Maneuvering Target Tracking in the Presence of Glint using the Nonlinear Gaussian Mixture Kalman Fil-xq4.pdf

I. INTRODUCTION

Maneuvering Target Tracking in

thePresenceofGlintusingthe

Nonlinear Gaussian Mixture

Kalman Filter

Target tracking is a basic problem in radar, sonar,

and infrared applications. Since 1960 several methods

[1—2] have been proposed to solve the tracking

problem. The dynamic state-space (DSS) modeling

approach is widely used in radar target tracking. In

the DSS approach, the time-varying dynamics of

an unobserved state are characterized by the state

vector. It is usually assumed that the system state

is a first-order Markov process [3]. The Bayesian

approach provides a general framework for dynamic

state estimation. The Kalman filter (KF), which is

optimal in the minimum-mean-square error (MMSE)

sense for linear, Gaussian systems, is the most popular

tracking algorithm [4, 5].

Target tracking in practical radar systems is

nonlinear and non-Gaussian. The nonlinearity

behavior is due to the fact that the target dynamics

are usually modeled in Cartesian coordinates, while

the observation model is in polar coordinates.

There is no general analytic expression for the

posterior probability density function (pdf) in

nonlinear problems, and only approximated estimation

algorithms have been studied [6]. The extended

Kalman filter (EKF) is the most popular approach

for recursive nonlinear estimation [4, 7]. The main

idea of the EKF is based on a first-order linearization

of the model where the posterior pdf and the system

and measurement noises are assumed to be Gaussian.

The nonlinearity of the measurement model leads

to a non-Gaussian, multi-modal pdf of the system

state even when the system and the measurement

noise are Gaussian. The Gaussian approximation of

this multi-modal distribution leads to poor tracking

performance. The unscented Kalman filter (UKF)

approximates the pdf at the output of the nonlinear

transformation using deterministic sampling [8—10].

The advantage of the UKF over the EKF stems from

the fact that it does not involve approximation of the

nonlinear model per se [11, 12]. The UKF provides

an unbiased estimate; however, its convergence is

slow [12].

The Singer model [13] is widely used in

maneuvering target tracking applications. In this

model, the target acceleration is assumed to be a

zero-mean, first-order Markov process. Maneuvering

targets are characterized by changes of acceleration

due to sudden breaking or steering [14]. Usually, a

heavy-tailed distribution is used to model the abrupt

changes of the system state in maneuvering target

tracking applications [14].

In radar systems, changes in the target aspect

toward the radar may cause irregular electromagnetic

wave reflections, resulting in significant variations

of radar reflections [15]. This phenomenon gives

rise to outliers in angle tracking, and it is referred

to as target glint. The concept of angular glint was

I. BILIK, Member, IEEE

University of Massachusetts, Dartmouth

J. TABRIKIAN, Senior Member, IEEE

Ben-Gurion University of the Negev

Beer-Sheva, Israel

The problem of maneuvering target tracking in the presence

of glint noise is addressed in this work. The main challenge in this

problem stems from its nonlinearity and non-Gaussianity. A new

estimator, named as nonlinear Gaussian mixture Kalman filter

(NL-GMKF) is derived based on the minimum-mean-square error

(MMSE) criterion and applied to the problem of maneuvering

target tracking in the presence of glint. The tracking performance

of the NL-GMKF is evaluated and compared with the interacting

multiple modeling (IMM) implemented with extended Kalman

filter (EKF), unscented Kalman filter (UKF), particle filter

(PF) and the Gaussian sum PF (GSPF). It is shown that the

NL-GMKF outperforms these algorithms in several examples with

maneuvering target and/or glint noise measurements.

Manuscript received December 20, 2006; revised January 30, May

16, August 8, and September 10, 2008; released for publication

October 3, 2008.

IEEE Log No. T-AES/46/1/935940.

Refereeing of this contribution was handled by W. Koch.

Author’s addresses: I. Bilik, Dept. of Electrical and Computer

Engineering, University of Massachusetts, Dartmouth, MA;

J. Tabrikian, Dept. of ECE, Ben-Gurion University of the Negev,

ECE, Beer-Sheva, 84105, Israel, E-mail: (joseph@ee.bgu.ac.il).

