Practical Fusion of Quantized Measurements via

Particle Filtering 1

Yanhua Ruan, Peter Willett

ECE Department

University of Connecticut

Storrs, CT 06269

Alan Marrs

Qinetiq Ltd.

Malvern Technology Centre

Great Malvern, UK WR14 3PS

Francesco Palmieri

Dipartimento di Ingegneria

dell’Informazione, Seconda Universita degli Studi di Napoli

Casa Reale dell’Annunziata, via Roma 29

I-81031 Aversa (CE), Italy

Stefano Marano

DIIIE

Universita degli Studi di Salerno

via Ponte don Melillo

I-84084, Fisciano (SA), Italy

Submitted: January 2003

Revision: September 2005

Second Revision: June 2006

1This work was supported by the UK Ministry of Defence Corporate Research Program. The authors

also wish to acknowledge Murat Efe, Orhan Arikan and Yaakov Bar-Shalom for sharing their useful ideas.

Abstract

Most treatments of decentralized estimation rely on some form of track fusion, in which local track

estimates and their associated covariances are communicated. This implies a great deal of commu-

nication; and it was recently proposed that by an intelligent quantization directly of measurements,

the communication needs could be considerably cut.

However, several issues were not discussed. The first of these is that estimation with quan-

tized measurements requires an update with a non-Gaussian distribution, reflecting the uncertainty

within the quantization “bin”. In general this would be a difficult task for dynamic estimation,

but Markov-Chain Monte-Carlo (MCMC, and specifically here particle filtering) techniques appear

quite appropriate since the resulting system is, in essence, a nonlinear filter. The second issue is that

in a realistic sensor network it is to be expected that measurements should arrive out-of-sequence.

Again, a particle filter is appropriate, since the recent literature has reported particle filter mod-

ifications that accommodate nonlinear-filter updates based on new past measurements, with the

need to re-filter obviated. We show results that indicate a compander/particle-filter combination

is a natural fit, and specifically that quite good performance is achievable with only 2-3 bits per

dimension per observation.

The third issue is that intelligent quantization requires that both sensor and fuser share an

understanding of the quantization rule. In dynamic estimation this is a problem since both quantizer

and fuser are working with only partial information; if measurements arrive out of sequence the

problem is compounded. We therefore suggest architectures, and comment on their suitabilities for

the task — a scheme based on delta-modulation appears to be promising.

1 Introduction

1.1 Measurement Fusion

It would appear that, for reasons of price, of survivability, and of performance, future generations

of military (and civilian) surveillance systems will comprise many heterogeneous sensors. At its

simplest level one might envision a pair or small suite of sensors each checking one another’s reports

and thereby avoiding the masking of detections that is inevitable with a fluctuating target; and

indeed, even with co-located and homogeneous sensors, there is some benefit to splitting one’s

remote sensing resources in such a way [22]. But a more adventurous interpretation of multi-sensor

surveillance involves many very cheap and not co-located sensors, perhaps of the DADS (deployable

autonomous distributed system) sort discussed in [11, 15]: there are obvious advantages in terms

of robustness, and creation of a decentralized “field” of sensors is a nice way to overcome the r−4

power-return law.

Along with the vision of a smart “web” of sensors come a number of issues, however: there is that

of deployment and layout [15], control [19], and, of greatest interest here, application to estimation.

Although distributed detection has been studied reasonably extensively (e.g., [6, 18, 20]), in most

applications the focus is, and should be, on the acquiring and tracking of threats using a distributed

array of sensors. Issues of concern here include the method for transport of information between

sensors, and the eventual fusion of information from disparate sources.

Ideally, there would be no problem at all in either respect: communication would be complete

and perfect, and there would likewise be no need for specialized estimation algorithms, since all

information from all sensors would arrive promptly. A fused estimate would therefore be no different

from that of a normal “centralized” tracking system, just with a richer set of observations on which

it would be based. Realistically, however, communication is over bandlimited channels.

What should be communicated? Blair in [4] gives a helpful taxonomy of several different fusion

approaches:

• Reporting Responsibility. Each target is assigned to a particular sensor, which makes allthe measurements and maintains the track. Presumably this sensor is that having the best

measurements (i.e., probably, is nearest), and otherwise is not so heavily loaded that such a

task is beyond its means; responsibility may be handed off to another sensor when appropriate.

Reporting responsibility is simple. But it is not really data fusion at all, saving that the

responsible sensors’ estimates can be shared; and it has very few of the above advantages of

fusion.

• Centralized Composite Tracking. Here each sensor’s observations – more likely, associated

measurement reports (AMRs) – are shared with a centralized “fusion center” tracker. The fu-

sion center digests these and broadcasts its current state estimates (composite tracks), prepara-

tory to association by the sensors’ next scan. This is ideal data fusion; its disadvantage, and

perhaps unreality, is its reliance on an excellent communications backbone.

• Distributed Track Fusion. Sensors each maintain all tracks, and information about tracks (i.e.,state estimates and covariances) is shared. This is, presumably, the lightest-loaded system as

1

regards communication. Its main disadvantage is that the state estimates thereby maintained

are necessarily correlated with each other (although it is possible to reduce this via the tracklet

approach 1[8]), and it is computationally complicated to remove this even in the idealized case

of a distributed Kalman filter.

• Distributed Composite Tracking. In this case each sensor maintains its own track of all targets,but, unlike Distributed Track Fusion, AMRs are shared.

Measurements 1 Measurements 2

Tracker 1 Tracker 2

False alarm

Platform 1 Platform 2

True detection

Measurements 1 Measurements 2

Tracker 1

Tracker 2

False alarm

Platform 1 Platform 2

True detection

Figure 1: Illustration of two fusion philosophies. Left: Distributed Track Fusion (3rd bullet); Right:

Distributed Composite Tracking (4th bullet).

The idea of the final two fusion schemes are illustrated in Figure 1. Most research seems to have

been on the penultimate scheme, despite its disadvantages (e.g. [12]). One suspects that this may

be a “cultural” bias, in that within a community whose specialty is tracking, it is not unexpected

that the item to be communicated is most comfortably a track. And, indeed, it would at first appear

that the sharing of track estimates is logical since they comprise (ideally, in a linear/Gaussian world)

a sufficient statistic of measurement history – the complexities of their combination are not too

onerous a price for miserly communication.

