† trials for the filter without extra particle diffusion and ⠆#† trials for the filter with extra diffusion, where the triÙ X als complete after time units, which is †#† observam  tions. For each trial, the initial number of particles â is †Z†Z†#†Z† .

100 80 60 40 20 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 5: Error in position for three targets

Figures 6-9 display the proportion of particles in the filter that exhibit the correct number of targets, in the case with no extra diffusion, broken down by the initial number of signal targets.

Figures 10-13 display the penalized average filter positional error for particle evolution simulated with extra diffusion, broken down by the initial number of signal targets.

1

100

0.8

80

0.6

60

0.4

40

0.2

20

0

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 6: Target number determination for zero targets

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 10: Error in position for zero targets

1

100

0.8

80

0.6

60

0.4

40

0.2

20

0

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 7: Target number determination for one target

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 11: Error in position for one target

1

100

0.8

80

0.6

60

0.4

40

0.2

20

0

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 8: Target number determination for two targets

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 12: Error in position for two targets

1

100

0.8

80

0.6

60

0.4

40

0.2

20 0

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 9: Target number determination for three targets

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 13: Error in position for three targets

Figures 14-17 display the proportion of particles in the filter that exhibit the correct number of targets, in the case with extra diffusion, broken down by the initial number of signal targets. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 14: Target number determination for zero targets

7

Conclusion

The branching particle-based filter has been found to be effective when extended to estimate the conditional distribution of multi-target signals for unknown, varying, but small numbers of targets. However, the computational resources required to obtain useful results can be large. In the figures above, the standard filter without extra diffusion in the particle evolution exhibited mediocre results in the case of two initial targets and poor results for three initial targets. Simulations with extra particle diffusion indicate a much more successful filter. Therefore, we see that tricks to ameliorate the difficulties inherent in discrete approximations can aid greatly in reducing the computational expense.

8

Acknowledgements

We thank Lockheed Martin, the Pacific Institute for the Mathematical Sciences, and the National Sciences and Engineering Research Council of Canada for their support of this work.

1 0.8 0.6

References

0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 15: Target number determination for one target 1 0.8

[1] D. Ballantyne, H. Chan, and M. Kouritzin, A novel branching particle method for tracking, Signal and Data Processing of Small Targets, O. Drummond, SPIE Proc. Vol. 4048, pp. 277-287, 2000. [2] D. Ballantyne, J. Hoffman, and M. Kouritzin, Practical application of a branching particle-based nonlinear filter, Signal Processing, Sensor Fusion, and Target Recognition X, I. Kadar, SPIE Proc. Vol. 4380, 2001. [3] P. Del Moral and G. Salut, Non-linear filtering using Monte Carlo particle methods, C.R. Acad. Sci. Paris, Volume 320, S´erie I, pp. 1147-1152, 1995.

0.6 0.4 0.2

[4] N. Gordon, Bayesian methods for tracking, PhD thesis, Imperial College, University of London, 1994.

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 16: Target number determination for two targets 1

[5] C. Hue, J.-P. Le Cadre, P. P´erez, Tracking multiple objects with particle filtering, Rapport de Recherche IRISA, No. 1361, October 2000. [6] M. Kouritzin, A pathspace branching particle filter, Preprint.

0.8

[7] T. G. Kurtz and J. Xiong, Particle representations for a class of nonlinear SPDEs, Stochastic Processes and their Applications, Volume 83, pp. 103-126, 1999.

0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 17: Target number determination for three targets

[8] D. Salmond and N. Gordon, Group and extended target tracking, SPIE Proc. Vol. 3809, pp. 284-296, 1999.