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PMHT with the True Association Probability

Darin T. Dunham Vectraxx, Inc.

12131 Howards Mill Road Glen Allen, VA 23059

804-749-8750 darin@vectraxx.com

Abstract—The Probabilistic Multi-Hypothesis Tracker (PMHT) offers a balance between the single-frame approach of the Probabilistic Data Association Filter (PDAF) and the multiple frame approach of the Multiple Hypothesis Tracker (MHT). With single-frame tracking algorithms, only information that has been received to date is used to determine the association between tracks and measurements. These decisions are made based on available data and are not changed even when future data may indicate that the decision was incorrect. On the other hand, in multi-frame algorithms, hard decisions are delayed until some time in the future, thus allowing the possibility that incorrect association decisions may be corrected with more data. This paper1,2 presents the ongoing results of research using the PMHT algorithm as a network-level composite tracker on distributed platforms. In particular, this paper discusses and explores different approaches to calculating the association probabilities within the PMHT algorithm. The results are presented for multiple targets with just a single sensor at this point in the research.

TABLE OF CONTENTS

1. INTRODUCTION......................................................1 2. ORIGINAL PMHT ALGORITHM............................2 3. RESEARCH METHOD .............................................3 4. RESULTS AND ANALYSIS .......................................3 5. CONCLUSIONS AND FUTURE WORK......................6 ACKNOWLEDGEMENTS .............................................6 REFERENCES .............................................................7 BIOGRAPHY ...............................................................7

1. INTRODUCTION

Traditional tracking algorithms typically use some form of a normal distribution to determine the association decisions between measurements and tracks. That is, as measurements are received, a distance cost is calculated based on a normal distribution between the current track estimates and the position of the measurement. This calculation involves both the covariance associated with the track estimate and the covariance associated with the measurement. From this calculation, the measurements are paired up with the existing tracks. Then the measurement 1 1-4244-0525-4/07/$20.00 ©2007 IEEE. 2 IEEEAC paper #1013, Version 3, December 18, 2006.

information is used to update the track with which it is paired. This process can either be done in a single-frame algorithm like a Nearest Neighbor or in a multiple-frame algorithm, like the MHT.

Another approach for tracking algorithms is to form a probabilistic weighting of the measurement to track possibilities. The PDAF does this in one form as a single-frame tracking algorithm, and the PMHT does this in another form as a multiple-frame tracking algorithm. However, the PDAF uses a normal distribution involving both the covariance of the track estimate and the covariance of the measurement. Furthermore, the PDAF weights the updated tracks with the measurements that fell within the ellipsoidal gate and then forms the track estimate based on the weighting of the possible track updates.

On the other hand, the original PMHT uses a normal distribution involving only the measurement covariance to weight the innovations between the track estimate and the measurement. The measurements are weighted accordingly and a pseudo-measurement made of the weighted possible measurements is used to update the track. In this manner, the PMHT has the measurements choosing the tracks they belong to instead of the tracks choosing the measurements. In the author’s research to date involving the PMHT [2, 3], he has used both the covariance of the track estimate and the covariance of the measurement to weight the innovations and calculate the weight of each possible measurement. The research shown in this paper explores the difference between these two approaches with the PMHT and presents some results.

This paper is divided into the following sections: Section 2 covers the details of the PMHT algorithm, focusing on the differences of calculating the weights for each measurement. Section 3 discusses the simulation environment used and explains the method for the research. Section 4 presents the results from this research to date along with some analysis of the results. Finally, Section 5 summarizes this paper, states some conclusions, and discusses the direction of future work.

2

2. ORIGINAL PMHT ALGORITHM

This algorithm is taken from Streit and Luginbuhl [1] using the linear Gaussian case. Measurement probabilities, Π(0) = {πtm

(0)} must be assigned so that πtm(0) > 0. It is not critical

what values are assigned to these measurement probabilities because in the first iteration they will be recalculated, and they do not have an adverse effect before they are. The πtm

(i) values specify the estimated probability that a measurement at scan t is assigned to target model m after i iterations of the algorithm. An initial target state (x0m

(0),x1m(0),…,xTm

(0)) for each time increment and each of the M target models must be assigned. Our experience has shown that these initial estimates must be extremely accurate. In this paper, m specifies the target model (m = 1,...,M); t specifies the discrete time index (t = 0,...,T); r specifies the index for measurements within a scan (r = 1,...,nt); and the superscript i specifies the iteration index (i = 0,1,...). For every target and measurement combination at each scan a weight is computed. The value of the likelihood function (assuming normal distributions) evaluated at the error between the current estimated position and each measurement is used for the weight.