However, it must be recalled that in a dynamic estimation situation with measurements of

uncertain provenance, covariances as well as state estimates must be shared. Thus, in a system

estimating an N -dimensional state, there are N +N(N + 1)/2 numbers to be communicated each

time: for example, with position, velocity and acceleration being tracked in three dimensions, this

means 54 separate numbers require transmission whenever data is fused2. Further, although it may

not be necessary to transmit each as a 32-bit number, due to the processing it is not clear how

relevant each number (for example, the (2,4) element of the covariance matrix) is to the eventual

fused estimation accuracy, and hence the best strategy to use for quantization is murky.

This appears to have been noticed in [5] by Blair, who compares the communication load of

Distributed Track Fusion (3rd bullet) to that of Distributed Composite Tracking (4th bullet) –

1There are limitations for the tracklet approach in track initialization and tracking maneuvering targets.2In some track fusion schemes, communication load can be lessened in that local track information does not need

to be shared for all scans; an information filter can be used if the state dimension is much larger than the measurement

dimension; information other than track estimate can be shared in non-standard distributed fusion.

2

the scheme in which associated measurements are directly shared. He finds that they are, rather

surprisingly, comparable when each represents its numbers with the same high (32 bit) precision.

Aside from a number of studies of decentralized detection, comparatively little research has

focused on the architecture in which measurements are shared directly. However, some recent work

[16] has found that the data communication requirements of the two schemes could be similar

and that the latter could achieve further bandwidth improvement through the use of an intelligent

quantizer. Therefore, the measurement-sharing schemes may be preferable, since data fusion and

tracking is in principle straightforward. Further, it is intuitively appealing, since measurements

which might appear locally at an individual sensor to be spurious, can, if several sensors report a

measurement near the same “surprising” location, be accorded credibility.

1.2 Issues and Plan of Paper

In the preceding section we have tried to make the case that direct sharing of measurements may

be preferable to the sharing of local estimates: in the latter case fusion is at first blush “easy”

since it amounts to estimation of a Gaussian process with Gaussian measurement errors, but in

fact the correlation due to process noise roils the matter when out-of-sequence measurements are

incorporated – this putative advantage may become moot in the case of a nonlinear model.

There is a tendency to think of measurement fusion as a “bandwidth hog”, since a great many

observations need to be shared. We disagree, and counter that although there are many measure-

ments, the direct coupling of each measurement’s location to its information content leads naturally

to schemes by which it can be communicated cheaply via an appropriate quantization. Specifically,

we intend to show that even with transmission of only a few bits per dimension per measurement,

quite good performance is achievable.

Now, even beyond standard multi-sensor concerns such as gridlock and registration, there are

most certainly issues:

• sensors with limited energy and communication capability – paradigms for appropriate quan-

tization and (lossy) data sharing must be explored

• data fusion – the admission that quantized data is being fused calls into question the use of

fusion schemes based on a Gaussian assumption

• architecture – in a network of sensors one is left more free to decide on the structure of

information communication (star, ring, tree, etc.)

• measurements arriving out-of-sequence – it is inevitable, given the communication delays

within even a moderate-scale fusion network, that occasionally a measurement will arrive to

be fused after another measurement with a later time stamp.

There has been little treatment in the literature of practical low-bandwidth schemes for estimation.

Much is known about moderate-bandwidth track fusion, and about low-bandwidth decentralized

detection. There are a number of clever results on low-bitrate decentralized estimation – well

represented in [10], for instance – that promise to be helpful. However, most results on distributed

3

estimation are at least one of: asymptotic, static or Gaussian. In tracking the situation is non-

Gaussian due to measurement-origin uncertainty, it is dynamic (since interesting targets tend to

move), and the number of sensors/observations is very finite.

We shall try to visit each of the four bulletized concerns above, as follows:

• Section 2. The first concern in a nonuniform quantization scheme is the quantizer itself,

and here we propose to use the robust compander idea from communications. We explore

its performance analytically: when the measurement noise is small (compared to the process

noise and to the uncertainty from quantization) its benefits can be substantial.

• Section 3. The analytical efforts from the previous section are not based on a particular fusionscheme. However, quantized measurements yield measurement noise that is non-Gaussian, and

a practical means of fusion is required. Particle filters offer a natural means to estimate in

non-Gaussian noise, and we explore this. We further note that in a practical sensor network

measurements arrive out-of-sequence; fortunately, recent research has shown that particle

filters lend themselves to this problem as well.

• Section 4. A nonuniform quantizer implies a subtlety: both quantizer and fusion center mustagree on where the quantizer is centered. Note that this is not (necessarily) the registration

problem, and ideally does not arise if the quantization is uniform. We suggest several archi-

tectures, of which our favorite is that based on the communications concept of ∆-modulation

[17].

• Section 5. The compander/particle-filter/∆-modulator idea is investigated by simulation.As was found analytically for the compander alone in Section 2, adequate performance is

achievable with only a few bits per dimension per sensor.

We do not pretend to give comprehensive results on a low-bandwidth measurement-fusion tracker

in this paper. But by our analytic treatment of quantization under measurement uncertainty and

our suggestion for an integrated scheme we hope to stimulate interest in this practical viewpoint.

2 Benefits of Non-Uniform Quantization

2.1 Analysis of the Companding Quantizer

We shall adopt the notation that the random variable to be estimated is X, that the observations

of X are {Yi}, and that the quantized versions of these are {Zi}, where the subscript i denotes theindex of the measurement. Specifically, we have that

fX(x) =1√2πe−x

2/2 (1)

for the pdf of the target variable — note that we shall assume this to be unit normal without loss

of generality, since any other Gaussian situation can be modeled via translation and scaling. We

further have that

Yi = X + wi (2)

4

in which {wi} are iid Gaussian with mean zero and variances σ2w. We further have, assuming thatthere is no measurement-origin uncertainty,

Zi = Q [Yi] = Yi + vi (3)

where Q[·] denotes a quantization operation of the observation and vi denotes the quantization error(or noise). We anticipate by assuming the quantization is sufficiently fine that an additive noise

models its effect adequately.

The approach taken here is as follows.

1. All work will be in one dimension. It is not expected that extension to several dimensions

(the more realistic case, and that explored in the previous section) will present qualitative

differences, but we leave that for further research.