0018-9251/10/$26.00 ° 2010 IEEE

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initially proposed in [16] and was explained as the tilt

of wave-front normal, resulting from the distortion of

target echo signal phase. It was found that glint has a

long-tailed pdf [15]. The statistical characteristics of

the glint noise and its mathematical models have been

studied in [15], [17], and [18]. The glint has been

modeled by the student’s t distribution in [18], based

on theoretical studies. In [17], the glint noise was

alternatively modeled by a mixture of a zero-mean,

small-variance Gaussian and a heavy-tailed Laplacian.

The Gaussian mixture model (GMM) with two

mixture components is widely used in the literature

for heavy-tailed, non-Gaussian pdfs. This model

consists of one small-variance Gaussian with high

probability and one large-variance Gaussian with low

probability of occurrence. The GMM is commonly

used to cope with abrupt changes of the system state

and glint noise modeling [15, 19].

Many researchers have addressed the problem

of filtering in non-Gaussian models. One of the

effective algorithms in the non-Gaussian problems is

the Masreliez filter [20, 21] that employs a nonlinear

“score-function,” calculated from known a priori

noise statistics. The score-function is customized for

the noise statistics and has to be redesigned for each

application. The main disadvantage of this approach

is that it involves a computationally expensive score

function calculation [17]. In [22], the Masreliez filter

was used in the target tracking problem with glint

noise.

Recently, a few filtering approaches have been

proposed. One of them is the multiple modeling

(MM) approach, in which the time-varying motion

of the maneuvering target is described by multiple

models [23]. In this approach, the non-Gaussian

system is represented by a mixture of parallel

Gaussian-distributed modes [4]. Using the Bayesian

framework, the posterior pdf of the system state

is obtained as a mixture of conditional estimates

with a priori probabilities of each mode [1].

Various filters are used for mode-conditioned state

estimation. For example, the Gaussian sum filter

(GSF) was implemented in [4], [24] using a bank

of KFs. The EKF and Masreliez filters were used as

mode-conditioned filters for the nonlinear problems of

target tracking in [17], [22], [25]. The main drawback

of the MM approach is the exponential growth of the

number of the modes, and exponentially increasing

number of mode-conditioned filters [1, 26]. Therefore,

optimal algorithms such as the GSF are impractical.

Suboptimal techniques for model order reduction

based on merging and pruning were summarized

in [26]. The generalized pseudo-Bayesian (GPB)

merging algorithm combines modes with similar

recent histories that differ mainly in old time instances

[1, 27]. A suboptimal, computationally-efficient

interacting MM (IMM) algorithm was successfully

applied to the maneuvering target tracking problem

[1, 28, 29]. In [19], [22], [25] the IMM algorithm

with EKFs and Masreliez filters were implemented for

maneuvering target tracking in the presence of glint

noise. The IMM algorithm with a greater number of

modes was proposed in [30] for non-Gaussian system

and measurement noises.

Another class of filtering algorithms is based

on the sequential Monte Carlo (MC) approach. The

sequential importance sampling technique forms

the basis for most MC techniques [31], in which

the filtering is performed recursively generating

MC samples of the state variables. These methods

are often very flexible in non-Gaussian problems

due to the nature of the MC simulations [32]. One

of the popular techniques of this approach is the

particle filter (PF), which is a suboptimal estimator

that approximates the posterior distribution by a

set of random samples with associated weights.

The PF models the state posterior distribution

using discrete random samples rather than using an

analytic model [33]. The Gaussian sum particle filter

(GSPF) [34] implements the PF assuming Gaussian

mixture distributions for the system noise. The GSPF

introduces a new model order reduction method based

on the importance resampling method. Thus, the

model order of the state vector posterior pdf remains

constant over iterations, discarding mixands with

small weights.

The PF has been extensively used for maneuvering

target tracking (e.g., [14]). In [35] the PF was applied

to the problem of tracking in glint noise environment.

As shown in [36] and [37], the PF outperforms the

IMM algorithm when the likelihood function is

multi-modal. Different application-driven PFs are

presented in the literature, but there is no precise

rule about which type of PF should be used in each

application. This implies that no rigorous PF exists,

which is one of the disadvantages of the PF approach.

