Robust Target Tracking with Quantized Proximity

2010 5th International Symposium on Wireless Pervasive Computing (ISWPC)

Robust Target Tracking with Quantized Proximity Sensors Majdi Mansouri, Hichem Snoussi and Cedric Richard ICD/LM2S, University of Technology of Troyes, FRANCE Abstract—As low energy, longevity and low cost transmitters are major requirements for wireless sensor networks (WSN), optimizing their design under energy constraints is of a great importance. To this goal and in order to solve the target tracking problem in WSN, we develop an algorithm to jointly, estimate the target position and optimize the quantization level where the transmitting power could be controlled by the sensors. At each sampling time, the adaptive algorithm provides not only the estimation of the target position by using the variational filtering (VF) but also, it gives the optimal quantization level. This optimal is obtained by minimizing the transmitting power under predicted Cram´er-Rao bound constraint. Performance analysis shows that the proposed method outperforms both the VF algorithm with uniform quantization strategy and the VF algorithm based on binary sensors. Compared to the uniform quantization strategy, numerical examples show that energy saving up to 80% can be achieved when each sensor generates the same number of bits.

I. I NTRODUCTION A wireless sensor network (WSN) consists of a large number of geographically distributed nodes, each of which equipped with a finite-life battery and limited computation, communication and mobility capabilities. These considerations motivate the employ of effective energy-saving and the design of energy-efficient WSN. One approach to prolong battery lifetime is to jointly estimate the target position and optimize the number of quantization bits per observation. The problem of quantizing observations to estimate a parameter, such as the target position or any other physical field (temperature, humidity, ...), is different from the problem of quantizing a signal for later reconstruction [1]. Instead of recovering a signal, our objective is rather estimating the target trajectory using quantized observations. Target tracking using quantized observations is a nonlinear estimation problem which can be solved using e.g., unscented Kalman filter (KF) [2], particle filters [3] or variational filtering (VF) algorithm [4]. To solve this problem, in [5], the authors have derived the optimal number of quantization levels as well as the optimal energy allocation across bits, but they have neglected the information content relevance of measured data. While the work in [6], Majdi et al. have used VF algorithm to estimate the target position but the transmitting power is assumed to be constant for all sensors. In this paper, we investigate the problem of target tracking using WSN composed by quantized proximity sensors. As in power and bandwidth efficient solution, the quantized proximity sensor can only make a quantized decision based

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on the strength of the sensed signal. The realistic quantized sensor network (QSN) suffers from several problems related to the noisy link and low estimation precision. To overcome these problems, we have developed the quantized variational filtering (QVF) algorithm, which perfectly fits the highly nonlinear conditions and eliminates the transmission error. Our contribution lies in the following aspects: 1) We improve the use of VF algorithm by a good precision of target tracking and better prediction of its evolution: we reduce errors in decision taken by the cluster head (CH) for next computations; 2) We investigate the impact of the choice of a fixed quantization level and uniform power on the VF algorithm performances and the importance of adaptive quantization; 3) We optimize the quantization level to minimize the transmission energy consumption in WSN. The proposed algorithm provides not only the Bayesian filtering distribution of the target position but also optimizes the quantization level as well as the minimal of the transmitting power. The rest of the paper is organized as follows. Section II presents the quantized proximity observation model. The bayesian estimation approach via a QVF algorithm is presented in Section III. Then, the main contribution of this paper which is the adaptive quantization is detailed in Section IV. Numerical simulations are shown in Section V. Finally, Section VI concludes the paper. II. M ODELING AND PROBLEM STATEMENT A. Quantized Proximity Observation Model QPOM In this work, we consider a sensor network with known sensor locations si = (si1 , si2 ), i = 1, 2, . . . , Ns , where Ns is the total number of sensors. Our objective is to estimate the target position xt = (x1,t , x2,t )T at each instant t (t = 1, . . . , N ), where N denotes the number of observations. At the sampling time t, the information, γti , observed by the ith sensor can be expressed as γti = K!xt − si !η + "t ,