2. The quantizers will function as companders [17]. That is, with reference to (3), we have

Zi = g−1 (Qu [g(Yi)]) (4)

in which Qu denotes a uniform quantizer with resolution δ.

3. As indicated previously, the quantization is assumed to be of reasonable fidelity that the quan-

tization operation can be modeled as adding independent noise, and that standard companding

analysis techniques can be employed.

4. There must be a “gate”: measurements within this gate are quantized, coded, and transmitted,

and those outside are ignored. We take this gate as −A ≤ Y ≤ A, with a typical3 value A = 6.

5. The criterion to be optimized (minimized) is the Cramer-Rao lower bound (CRLB) [21],

reflecting the eventual accuracy of the estimation of X.

Our goal is to see whether there is some benefit to a non-uniform quantizer (versus a uniform A/D)

in estimation of a random quantity laboring under missed detections and false alarms. Ideally we

would have an explicit performance metric for an optimal quantizer and associated fusion scheme.

It may be possible to specify the optimal quantization, but experience suggests that its per-

formance gains versus a merely “good” scheme are evanescent. Borrowing from communication

practice, we adopt as a good, easy and robust approach: the logarithmic compander. The compan-

der has the right shape, in that its levels are finest where the measurement is most probable and

informative, but similar to the communication application there is no implication of optimality. In

the following we simplify by assuming that whatever the optimal fusion scheme may be, its perfor-

mance is close to the CRLB. We find that the performance gains from an intelligent non-uniform

quantization scheme can be impressive.

The idea of a companding quantizer is illustrated in Figure 2. With reference to that figure, we

note that

g(y +∆y) − g(y) = δ (5)

3The necessity of this gate is in our opinion a weakness of our work, in that one would prefer an algorithm that

gated automatically.

5

in which y is the variable to be quantized, g denotes a nonlinear function, δ and∆y are the resolution

of the uniform quantizer and the quantization resolution in y domain, respectively. Using standard

companding analysis, we get

∆y ≈ δ

g(y)(6)

where g denotes differentiation. The error arising from quantization has a uniform distribution with

variance ∆2y/12. Thus, we have

Var(Zi|X) = σ2w +∆2Yi12

(7)

We approximate this as

Var(Zi | X) ≈ s ≡ σ2w +∆2X12

= σ2w +δ2

12g(x)2(8)

We shall note that the error {Zi −X} is generally non-Gaussian. However, to derive the analyticexpression, we assume in the next section that {Zi −X} is Gaussian and independent.

Figure 2: Above: illustration of the companding idea. An observation is nonlinearly transformed and

then uniformly quantized; following this the nonlinearity is undone. Below: a typical “compressive”

nonlinearity, with quantization resolution in the observation domain (i.e. ∆) defined.

The measure we shall adopt to represent the quality of a quantizer is the CRLB. Based on (1,2,3)

and (8), plus our free assumptions of Gaussianity, we get

CRLB = EX½s2

2s2+1

s+ 1

¾−1(9)

where

s = σ2w +δ2

12g(x)2

s = −δ2g(x)

6g(x)3. (10)

Detail of the derivation is given in [16].

6

00.5

11.5

22.5

3

2

3

4

5

60

5

10

15

20

25

σw

for measurement# bits per measurement

CR

LB

no

rma

lize

d t

o u

nq

ua

ntize

d (

dB

)

2

3

4

5

6

0

2

4

6

8

100

5

10

15

# bits per measurement

A = 6, Pr(valid)= 0.1, σw

= 0.05

log2(µ) for µ−law compander

CR

LB

ga

in r

ela

tive

to

un

iform

A/D

(d

B)

Figure 3: Left: Loss in terms of CRLB for use of uniform quantization; Right: Gain from µ-law

companding, σw = 0.05 and πv = 10%.

2

3

4

5

6

0

2

4

6

8

100

1

2

3

4

5

6

7

# bits per measurement

A = 6, Pr(valid)= 0.1, σw

= 0.25

log2(µ) for µ−law compander

CR

LB

ga

in r

ela

tive

to

un

iform

A/D

(d

B)

2

3

4

5

6

0

2

4

6

8

100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

# bits per measurement

A = 6, Pr(valid)= 0.1, σw

= 1

log2(µ) for µ−law compander

CR

LB

ga

in r

ela

tive

to

un

ifo

rm A

/D (

dB

)

Figure 4: Gain from µ-law companding. Left: σw = 0.25 and πv = 10%; Right: σw = 1 and

πv = 10%.

7

2.2 Results

Before examining any specific companders, let us ask where there may be improvement to be had

at all. In Figure 3 (Left) we show the quantization loss in terms of the CRLB (9) as a function of

the number of bits assigned per measurement and in terms of the relative measurement standard

deviation (recall that the prior standard deviation is unity). The quantization loss is defined as

the ratio of the CRLB with a uniform quantizer to the CRLB without quantization; if there is any

benefit to be derived from a non-uniform quantizer it must be less than this, since an unquantized

system provides an upper-bound on performance. It is readily seen that the “interesting” cases are

those in which the number of bits is small (else the quantization noise is negligible) and/or when

the measurement noise standard deviation σw is small — for a large σw any additional error accruing

from quantization is a minor concern4.

We first consider the popular µ-law companding structure. In this case we have for z ≥ 0

g(z) =A ln (1 + µz/A)

ln (1 + µ)(11)

with g(−z) = −g(z). Results are plotted in Figures 3 (Right) and 4 for various values of σw anda fixed πv, where πv denotes the a-priori probability that a given measurement is target-generated.

Note that if the measurement does not arise from a target, then it is considered a false alarm

and afforded a uniform distribution over the “gate”. As expected from Figure 3 (Left), gains are

interesting only for few bits and small σw; but these gains can be quite substantial. Note that a

reasonably large value of µ is most appropriate; in many companding systems the value µ = 255 is

used.

3 Tracking via the Particle Filter

The non-Gaussian error introduced by the measurements’ quantization presents a difficult task for

dynamic estimation. However, the particle-filtering techniques appear quite appropriate for dealing

with this kind of problem. In the sequel we present the basic idea of particle filtering and address

some important issues that come up in implementation.