( )

w2 det(mtr

i mtr(i)T

tm-1

mtr(i)

tm

*( )exp ~ ~

)+ =

−1

12 z zΣ

Σπ (1)

w w

wmtri+1) mtr

i+1)

ts(i)

stri+1)

s=1

M(

*(

*(=∑π

(2)

where,

(i)tmtmtr

i)(mtrtr

(i)mtr ˆ~ xHzzzz −=−= (3)

is the error between the current estimate and a measurement. Further, Σ is the weighting matrix defined as:

tmtm R=Σ (4)

However, in past research, the weighting matrix has been:

tmTtmtmtmtm RHPH +=Σ (5)

Here Ptm is the covariance matrix associated with the target estimate, and Htm is the measurement matrix, defined as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

001000000000001000000000001

=tm HH (6)

for the standard Kalman algorithm. Next, the measurement centroids are computed. First, the mean measurement weight for each target model m at time t is defined as:

w = wmt(i+1) 1

n mtri+1)

r=1

n

t

t (∑ (7)

Next, the measurement centroid is computed as:

z ztm(i+1)

t tm(i+1) mtr

(i+1)tr

r=1

n

n ww

t= ∑

1 (8)

This measurement centroid will be used in the Kalman Smoothing step below. The next step before the Kalman Smoother is to update the target measurement probabilities. This is computed as:

π πtm(i+1)

mt(i+1)

tm(i)w= (9)

The target state sequences are updated via the Kalman Smoother using the measurement centroids as the inputs. First, the intermediate variables of the forward recursion are initialized as:

y x0|0 0m(i)= (10)

P P0|0 0m= (11)

Here with these dummy variables, the model m and iteration index i have been suppressed for notational simplicity. The forward recursion is defined for t = 0,1,...,T-1 as:

QPP qTt|tt|1+t +ΦΦ= (12)

[ ] 1m1,+t

Tm1,+tt|1+tm1,+t

1)+(im1,+t1+t

Tm1,+tt|1+t

1)+(im1,+t1+t1+t

n

...n−+

=

RHPH

HPK

π

π (13)

t|1+tm1,+t1+tt|1+t1+t|1+t - PHKPP = (14)

[ ]t|tm1,+t1)+(im1,+t1+tt|t1+t|1+t ˆˆˆ yHzKyy Φ−+Φ= (15)

For these equations, Q is the plant noise and the scalar variable, q, is used as a tunable parameter. Next, the updated target state estimate for model m at time T, which is the initial state for the backwards recursion, is:

x yTm(i+1)

T|T= (16)

and the updated target state estimates for t = T-1,...,1,0 are computed via the backward recursion as:

3

[ ]x y P P x ytm(i+1)

t|t t|tT

t +1|t-1

t +1,m(i+1)

t|t= + −Φ Φ (17)

The equations in this subsection make up a bank of M Kalman Smoothers, which can be run in parallel, although these filters are not independent because they are linked by the weights.

3. RESEARCH METHOD

The simulation environment used in this research is the Ballistic Missile Defense (BMD) Benchmark. This is a product of the Missile Defense Agency and is primarily managed by Dr. William D. Blair at Georgia Tech. During this effort, just a single sensor is used—a THAAD-like X-band radar. Two different missiles are used. One is a single stage missile with a range of about 600 kilometers. The other is a multiple-state missile with a range of about 1,250 kilometers.

The simulation environment is extremely realistic with many different factors included. There are biases, gridlock errors, communication delays, Swerling cases, and multi-path returns.