(1)

where "t , is a Gaussian noise with zero mean and variance σ"2 and η and K are a known constants. The ith sensor transmits its observation if and only if Rmin ≤ !xt − si ! ≤ Rmax , where Rmax denotes the maximum distance at which the sensor can detect the target, and Rmin is the minimum distance from which the sensor can detect the target. Before being transmitted, the observation is quantized by partitioning the observation space into Nti intervals Rj = [τj (t), τj+1 (t)], where j ∈ {1, . . . , Nti }. Nti denotes the quantization level used

278

i

by ith sensor (Nti = 2Lt where Lit represents the number of quantization bits per observation) which is to be determined. The quantization level, Nt , is sub-indexed by the sampling instant t since it will be optimized online and jointly using online target position estimation. The quantizer, assumed unij=N i +1 form, is specified through: (i) the thresholds {τj (t)}j=1 t , η where (if η ≥ 0): τ1 (t) = KRmin , τj (t) ≤ τj+1 (t) and η τNt +1 (t) = KRmax ; and (ii) the quantization rule yti = dj if

γti ∈ [τj (t), τj+1 (t)],

(2)

where dj is the centroid of jth cell. Then, the signal received by the cluster head from the sensor i at the sampling instant t is written as, zti = βti yti + nt , (3) where βti = riλ is the ith sensor channel attenuation coefficient at the sampling instant t, ri is the transmission distance between the ith sensor and the CH, λ is the pass-loss exponent and nt is a random Gaussian noise sensor with a zero mean and a variance σn2 . Figure.1 summarizes the transmission scheme. B. General state evolution model In this paper, we employ a general state evolution model GSEM [7] instead of the kinematic parameter model [8] usually used in tracking problems. In fact, the GSEM model, introduced in [9] for visual tracking, is more practical to nonlinear and non-gaussian situations where no a priori information on the target velocity. Considering a planar geometry, the target position xt at instant t is assumed to follow a Gaussian model, where the expectation µt and the precision matrix λt are both random. Gaussian distribution for the expectation and whishart distribution for the precision matrix form a practical choice for these distributions. The hidden state xt is extended to an augmented state αt = (xt , µt , λt ), yielding an hierarchical model as follows,  ¯ µt ∼ N (µt | µt−1 , λ)      ¯ λt ∼ Wn¯ (λt | S) (4)      xt ∼ N (µt , λt ) ¯ n where the fixed hyperparameters λ, ¯ and S¯ are respectively the random walk precision matrix, the degrees of freedom and the precision of the Wishart distribution. Note that assuming random mean and covariance for the state xt leads to a prior probability distribution covering a wide range of tail behaviors allowing discrete jumps in the target trajectory. In fact, the marginal state distribution, p(xt | xt−1 ), is obtained by integrating over the mean and precision matrix % p(xt | xt−1 ) = N (xt | µt , λt )p(µt , λt | xt−1 )dµt dλt

(5) where the integration with respect to the precision matrix leads to the known class of scale mixture distributions introduced by Barndorff-Nielsen [10]. Low values of the degrees of

freedom n ¯ reflects the heavy tails of the marginal distribution p(xt | xt−1 ). The next section describes the Bayesian estimation approach via the VF algorithm. III. BAYESIAN

ESTIMATION APPROACH VIA THE

VF

ALGORITHM

A. Overview of the VF algorithm According to the model (4), the augmented hidden state is now αt = (xt , µt , λt ). The distribution of interest for target tracking takes the form of marginal posterior distribution p(αt | z1:t ), where z1:t = {z1 , z2 , . . . , zt } denotes the collection of observations gathered until time t by the CH (which is activated according to the prediction made by the variational filtering algorithm). The variational approach consists in approximating p(αt | z1:t ) by a separable distribution q(αt ) = Πi q(αit ) that minimizes the Kullback-Leibler (KL) divergence between the true filtering distribution and the approximate distribution, % q(αt ) DKL (q||p) = q(αt ) log dαt (6) p(αt | z1:t ) & & under the constraint q(αt )dαt = Πi q(αit )dαit = 1. The distribution q(αt ) is approximated by: q(αit ) ≈ exp (log p(zti , αt ))Πj!=i q(αj ) t