3.1 Particle Filter

The particle filtering techniques, also known as the sequential or Markov chain Monte Carlo (MCMC)

methods, have drawn an increasing level of interest in recent years. The essence of the method is

that the pdfs of interest are represented by a set of random numbers, namely the “particles” –

essentially any inference about the state, including of course its mean value and the correspond-

ing covariance, are fully derivable given the pdfs. The particles can be propagated according to

the system’s dynamic equation and updated to form a new set of posterior particles using newly

arrived observations, and this process can continue. Particle filtering methods are well suited to

non-Gaussian/non-linear estimation problems in which results from the traditional approaches such

as the extended Kalman filter (EKF) are unsatisfactory. In the sequel, we present our view of the

method.4In the former case (3) is of concern.

8

Consider a general nonlinear filtering problem. It is desirable to find the pdf f (xn|Zn1 ), whereZn1 ≡ {z1, z2, . . . , zn}. In the linear/Gaussian case this is “easy”, since the sufficient statistic for¡xn−1|Zn−11

¢is the mean xn−1|n−1 and covariance Pn−1|n−1 from the Kalman filter. But in general,

this is not so. We have

f(xn|Zn1 ) =f (zn|xn)f¡zn|Zn−11

¢ Z f(xn|xn−1)f(xn−1|Zn−11 )dxn−1 (12)

where f (zn|xn) is the observation pdf ; f¡zn|Zn−11

¢does not depend on xn; f(xn|xn−1) is the state

transition pdf and f(xn−1|Zn−11 ) is available from previous iteration.

The idea of the particle filter begins with a set of point-mass estimates of the posterior state

pdf :

f(xn−1|Zn−11 ) =1

N

NXi=1

δ(xn−1 − xn−1[i]) (13)

where N denotes the number of particles. Our goal is to generate a set of particles {xn[i]}Ni=1 thatrepresents the pdf of f(xn|Zn1 ) from the current particles {xn−1[i]}Ni=1. The Metropolis-Hastingsalgorithm provides us a tool to generate random samples which follow a specific pdf from a Markov

chain. We apply a Metropolis-Hastings special case known as the “independence sampler” to the

generation of xn: Let the transition pdf be

q(xn) = f(xn|Zn−11 ) (14)

which can be approximated by choosing j ∼ Uniform(1,N) and generating xn from f(xn|xn−1 =xn−1[j]). The true pdf is

π(xn) = f(xn|Zn1 ) ∝ f(xn|Zn−11 )f(zn|xn) (15)

and fortunately we do not need to deal with this equation directly to obtain the samples of π(xn).

Instead, we apply the Metropolis-Hasting algorithm and the rejection test is defined by:

α(x,xn) = min

½1,π(xn)q(x)

π(x)q(xn)

¾= min

½1,q(x)

π(x)κf(zn|xn)

¾(16)

Since xn does not depend on x, we can choose x however we want, so let

Pr(Reject) = 1− κZxn

f(zn|xn)dxn (17)

and choose κ to keep exactly N particles after this new scan n. In fact, we use a slightly modified

implementation:

1. For j = 1, 2, . . . , N , propagate xn−1[j] to xn[j] using f(xn[j] | xn−1[j]).

2. For j = 1, 2, . . . , N , calculate

wn[j] = κf(zn|xn[j]) (18)

where κ ensures that wn’s add to unity.

9

3. Draw from xn[j]’s N times (with replacement) according to probabilities wn[j].

This is, of course, the importance sampling/resampling or “bootstrap” filter of Gordon, Salmond

and Smith [9].

The insertion of the quantized measurement to the particle filter is straightforward. All remains

intact, except that we simply update the current particles by the quantized measurements and take

into account the extra error introduced by the quantization given in (7). This latter error is of

course non-Gaussian but is easily inserted to (18).

3.2 Out of Sequence Measurements

Out of sequence measurements (OOSMs) must be accounted for in a multisensor tracking system

where measurements from multiple sensors are sent to a processing center. Due to (random) delays

in ad-hoc network transmission, or indeed due to polling delays in more deterministic communication

networks, an earlier measurement relating to a specific target may arrive at the central processor

after a later one. In other words, the central processor receives a measurement from a particular

target with time tag tk, and it updates its state estimate using this. Later, another observation

with time tag tj (j < k) from the same target arrives at the processing center. The question is how

to incorporate the measurement taken at an earlier time in a track that has already been updated

with a later observation.

There are several potential approaches that traditional tracking algorithms use to address this

problem [3]. Among these, probably the simplest is simply to discard the delayed measurement

and hope that the lost information is not substantial. Other approaches involve using the stan-

dard Kalman filter and make a backward prediction to the time of delayed measurement. These

approaches are suboptimal as they neglect the process noise in retrodiction. In [1, 2] an optimal

OOSM solution, which took process noise into account, was developed and compared with two of the

suboptimal algorithms on some realistic cases. More recently, interest has been drawn to a general

approach to cope with OOSMs on nonlinear and non-Gaussian models – including, of course, a

model with data association. One idea is to absorb the OOSMs in the particle filter framework [13],

and we now give a brief overview of that.

Suppose at time tn we have an ordered set of in-sequence measurements Zn1 . The next measure-

ment zκ arrives and is found to be out-of-sequence, such that τ < tn (τ is the time with which the

OOSM zκ was tagged). It is further assumed that

tb < τ < ta (19)

where b and a are the contiguous time indices (tags) respectively before and after the OOSM in the

ordered sequence Zn1 . Then the updated posterior is

p∗(Xn0 |Zn1 ) = p(Xb

0|Zb1)p(zκ|xκ)p(xκ|xb)p(xa|xκ)

p(xa|xb)p(zκ) p(Xna |Zna) (20)

which can be simplified to

p∗(Xn0 |Zn1 ) = p(Xn

0 |Zn1 )p(zκ|xκ)p(xκ|xb)p(xa|xκ)

p(xa|xb)p(zκ) (21)

10

Then the particle weight (18) can be updated by

w∗n[i] = wn[i]p(zκ|xκ[i])p(xκ[i]|xa[i],xb[i])

π(xκ[i]|Xn0 [i],Z

n1 )

(22)

In [14], tracking performance was investigated in a simulated wireless network for a wide range of

communication delays. Results show that position estimation accuracy close to the lower bound

should be possible for communication intervals up to four times the average inter-scan time. How-

ever, the authors did not consider the communication bandwidth problem: they assumed that

measurements were shared without loss of quality due to communication.