Programming Environment

The BMD Benchmark is written in Matlab. The PMHT tracker used in this research is coded in Fortran. The interface is made by mexing the Fortran code and then calling the mexed file from the Matlab when there are measurements to be passed from the sensor to the tracker. Contact the author for further details on how this was accomplished, including debugging the Fortran through the Matlab.

Iterations and Convergence

As is outlined in section 2, the PMHT is an iterative algorithm that uses the Expectation Maximization (EM) theory to converge on the solution. Sometimes this process will converge on a local maximum rather than the global maximum. This is one of the problems with the PMHT algorithm [4]. With the iterative process, it has been found that three iterations are usually sufficient to reach convergence. With the Σ approach (equation 5), three iterations were all that were needed. However, for the R approach (equation 4), the initial results showed that just using R was not providing a large enough ellipsoid at the beginning of each batch of iterations. This caused the algorithm to lose tracks because they were not falling within the ellipse. Therefore, a hybrid approach was used between the Σ approach and the R approach. Σ was used for the first two iterations and then three more iterations used the R as the weighting factor in equation 1. This allowed the algorithm to maintain track while using the correct weighting calculation for the last three iterations.

4. RESULTS AND ANALYSIS

The results will be shown for both approaches. The different metrics shown will display results from both approaches. First the results for the Σ approach will be shown and then the results for the R approach (really the hybrid Σ/R approach) will be shown and compared. Ten Monte Carlo runs were completed for all of the metrics that are shown.

The first set of results show the completeness, spurious, and redundant metrics.

Figure 1 – Completeness, Redundant, Spurious using Σ

Figure 2 – Completeness, Redundant, Spurious using R

For this first pair of plots, the only difference is that the Σ approach has a scored track ratio of 1.3 as opposed to 1.2 for the R approach. This means that over the 10 Monte Carlo runs, the Σ approach maintained track 65% of the time, and the R approach maintained track 60% of the time. There are essentially two tracks in the scenario, so that means there were 20 tracks throughout the 10 Monte Carlo runs.

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Figure 3 – Position RMSE on the M600 using Σ

Figure 4 – Position RMSE on the M600 using R

These results for the position Root Mean Squared Error (RMSE) show that both approaches are performing just as well.

Figure 5 – Velocity RMSE on the M600 using Σ

Figure 6 – Velocity RMSE on the M600 using R

Nothing really conclusive can be determined from this plot, as both approaches seem to be doing well.

Figure 7 –Covariance Consistency on the M600 using Σ

Figure 8 – Covariance Consistency on the M600 using R

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With these covariance consistency plots shown in figures 7 and 8, there appears to be a slight improvement with the R approach. The ideal value here is one as the covariance over the position and velocity has been normalized.

Figure 9 – Position RMSE on the M1250 using Σ

Figure 10 – Position RMSE on the M1250 using R

Figures 9 and 10 show the position RMSE on the primary object from the 1,250-kilometer missile. The primary object starts out as the full stack then becomes the ACM / RV module, and finally is the RV.

The results show that the two approaches are again very close, but the R approach is showing a slight advantage. The next two figures compare the velocity accuracy.

Figure 11 – Velocity RMSE on the M1250 using Σ

Figure 12 – Velocity RMSE on the M1250 using R

Once again, the approaches show very similar results, but the R approach seems to be slightly better.

Figure 13 – Covariance Consistency on the M1250 using Σ

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Figure 14 – Covariance Consistency on the M1250 using R

The results shown in the covariance consistency metric plots (figures 13 and 14) follow all of the previous results shown in this paper. The R approach is show a slight improvement on the accuracy plots.

5. CONCLUSIONS AND FUTURE WORK

The difference between these two approaches for the PMHT is negligible based on the scenarios performed here. However, the PMHT was designed for two things—target estimation in an underwater environment. This research has explored the effects of changing the weighting matrix for the calculation of the probability of measure-to-track association in a radar environment.

The PMHT in its original form has a few shortcomings, which are described very well in [4]. Its original form is optimal in that it uses all of the available measurements, and it is very elegant in its theory and simplicity. However, it still has to overcome the known shortcomings. One of the issues is converging to a local maximum instead of the global maximum. By using the Σ in conjunction with the R matrix, a deflation technique was basically employed. This helps some for this problem, but the PMHT is still fairly narcissistic.