A recursive estimator, denoted as Gaussian mixture

KF (GMKF), for linear non-Gaussian problems was

derived in [38], [39]. The main idea of the GMKF

is that any pdf can be closely approximated by a

mixture of a finite number of Gaussians [40]. The

GMKF is based on the GSF, where the problem of

exponential model order growth was solved using

a greedy expectation-maximization (EM)-based

method. In [38], [39], it was shown that the GMKF

outperforms the PF, the IMM-KF, and the GSPF in

linear non-Gaussian problems.

This work addresses the nonlinear and

non-Gaussian problem of maneuvering target tracking

in the presence of non-Gaussian glint noise. A

recursive algorithm named as nonlinear GMKF

(NL-GMKF), which extends the GMKF to nonlinear

models, is derived based on the MMSE criterion.

The proposed NL-GMKF considers the case of

non-Gaussian system and measurement noises as

well as the non-Gaussian posterior pdf of the state

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vector. The NL-GMKF differs from the GSPF [34]

and the mixture Kalman filter (MKF) [32] in the

following aspects. First, the NL-GMKF assumes

general non-Gaussian distributions for the system and

measurement noises, rather than GMM distributions.

Second, in the NL-GMKF, the posterior distribution

of the state vector is modeled by GMM whose

parameters are determined to minimize its estimated

Kullback-Leibler divergence (KLD) from the “true”

distribution, while in the GSPF and the MKF, the

GMM parameters are determined by resampling,

which throws out mixands with insignificant weights.

Usually, the parameters of the mixture model are

estimated via the EM algorithm. The performance

of this algorithm depends on its initialization, due

to its tendency to converge to local maxima. An

elegant solution for the initialization problem is

provided by the greedy learning of GMM [41], which

is implemented in this work.

The expected significance of the proposed

NL-GMKF is in practical applications of low

to moderate maneuvering target tracking, when

maneuver detection is difficult. The advantage

of the NL-GMKF over other tracking algorithms

is significant especially in the presence of glint

measurement noise with small probability of detection

and high significance. The correlation between

the statistics of glint noise and maneuver (that

characterizes a maneuvering target consisting of

multiple scattering centers) makes the problem

of maneuvering target tracking in the presence

of glint noise extremely challenging due to the

difficulty of maneuver and glint detection and filtering

simultaneously. The proposed NL-GMKF does not

require prior knowledge of the target dynamics

such as coordinated turn model; therefore, it might

be useful when tracking targets with complicated

maneuvering profile that cannot be modeled by a

finite set of simple dynamic models.

The paper is organized as follows: The NL-GMKF

is derived in Section II. The NL-GMKF computational

complexity is analyzed in Section III. The DSS

model for radar target tracking is stated in Section IV.

Estimation performances of the NL-GMKF for various

target tracking cases are evaluated in Section V. Our

conclusions are summarized in Section VI.

where the nonlinear transition function a ( ¢ , ¢ )and

the observation function h ( ¢ , ¢ )areassumedtobe

known. The system and measurement noises are

non-Gaussian with known pdfs. The driving noise

u [ n ] and the measurement noise w [ n ] are temporally

independent, i.e., u [ n ]and u [ n 0 ], and w [ n ]and w [ n 0 ]

are mutually independent for any time instances

n =0,1,2, ::: ; n 0 =0,1,2, ::: ; n6 = n 0 . The initial state

s [ ¡ 1], the driving noise u [ n ], and the measurement

noise w [ n ] are statistically independent. The initial

state distribution is modeled by

s [ ¡ 1] » GMM( ® s l [ ¡ 1], ¹ s l [ ¡ 1], ¡ s l [ ¡ 1]; l =1, ::: , L )

(3)

where GMM( ® y j , ¹ y j , ¡ y j , j =1, ::: , J ) denotes

a J th-order proper complex Gaussian mixture

distribution whose density function is given by

f y ( y )=

® y j © ( y ; ° y j )

( 4 )

j =1

where ® y j is the mixture weight, © ( y ; ° y j )isthe

j th mixture component, and ° y j contains the mean

vector ¹ y j , and the covariance matrix ¡ y j of the j th

Gaussian.