4 Architecture of the Sensor Network

The measurements of all the sensors among the wireless sensor network are shared directly with

the fusion center, and communication issues must be considered. A practical protocol known as

the Bluetooth architecture [7] has been developed. In that case a frequency hopping Time Division

Multiple Access (TDMA) approach is taken and the sensor nodes form piconets where one of the

nodes becomes the “master”. The master node, which plays the role of fusion center, broadcasts and

other nodes (sensors) receive during odd time slots and the master receives while the other nodes

broadcast during even time slots. Intra-piconet collisions are avoided by only permitting one node

to communicate with the master during each time slot. This leads to the concept of the master

polling a node, which then replies in the following time slice, which in turn implies that a node

must in general wait for several time slots before being able to communicate with its master again.

Piconets can be networked together by forming their master nodes into a net with one node being

nominated as the overall master.

Intelligent quantization requires that both sensor and fuser share an understanding of the quan-

tization rule, and in dynamic estimation this is a problem, particularly so if measurements arrive out

of sequence. There must be a “measurement gate” to implement quantization schemes: measure-

ments within this gate are quantized, coded, and transmitted, and those outside are ignored. We

take this gate as xn−A ≤ Y ≤ xn+A, and use the typical value A = 6σm and x the estimate of thetarget state. Aside from the gate, the quantization rule consists of quantization scheme selection,

which in the case of a compander means the nonlinearity. Now, in a dynamic system all these can

be pre-defined and known to all sensors in the network; all, that is, except for the center of the

gate – the location of the “middle” of the non-uniform quantizer.

This may seem at first trivial. However, since all information is processed at the fusion center

and only there is the repository of the global estimate of target state. Sensors need to be constantly

informed of this information – presumably by the fusion center – in order to implement the

quantization schemes. The fusion center could broadcast this information over its allocated time

slots; however, such transmission would consume bandwidth, and would hence appear incompatible

with the original intent to save on bandwidth. There are probably many approaches to deal with

this. Here we propose four: the first three do not require “feedback” broadcast from the fusion

center to the sensors, while the last one requires a low-bandwidth broadcast.

• In the first scheme (A), a fusion center broadcast is avoided by arranging each sensor itself totrack based upon the (local) measurements available to it, with the center of its gate obtained

11

from this local estimate. The fuser uses the global estimate as an approximate center of the

gate to decode the quantized measurements. This scheme would appear to work well if the

difference between the local and the global estimates are much less than the error introduced

by the process noise, measurement noise or quantization noise; that is, when the local track

is decent.

• The second scheme (B) is an overlay to the first. Since the fusion center has access to allmeasurements and is, presumably, powerful in computational capability, it can not only calcu-

late the global estimate of the targets, but also infer what each sensor knows about the target

state, i.e., the local estimate of each sensor. Thus, when decoding the quantized measurements

the fuser can use its estimate of each local estimate as the center of each local gate, instead

of its own global estimate.

• A third approach (C) is suggested by the ∆-modulation scheme [17] from communication

theory: each sensor will use its local track estimate to update the center of its gate, and

will provide this information to the fusion center along with its measurement data. This

communication is most efficient when it is specified in terms of movement in integer numbers

of the local sensor’s own quantization grid, and the special case of single-bit transmission is

perhaps the most interesting.

• In the fourth idea (D) the ∆-modulation scheme of the previous is altered such that the fusioncenter commands the movement (gate re-alignment) of each sensor by a broadcast message.

We note that in schemes C & D there is no need to constrain grid-movement communication

to be only of a single bit. There is some comparison of schemes in Table 1. The last row of that

table, the expected performance, is subjective: we would expect that schemes A & B will suffer

when the fusion center’s understanding of the quantization grid deviates significantly from those of

the local sensors, and that scheme D is preferable to C when tracking is challenging due to the grid

movement’s being controlled by more-accurate fused-track information. Scheme C is perhaps the

“cleanest” of the group, since grid-movement information arrives at the fusion center along with all

data; however, an OOSM will perturb estimates, since already-arrived data must be re-examined in

light of a previously undeclared grid movement. In the following sections we pay particular attention

to scheme D, but we note that the other schemes are interesting and deserve further study.

Scheme: A B C D

FC → LS no no no yes

LS → FC no no yes no

local tracker yes yes yes no

performance poor ? fair good

Table 1: Comparison of four schemes to manage fusion center grid movement, as given in the text.

The first and second rows relate to the need for fusion center (FC) to local sensor (LS) and reverse-

path communication of grid-movement information. The third row shows whether there is a need

for a local tracker. The fourth row gives expected level of performance in challenging tracking

situations.

12

5 Simulation Results

The target motion is modeled as constant velocity using the standard kinematic equations

xn+1 = Fxn + vn

yn = h(xn, tn) +wn (23)

for n = 1, 2, . . . , N . Here, as usual, xn represents the state of the target at time tn, and yn its

corresponding observation; the matrix F and function h(·) are known, and are assumed to representan observable and controllable system. The random sequences {vn,wn} are assumed white, zero-mean, Gaussian, and mutually independent, with E{vnvTn} = Qn, E{wnwTn } = Rn.

Suppose the time interval between two consecutive sampling point is T , then for one dimension

F =

"1 T

0 1

#, Q = q

"T 3

3T 2

2T 2

2 T

#(24)

and for two dimensions

F =

1 T 0 0

0 1 0 0

0 0 1 T

0 0 0 1

, Q = q

T 3

3T 2

2 0 0T 2

2 T 0 0

0 0 T 3

3T 2

2

0 0 T 2

2 T

(25)

where q is the (continuous-time) process noise power spectral density. The false alarms, if present,

are assumed uniformly distributed in the surveillance region V and their number follows a Poisson

distribution with mean λV , where λ denotes the spatial density.

We test the algorithms in two different types of scenarios:

1. one dimensional motion, linear measurement model and two asynchronous (random) sensors,

with and without false alarms.

2. two dimensional motion, bearing only measurement model and in the wireless sensor network

of [14], without false alarms.

In the following we present the simulation results obtained from these scenarios and take the mean

squared error (MSE) of the estimate as the measure of the tracking performance.

5.1 Linear Motion and Linear Measurement

We first test the algorithm in the simplest case — one target moves in a one dimensional Cartesian

coordinate, and its position is observed directly. We can write the measurement equation as

yn = Hxn +wn (26)

and

H = [ 1 0 ] , R = σ2m (27)

Two sensors are at different locations and are arranged to take scans alternately. Sensor 1 is

designated as a local sensor (relative to the fusion center) and sensor 2 is a remote sensor. We

13

assume that every measurement from sensor 2 is sent to the fusion center and the transmission is

subject to a random delay t, where t follows an exponential distribution with mean τ .