Future Work

The covariance-weighting matrix appears not to be a critical factor in the success of the PMHT. Work from this point will focus on improving the algorithm in other ways. For instance, there is work currently underway to develop a covariance estimation technique utilizing PDA methods. This work is showing promise and should be reported on in the future.

Another enhancement that is being worked on the PMHT is implementing what is being called a multiple frame PMHT. This enhancement will link up measurement data sequences to be weighted over the processing window instead of independent measurements [6].

There is still much more research that can be conducted on utilizing the PMHT as a network-level tracker in a ballistic missile defense scenario. The scenarios can be made more challenging by adding more complex missiles. The missile used in this research was a single-stage missile with no debris. In the future, multiple-stage missiles with debris will be used. Also, many missiles in the same vicinity will be used in order to develop better tracking methods for dealing with unresolved closely spaced objects.

Other planned improvements and enhancements to the PMHT algorithm will include a new initialization scheme and separation point prediction. The new initialization scheme that is envisioned will enable tracks in the PMHT to be initialized by more than one sensor. Moreover, if the new initialization scheme works as anticipated, it will also organically determine when tracks need to be deleted. This will round out the track management for the PMHT as a network-level composite tracker.

In addition, a higher order prediction step that accounts for gravitational effects should be utilized with any tracker trying to estimate the position and velocity of ballistic missiles. In this research only a second-order solution was used, and it is hoped that a fourth-order or fifth-order Runge-Kutta solution will be implemented in future work.

It is hoped that these improvements to the PMHT algorithm will make the tracker more robust and improve accuracy as well as covariance consistency. If these improvements show progress, then the PMHT could become a great target tracker that is a viable option against not only the single frame tracking approaches, but also the other multiple frame approaches like MHT.

ACKNOWLEDGEMENTS

This work was aided significantly by discussions the author had with Dr. Peter Willett of the University of Connecticut.

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REFERENCES

[1] R. L. Streit and T. E. Luginbuhl, “Probabilistic Multi-Hypothesis Tracking,” NUWC-NPT Technical Report 10,248, Naval Undersea Warfare Center Division, Newport, RI, 15 February 1995.

[2] D. T. Dunham, “Distributed multiple sensor tracking with the PMHT,” Proceedings for the 2006 SPIE Conference on Signal and Data Processing of Small Targets, Orlando, FL, 18-20 April 2006.

[3] D. T. Dunham, R. J. Dempster, and S. S. Blackman, “A Survey of the Advances in the PMHT with Run Time Comparisons to the MHT,” Proceedings of the Workshop on Multiple Sensor Target Tracking: A Tribute to Oliver E. Drummond, Key West, FL, 24 June 2004.

[4] P. Willet, Y. Ruan, and R. Streit, “The PMHT: Its Problems and Some Solutions,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 38, No. 3, July 2002.

[5] Y. Ruan and P. Willett, “The Multiple Model PMHT and its Application to the Second Benchmark Radar Tracking Problem,” submitted to IEEE Transaction on Aerospace and Electronic Systems, March 2002.

[6] R. Streit, “PMHT Algorithms for Multi-Frame Assignment,” Proceedings of the 9th International Conference on Information Fusion, Florence, Italy, 10-13 July 2006.

BIOGRAPHY

Darin Dunham is the President and Senior Research Engineer with Vectraxx, a company he started after leaving the United States Marine Corps in March of 2001. Mr. Dunham served almost 10 years in the Marine Corps as an infantry platoon commander, company executive officer, and company operations officer in Hawaii and then as a project officer at the Marine Corps Systems Command. He received his MSEE from the Naval

Postgraduate School in 1997 and his BSEE from Carnegie Mellon University in 1991. His thesis work focused on the PMHT and comparing its performance to the PDAF and MHT in an underwater, littoral region scenario. His recent work has involved target tracking algorithms and sensors within the Ballistic Missile Defense (BMD) Benchmark simulation. In particular, he has done extensive research on improving SIAP across distributed multiple platforms with both the MHT and the PMHT. Mr. Dunham is actively

involved with the continued development of the BMD Benchmark simulation.