Let ˆ s [ njp ] denote the MMSE estimator of s [ n ]

from X [ p ] =( x T [0], x T [1], ::: , x T [ p ]) T . The notation

ˆ s [ njn¡ 1] stands for one-step prediction of the state

vector s [ n ] from data X [ n¡ 1]. The objective of this

paper is to derive a recursive method for estimation

of s [ n ] from the observed data X [ n ]. To this end, the

MMSE criterion resulting in the conditional mean

estimator

ˆ s [ njn ] = E [ s [ n ] jX [ n ]]

(5)

is employed.

B. NL-GMKF Derivation

In this section, the recursive NL-GMKF is derived

based on the MMSE criterion. Let x [ njn¡ 1] denote

the MMSE estimator of x [ n ]from X [ n¡ 1] using (2).

Then x [ njn¡ 1] is given by

x [ njn¡ 1]= E ( x [ n ] jX [ n¡ 1])

II. NL-GMKF

= E ( h ( s [ n ], w [ n ]) jX [ n¡ 1]) (6)

A. DSS Model

and the innovation process defined as x [ n ]isgivenby

The DSS model is widely used for maneuvering

target tracking problems where the system state and

observation vectors f s [ n ], x [ n ], n =0,1,2, :::g are

described by the following nonlinear, non-Gaussian

DSS model:

x [ n ]= x [ n ] ¡ x [ njn¡ 1]

= h ( s [ n ], w [ n ]) ¡ x [ njn¡ 1] : (7)

If the transformation X [ n ] $ [ X T [ n¡ 1], x T [ n ]] T

is one-to-one, then the conditional distribution of

s [ n ] jX [ n ] is identical to the conditional distribution

of s [ n ] jX [ n¡ 1], x [ n ]. Since s [ n ]and x [ n ]given

X [ n¡ 1] are assumed to be jointly GMM of order

s [ n ]= a ( s [ n¡ 1], u [ n ])

(1)

x [ n ]= h ( s [ n ], w [ n ])

(2)

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L , the conditional distribution of s [ n ] j x [ n ], X [ n¡ 1]

given the random mixture indicator ´ l [ n ] [42] is

Gaussian. Therefore, the conditional distribution of

s [ n ] jX [ n ] is GMM of order L :

In the following, the parameters required in (14)

and (15) are obtained. Let

s [ n ]

x [ n ]

s [ n ]

x [ n ]

y [ n ]=

and y [ n ] =

s [ n ] jX [ n ] » GMM( ® s [ njn , ´ l [ n ]], ¹ s [ njn , ´ l [ n ]],

¡ s [ njn , ´ l [ n ]]; l =1, ::: , L ) :

Then using (1), (2), and (7) one obtains

4 s [ n¡ 1]

a ( s [ n¡ 1], u [ n ])

h ( a ( s [ n¡ 1], u [ n ]), w [ n ])

(8)

y [ n ]=

= G

u [ n ]

w [ n ]

In the following, the parameters of this conditional

distribution Ã s [ njn ]where

(16)

and

Ã s [ njn ]= f® s [ njn , ´ l [ n ]], ¹ s [ njn , ´ l [ n ]],

¡ s [ njn , ´ l [ n ]] g l =1

x [ njn¡ 1]

y [ n ]= y [ n ] ¡

(9)

4 s [ n¡ 1]

are derived.

Since the conditional distribution of s [ n ], x [ n ]

given the random mixture indicator ´ l [ n ], is jointly

Gaussian, then the mean vector ¹ s [ njn , ´ l [ n ]] and

covariance matrix ¡ s [ njn , ´ l [ n ]] can be obtained as

¹ s [ njn , ´ l [ n ]]= E [ s [ n ] j x [ n ], X [ n¡ 1], ´ l [ n ]]

= E [ s [ n ] jX [ n¡ 1], ´ l [ n ]]

+ ¡ sx [ njn¡ 1, ´ l [ n ]] ¡ ¡ 1

x [ njn¡ 1]

= G

u [ n ]

w [ n ]

: (17)

Since y [ n ], is a linear transformation of y [ n ], then

the vectors s [ n ]and x [ n ]given X [ n¡ 1] are jointly

GMM of order L , that is, the conditional distribution

of y [ n ]given X [ n¡ 1] can be modeled by an L th

order GMM with parameters:

Ã y [ njn¡ 1]= f® y [ njn¡ 1, ´ l [ n ]], ¹ y [ njn¡ 1, ´ l [ n ]],

¡ y [ njn¡ 1, ´ l [ n ]] g l =1

£ [ njn¡ 1, ´ l [ n ]]

£ ( x [ n ] ¡¹ x [ njn¡ 1, ´ l [ n ]]) (10)

¡ s [ njn , ´ l [ n ]]=cov( s [ n ] j x [ n ], X [ n¡ 1], ´ l [ n ])

=cov( s [ n ] jX [ n¡ 1], ´ l [ n ])

¡¡ sx [ njn¡ 1, ´ l [ n ]] ¡ ¡ 1

(18)

where

¹ y [ njn¡ 1, ´ l [ n ]]

¹ s [ njn¡ 1, ´ l [ n ]]

¹ x [ njn¡ 1, ´ l [ n ]]

(19)

£ [ njn¡ 1, ´ l [ n ]] ¡ sx [ njn¡ 1, ´ l [ n ]]

(11)

¡ y [ njn¡ 1, ´ l [ n ]]

¡ s [ njn¡ 1, ´ l [ n ]] ¡ sx [ njn¡ 1, ´ l [ n ]]

¡ sx [ njn¡ 1, ´ l [ n ]] ¡ x [ njn¡ 1, ´ l [ n ]]

where

¹ x [ njn¡ 1, ´ l [ n ]]= E [ x [ n ] jX [ n¡ 1], ´ l [ n ]]

¡ x [ njn¡ 1, ´ l [ n ]]=cov( x [ n ] jX [ n¡ 1], ´ l [ n ])

¡ sx [ njn¡ 1, ´ l [ n ]]=cov( s [ n ], x [ n ] jX [ n¡ 1], ´ l [ n ]) :

(20)

(12)

Using the properties of the jointly

GMM-distributed random processes, s [ n ] jX [ n¡ 1]

and x [ n ] jX [ n¡ 1], the mixture weights can be

obtained as

Following the conventions of the KF, the l th Kalman

gain corresponding to the l th mixture component is

defined as

® s [ njn , ´ l [ n ]]

K l [ n ] = ¡ sx [ njn¡ 1, ´ l [ n ]] ¡ ¡ 1

x [ njn¡ 1, ´ l [ n ]] :

(21)

(13)

Using (13), (10) and (11) can be rewritten as

where

° x l [ njn¡ 1]= f¹ x [ njn¡ 1, ´ l [ n ]], ¡ x [ njn¡ 1, ´ l [ n ]] g:

(22)

¹ s [ njn , ´ l [ n ]]

= ¹ s [ njn¡ 1, ´ l [ n ]]+ K l [ n ]( x [ n ] ¡¹ x [ njn¡ 1, ´ l [ n ]])

(14)

Therefore, one can calculate the pdf parameters of

s [ n ] jX [ n ] given in (14), (15), and (21) using the

parameters of the distribution y [ n ] jX [ n¡ 1] obtained

in the following.

¡ s [ njn , ´ l [ n ]]

= ¡ s [ njn¡ 1, ´ l [ n ]] ¡ K l [ n ] ¡ xs [ njn¡ 1, ´ l [ n ]] : (15)

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The conditional pdf of y [ n ] jX [ n¡ 1] is modeled

by GMM of order L with parameters

The measurement prediction is calculated as a

conditional mean estimator of x [ n ]given X [ n¡ 1],

using parameters obtained in (24) and (25) as follows

Ã y [ njn¡ 1]= f® y [ njn¡ 1, ´ l [ n ]], ¹ y [ njn¡ 1, ´ l [ n ]],

¡ y [ njn¡ 1, ´ l [ n ]] g l =1

x [ njn¡ 1]=

® y [ njn¡ 1, ´ l [ n ]] ¹ x [ njn¡ 1, ´ l [ n ]] :

(23)

l =1

where

(30)

The MMSE estimator in (5) is obtained using (14)

and (21) as follows:

¹ s [ njn¡ 1, ´ l [ n ]]

¹ x [ njn¡ 1, ´ l [ n ]]

¹ y [ njn¡ 1, ´ l [ n ]]=

(24)

and

ˆ s [ njn ]=

® s [ njn , ´ l [ n ]] ¹ s [ njn , ´ l [ n ]] : (31)

¡ y [ njn¡ 1, ´ l [ n ]]

l =1

¡ s [ njn¡ 1, ´ l [ n ]] ¡ sx [ njn¡ 1, ´ l [ n ]]

¡ sx [ njn¡ 1, ´ l [ n ]] ¡ x [ njn¡ 1, ´ l [ n ]]

This completes the derivation of the NL-GMKF.

Note that the GMM order of the conditional

distribution y [ n ] jX [ n¡ 1] might be obtained

using model order selection algorithms such as

the minimum description length (MDL) [43].

Alternatively, L can be set as an upper bound on the

number of mixture components in the conditional

pdf.

In this work, the vector parameter Ã y [ njn¡ 1]

is obtained from the data D 0 using the greedy EM

algorithm [44]. The greedy learning algorithm

controls the GMM order of the estimated pdf, which

varies over the iterations. In [41] it was shown

that the greedy EM algorithm is insensitive to the

initialization. The pdf estimation using the greedy EM

algorithm appears in [41], [44] and is summarized in

the Appendix.

(25)

Since x [ njn¡ 1] depends on X [ n¡ 1] only, then

from (17) we conclude that the conditional pdf of

y [ n ] jX [ n¡ 1] is shifted by

x [ njn¡ 1]

compared to the conditional pdf of y [ n ] jX [ n¡ 1]:

x [ njn¡ 1]

¹ y [ njn¡ 1, ´ l [ n ]]= ¹ y [ njn¡ 1, ´ l [ n ]] ¡

(26)

¡ y [ njn¡ 1, ´ l [ n ]]= ¡ y [ njn¡ 1, ´ l [ n ]]

(27)

® y [ njn¡ 1, ´ l [ n ]]= ® y [ njn¡ 1, ´ l [ n ]]

(28)

C. NL-GMKF Summary

and

This subsection summarizes the derived

NL-GMKF for recursive estimation of s [ n ] from X [ n ].

¹ x [ njn¡ 1, ´ l [ n ]]= ¹ x [ njn¡ 1, ´ l [ n ]] ¡ x [ njn¡ 1] :

(29)

Hence the mixture weights and covariance matrices

in Ã y [ njn¡ 1] and Ã y [ njn¡ 1] are identical and the

means are related in (29).

Since the function G ( ¢ ) is nonlinear, the parameters

of the conditional distribution of y [ n ] jX [ n¡ 1]

cannot be obtained analytically. Alternatively, the MC

approach can be implemented. Thus, an artificial data

set D is obtained from the conditional distribution of

4 s [ n¡ 1]

1) Initialization:

Initialize the L -order GMM parameters of the state

vector at time instance n = ¡ 1.

® s [ ¡ 1 j¡ 1, ´ s l [ ¡ 1]]= ® s l [ ¡ 1]

¹ s [ ¡ 1 j¡ 1, ´ s l [ ¡ 1]]= ¹ s l [ ¡ 1]

¡ s [ ¡ 1 j¡ 1, ´ s l [ ¡ 1]]= ¡ s l [ ¡ 1] :

Set n =0.

2) Mixture parameters of the state and measurement

prediction:

Generate an artificial data set D from the

conditional distribution of

4 s [ n¡ 1]

u [ n ]

w [ n ]

given X [ n¡ 1]. Next, the nonlinear function G ( ¢ )is

applied on the data set D to obtain a new artificial

data set D 0 , which is used to obtain the pdf parameters

of y [ n ] jX [ n¡ 1], i.e., Ã y [ njn¡ 1]. The statistical

parameters required for calculation of (13), (14),

and (15) can be obtained from the parameters of

Ã y [ njn¡ 1] in (18).

u [ n ]

w [ n ]

given X [ n¡ 1], according to the pdf of s [ n¡ 1] j

X [ n¡ 1] from the previous step and pdfs of u [ n ]

and w [ n ].

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