We attempt to achieve bandwidth improvement via intelligent quantization schemes and dealing

with the OOSMs. The parameters are:

• process noise power spectral density q = 0.01 m2/s3;

• measurement noise standard deviation σm = 30 m;

• sampling time interval T = 30 s;

• initial target state x0 = [0 5]T (m m/s);

• probability of detection pD = 1.0.

• clutter (false alarms) density λ = 0 or λ = 10−3 1/m.

In Figure 5 we compare the particle filter with the Kalman filter when the quantized measure-

ments are used. It can be seen that the particle filter outperforms the Kalman filter significantly

when the quantization bits are few. This is readily explained since the small number of quanti-

zation bits introduces large quantizational non-Gaussian errors into the measurement uncertainty

and the Kalman filter is designed only to handle Gaussian noise. When the non-Gaussian measure-

ments noise components are not ignorable, the performance deterioration of the Kalman filter is

inevitable. On the contrary, such a problem does not exist in the particle filter due to its capability

to incorporate the non-Gaussianity and nonlinearity of the model.

1 2 3 4 5 6 7 8200

250

300

350

400

450

500

550

600

MS

E

number of the quantization bits

PF w/ µ-law quantized measurementsKF w/ µ-law quantized measurementsKF w/ unquantized measurements

Figure 5: Comparison of the MSEs of the particle filter and the Kalman filter using µ-law quantizerin the false alarms free scenarios. (Pd = 100%, q = 0.01m

2/s3, σm = 30 m , T = 30 s.)

Simulation results are shown in Figures 6 (Left) and 7 (Left) for clutter-free scenarios and in

Figures 6 (Right) and 7 (Right) for scenarios with clutter. Figures 6 display the results of the µ-law

compander versus the uniform quantizer. We can see a similar pattern exhibited in both figures:

the µ-law compander outperforms the uniform quantizer in ranges from 2 bits to 6 bits, and the

improvement is most significant (around 10% difference) at 3 bits; the two schemes yield almost the

same MSEs at 1 bit and the ranges over 7 bits. To understand the figures, note the two quantizers

14

are identical for 1-bit quantization, and when bit-rates are high enough, the quantizations are very

fine and the errors introduced by them are negligible compared to the measurement uncertainty.

We also display the results for the unquantized (ideal) case, which means perfect transmission and

infinite bandwidth, with estimation via the Kalman filter (in Figure 6 (Left)) and the PDAF (in

Figure 6 (Right)), as the performance references. In Figure 6 (Left) we can see when the number

of bits increases, the performance of both quantizers approaches that of the Kalman filter.

1 2 3 4 5 6 7 8190

200

210

220

230

240

250

260

MS

E

number of the quantization bits

PF w/ uniform quantizerPF w/ µ-Law quantizerKalman filter w/o quantization

1 2 3 4 5 6 7 8220

240

260

280

300

320

340

MS

E

number of the quantization bits

PF w/ uniform quantizerPF w/ µ-Law quantizerPDAF w/ µ-Law quantizationPDAF w/o quantization

Figure 6: Comparison of the MSEs of the particle filters using µ-law and uniform quantizers,

(Pd = 100%, q = 0.01m2/s3, σm = 30 m , T = 30 s). Left: In the false alarms free scenarios. Right:

In the presence of false alarms with spatial density λ = 10−3 m−1.

We are also interested in how much improvement is offered through processing the OOSMs as

opposed simply to ignoring them; the result is shown in Figures 7. As upper and lower baselines,

we show in Figure 7 (Left) respectively the results of the Kalman filter that processes only local

measurements, and of the clairvoyant KF that has access to all measurements taken in sequence. In

Figure 7 (Right), the process is repeated except that we use the PDAF to replace the Kalman filter,

since false alarms are present; there is a similar pattern. In both cases the improvement of MSE

ranges from 5% when the average delay is 0.5T0 to around 15% when the average delay increases

beyond T0 – T0 denotes the time interval between two closest sensor scans. This indicates that we

can improve our estimation accuracy significantly by processing OOSMs. When τ gets larger, the

difference between with and without processing OOSMs diminishes, since both curves will converge

to the upper bound, that being the local-sensor-only case. Note that in the figures the MSEs of the

particle filter that processes OOSMs increase (in a relatively slow manner) with the increase of τ .

This is due to the following two factors:

1. The first and the dominant one, when τ increases, more measurements become out-of-sequence

and therefore fewer measurements are available to the current estimate. To amplify, suppose

the measurement yn reflects the target state at time tn but arrives at the processor at a later

time tn+k. Consequently yn fails to provide information to the estimate xn, although it can

be used by the estimate xn+k via OOSM processing. In other words, after all, OOSMs lead

to information loss, and any OOSM algorithm can only alleviate this.

2. Secondly, we set the maximum lag (8 seconds here) when dealing with OOSMs – the mea-

15

surements arriving with a time lag greater than the maximum lag will not be processed. Thus

when τ becomes larger than this, more measurements are likely to arrive “too late”, and will

be discarded and we lose more information.

With false alarms present, an interesting phenomenon is seen in Figures 6 (Right) and 7 (Right):

the particle filter outperforms the PDAF in the ranges of large number of bits or small τ . Presumably

this is due to the PDAF being a suboptimal algorithm, while the particle filter is (asymptotically)

optimal. In contrast, from Figures 6 (Left) and 7 (Left), we see that the particle filter always

converges in performance to the (optimal) Kalman filter (the small gap between them shown in

Figure 7 (Left) is due to the 6-bit quantizer used in the particle filter.)

0 0.5 1 1.5 2

210

220

230

240

250

260

270

280

290

6 bits quantizer

MS

E

average delay τ0 (nomalized by T0)

KF1 with local meas. inputPF w/ OOSM rejectedPF w/ OOSM processedPF w/ OOSM prcsd and ∆-modulationKF2 with all meas. input

0 0.5 1 1.5 2240

260

280

300

320

340

360

3808 bits quantizer

MS

E

average delay τ0 (nomalized by T0)

PDAF1 with local meas. inputPF w/ OOSM rejectedPF w/ OOSM processedPDAF2 with all meas. inputPF w/ OOSM prcsd and ∆-Modul.

Figure 7: Comparison of the MSEs of the particle filter with OOSM processed versus the particle

filter with OOSM discarded, (Pd = 100%, q = 0.01m2/s3, σm = 30 m , T = 30 s). Left: In the false

alarms free scenarios. Right: In the presence of false alarms, (λ = 10−3 m−1 and ∆-modulation isalso included).

In the above simulations we assume that the quantization rule is shared perfectly by the two

sensors. This is of course not the case in practice. We adopt ∆-modulation as the way of sharing

the estimate (scheme D), and the results are shown in Figures 7; they are also compared to the case

of perfect sharing of estimate (i.e., both sensors have the same understanding of the quantization

center). It can be seen the performance degrades by the surprisingly small factor of 4% through the

use of 1-bit-per-coordinate communication. This factor is scenario dependent and can presumably

be reduced even further by increasing the bandwidth usage; but it is encouraging for scheme D.

5.2 A Wireless Sensor Network

The plant equation in (23) is defined in Cartesian coordinates by xn ≡ [ px(tn) vx(tn) py(tn) vy(tn)],with observations yin ≡ [ θ(tn) ] in bearing for the ith sensor given in polar coordinates. The mea-surement model is given by

yin = arctan

Ãpy(tn)− siypx(tn)− six

!+wn (28)

where [siy, siy] denote the position of the i

th sensor in Cartesian coordinates.

16

We consider a small sensor network given in [14]. It comprises 11 sensors and a tree architecture

which essentially is a scatternet formed by three piconets. The master node for the scatternet

represents the “command” node where data fusion and tracking will be performed. The sensors are

modeled as simple bearings-only sensor types, representative of an acoustic array. The fusion center

obtains measurements from other sensors according to the Bluetooth polling protocol. A schematic

overview of the architecture is shown in Figure 8.

Fusion Center

Sensor

Figure 8: The architecture of the sensor network.

The observation rate for each of the sensor nodes in the scenario was fixed at one per second, and

the scenario was re-run for a variety of communication rates to investigate their effects on tracking

performance. The scenarios were also run for a variety of quantization rates. The master shares its

knowledge of target state estimate by 1 bit ∆-modulation transmission (i.e., scheme D).

The parameters of the scenario are:

• process noise power spectral density q = 0.1 m2/s3;

• measurement noise standard deviation σm = 1◦;

• sampling time interval T = 1 s;

• initial target state x0 = [88.3 − 4.0 766.6 − 4.0]T (m m/s m m/s);

• probability of detection pD = 1.0.

To minimize power consumption, we attempt to extend the time intervals between transfer of

data packets between nodes and to choose an appropriate intelligent quantization scheme. At the

same time, in order to maintain tracking performance, we need to process OOSMs. Whichever

quantizer we choose, the master node needs to share the updated target state estimate with all

sensors since this knowledge is vital to the quantization task. In order to inform all the sensors

of this information, we opt for the ∆-modulation idea. The fuser broadcasts 1 bit per coordinate

message to all sensors during odd time slots. Upon receiving this message, the sensors update their

current estimate of the target state by adding/subtracting a steplength and take it as the “window”

center for the quantization.

Simulation results are shown in Figures 9. In the left figure, we compare the uniform quantizer

with the µ-law compander for different quantization rates. The time intervals between transfer of

17

data packets is set to 1 second and the OOSMs are processed by the two schemes. Both quantizers

adopt the ∆-modulation scheme which needs 1 bit per coordinate bandwidth for communication:

the uniform quantizer requires grid-movement commands too, since to be efficient it must spread its

levels within a gate. We can see that the µ-law compander offers 20% and 6% MSE improvements

versus the uniform quantizer for 2 and 3 bits respectively; the difference becomes negligible when the

number of bits exceeds 3. In the right figure, we also show the MSEs of the particle filters with and

without processing of OOSMs in which µ-law compander is used and the number of quantization

bits is set to 6. It can be seen that the processing of the OOSMs improves the performance quite

significantly, and this is especially so in large communication-delay situations.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 625

30

35

40

45

50

55

MS

E

number of the quantization bits

PF w/ uniform quantizerPF w/ µ-Law quantizer

1 1.5 2 2.5 3 3.5 420

40

60

80

100

120

140

1606 bits µ-law quantizer

MS

E

Communication interval T

PF w/ OOSM rejected and ∆-Modul.PF w/ OOSM processed and ∆-Modul.

Figure 9: Left: Comparison of the MSEs of particle filters using µ-law and uniform quantizer.

(Pd = 100%, q = 0.1m2/s3, σm = 1

◦, T = 1 s). Right: Comparison of the MSEs of particle filterswith OOSM processed versus particle filter with OOSM discarded. (Pd = 100%, q = 0.1m2/s3,

σm = 1◦).

6 Summary

Here we consider the multi-sensor tracking problem with energy and/or bandwidth constraints –

both amount to a preference for reduced communication. We are also particularly interested in

large sensor networks of irregular (or ad-hoc) structure, since these offer attractive robustness and

economy. A concern with such networks, however, is the inevitability of out-of-sequence measure-

ments (OOSMs), due either to the randomness of information propagation or to the necessity for a

sensor-polling architecture.

For estimation fusion most researchers appear to favor track fusion architectures in which local

state estimates and their associated covariances are shared directly. As compared to measurement

fusion, track fusion seems appealing in its simplicity and parsimonious communication. However,

in track fusion, one has to address the cross-correlation due to process noise and prior information,

which is much harder than measurement-to-track association and fusion. Furthermore, OOSMs

are difficult to absorb, and it has recently been shown that the claim that track fusion has lower

communication needs may be open to debate.

18

In fact, we have indicated in the first part of this paper that via an intelligent quantization

scheme (such as a compander) the communication needs of measurement fusion can be cut still

further. Similarly, a recent paper by one of the authors found a neat procedure for the processing of

OOSMs based on particle filters – this scheme is applicable to the non-Gaussian measurement noise

accruing from the quantization operation, but also subsumes nonlinear models (such as bearings-

only measurements) and data association. Fusion of quantized measurement via particle filtering is

thus a natural marriage of the two ideas.

In the second part of this paper, we have discussed some practical problems of target tracking in

sensor networks and suggested solutions by making the particle filtering techniques to work with the

OOSMs and quantizers. We have implemented the particle filtering/OOSM/quantization algorithms

in two types of scenarios and the simulation results have shown that exploiting OOSMs in wireless

sensor network can effectively improve tracking performance. The improvement margin yielded by

OOSMs enables us to extend the communication intervals between sensors for the purpose of energy

saving while not compromising the overall performance. We have further shown that the intelligent

quantization of measurements is more effective than the uniform quantization in terms of bandwidth

efficiency.

The compander we used here is perhaps only “partially intelligent”. That is, we assumed a fixed

number of quantization bits as well as a fixed size (length, area or volume) of quantization gate.

The questions naturally arisen are:

• Should we make the quantizer data-adaptive?

• How much improvement in performance will that yield?

• When, presumably in terms of target maneuvering index, is the improvement most interesting?

• Is there a structure that is preferable to the one that we have adopted?

As regards the last, let us note that to use a non-uniform quantizer it is necessary that both quantizer

and fuser share a common understanding of the scheme: where should the quantization levels be

most dense and where most sparse? In this work we have adopted the ∆-modulation paradigm, in

which to save bandwidth the fusion center broadcasts changes in its state estimate using only one

bit per coordinate. This appears to work reasonably well, but other approaches are possible.

Here is a brief list of our findings (and musings):

• There is little loss from quite coarse quantization of measurements: only 2-3 bits per dimension.

• Particle filtering is an attractive way to allow for the non-Gaussian measurement noise accruingfrom quantization.

• Use of both non-uniform and uniform quantization requires a common understanding of a

quantization grid. This grid must be modified dynamically, and in a manner that is synchro-

nized between sensors and fusion center.

• There are a number of ways to achieve this synchronized movement. The one that we haveinvestigated, and which seems to hold most appeal, is that in which movements are according

to a ∆-modulation command from the fusion center.

19

• OOSMs ought not, in general, to be rejected. Again, the particle filtering methodology offersa simple means to incorporate them.

References

[1] Y. Bar-Shalom, “Update with Out-of-Sequence Measurements in Tracking: Exact Solution”,Proceedings of the 2000 SPIE Conference on Signal and Data Processing of Small Targets, April2000.

[2] Y. Bar-Shalom, M. Mallick, H. Chen and R. Washburn, “One-Step Solution for the GeneralOut-of-Sequence Measurement Problem in Tracking”, Proceedings of 2002 IEEE AerospaceConference, Big Sky MT, March 2002.

[3] S. Blackman and R. Popoli, Design and Analysis of Modern Tracking Systems, Artech House(Boston), 1999.

[4] W.D. Blair, “Practical Aspects of Multisensor Tracking”, in Multitarget-Multisensor Tracking:Applications and Advances (Volume III), Bar-Shalom & Blair eds., Artech-House, 2000.

[5] W.D. Blair, “Multiplatform/Multisensor Tracking System Architectures and Technical Issues”,presented at the Third ONR/GTRI Workshop on Target Tracking and Data Fusion, May 17,2000.

[6] R. Blum, “Necessary Conditions for Optimum Distributed Detectors Under the Neyman-Pearson Criterion”, IEEE Transactions on Information Theory, May 1996.

[7] J. Bray, C.F. Sturman, Bluetooth 1.1: Connect Without Cables, Prentice Hall.

[8] O. Drummond, “Track and Tracklet Fusion Using Data from Distributed Sensors”, Proceedingsof the Workshop on Estimation, Tracking and Data Fusion: A Tribute to Yaakov Bar-Shalom,Monterey CA, May 2001.

[9] N. Gordon, D. Salmond and A. Smith, “Novel Approach to Nonlinear/Non-Gaussian BayesianState Estimation”, IEE Proceedings F, Vol. 140, No. 2, pp. 107-113, 1993.

[10] T. Han and S. Amari, “Statistical Inference Under Multiterminal Data Compression”, IEEETransactions on Information Theory, pp. 2300-2324, October 1998.

[11] E. Jahn, M. Kaina and M. Hatch, “Fusion of Multisensor Information from an AutonomousUndersea Field of Sensors”, Proceedings of the International Conference on Information Fusion,Sunnyvale CA, July 1999.

[12] X. Li and K. Zhang, “Optimality and Efficiency of Optimal Distributed Fusion”, Proceedingsof the Workshop on Estimation, Tracking and Data Fusion: A Tribute to Yaakov Bar-Shalom,Monterey CA, May 2001.

[13] M. Orton and A. Marrs, “A Bayesian Approach to Multi-Target Tracking and Data Fusion withOut-of-Sequence Measurements”, IEE International Seminar on Target Tracking: Algorithmsand Applications, October 2001.

[14] M. L. Hernandez, A. D. Marrs, S. Maskell, M. R. Orton, “Tracking and Fusion for Wire-less Sensor Network”, Proceedings of the 5th International Conference on Information Fusion,Annapolis, MD, July 2002.

[15] M. Owen, D. Klamer and B. Dean, “Evolutionary Control of an Autonomous Field”, Proceed-ings of the International Conference on Information Fusion, Paris, July 2000.

[16] F. Palmieri, S. Marano, P. Willett, “Measurement Fusion for Target Tracking Under BandwidthConstraint”, Proceedings of the 2001 IEEE Aerospace Conference, Big Sky MT, March 2001.

[17] P. Peebles, Digital Communication Systems, Prentice-Hall NJ, 1987.

20

[18] A. Reibman and L.W. Nolte, “Optimal Detection and Performance of Distributed Sensor Sys-tems”, IEEE Transactions on Aerospace and Electronic Systems, January 1987.

[19] P. Shea, M. Owen, “Fuzzy Control in the Distributed Autonomous Deployable System”, Pro-ceedings of the SPIE Signal Processing, Data Fusion and Target Recognition, Orlando FL, April1999.

[20] J. Tsitsiklis, “Decentralized Detection”, in Advances in Statistical Signal Processing, vol. 2 —Signal Detection, Poor and Thomas, editors, JAI Press, 1990.

[21] H. Van Trees, Detection, Estimation and Modulation Theory, Part I, Wiley, 1968.

[22] P. Willett, M. Alford and V. Vannicola, “The Case for Like-Sensor Pre-Detection Fusion”,IEEE Transactions on Aerospace and Electronic Systems, October 1994.

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