(7)

where (.)q(αj ) denotes the expectation operator relative to the t distribution q(αjt ). The updated separable distribution q(αt ) has the following form q(xt ) q(µt ) q(λt ) q(µt | µt−1 )

≈ ≈ ≈ ≈

p(zt | xt )N (xt | (µt ), (λt )) N (µt | µ∗t , λ∗t ) Wn∗ (λt | St∗ ) N (µpt , λpt )

where the parameters are iteratively updated according to the following scheme µ∗t =λ∗−1 ((λt )(xt ) + λpt µpt ) t ∗ λt =(λt ) + λpt n∗ =¯ n+1 St∗ =((xt xTt ) − (xt )(µt )T − (µt )(xt )T + (µt µTt ) + S¯−1 )−1 µpt =µ∗t−1 ¯ −1 )−1 λpt =(λ∗−1 t−1 + λ (8) Hence, the QVF provides at the sampling instant t, the predicted target position xt/t−1 . Based on the predicted target position (xt )qt|t−1 , which is the information transmitted to the cluster head CHt−1 at sampling instant t − 1, to the cluster head CHt at sampling instant t ; the CHt is chosen. If the predicted target position (xt )qt|t−1 remains in the vicinity of CHt−1 , which means that, at least four of its salve sensors, can detect the target [11], then CHt = CHt−1 . Otherwise, if (xt )qt|t−1 is going beyond sensing range of the current cluster,

279

nt

!t

F

γt1

K||xt − s1t ||η

yt1

Q

zt1 nt

!t

xt

F

K||xt −

γt2

s2t ||η

yt2

Q

Cluster Head CH

zt2

Target

F

ztNs

nt

!t

γtNs

s η K||xt − sN t ||

ytNs

Q

Figure 1. Illustration of the communications path-ways accruing in a WSN. The 1st sensor measures its information relevance F (xt ) = K!xt − s1 !η and makes a noisy reading γt1 . The quantized measurement yt1 = Q(γt1 ) with L1t bits of precision is sent to the CH. The measurement zt1 is received by the CH, it is corrupted with additive white Gaussian noise nt .

then a new CHt is activated, based on the target position prediction (xt )qt|t−1 and its future tendency CHt = arg max k=1,...,K

'

cos θtk dkt

(

(9)

A. Predicted Cram´er-Rao bound (CRB) The Cram´er-Rao bound (CRB) is often used to evaluate the efficiency of a given estimator. In its simplest form, the bound states that the covariance of any estimator is at least higher as the inverse of the Fisher information (FI) matrix. The FI matrix is a quantity measuring the amount of information that the observable variable zti carries about the unknown parameter xt . At the sampling time t, the FI entries are given by ,

where dkt = !(xt )qt|t−1 − LCHtk ! )−−−−−−−−−−−→ −−−−−−−−→* and θtk = angle (xt−1 )(xt )qt|t−1 , (xt−1 )LCHtk

where K is the number of CHs in the neighborhood of CHt−1 and LCHtk is the location of the kth neighboring CHt . In [6], CHt is chosen to be the nearest to (xt )qt|t−1 . In this case, the angle between the cluster head direction and the predicted target position can be large, as the target is most probably sensed by it for a considerably long period. This decision rule, augments the CHs number, allows unnecessary operations, affects the estimation performances and increases the energy consumption. The next section presents our method optimizing adaptively the number of quantization bits per observation. IV. A DAPTIVE

QUANTIZATION FOR TARGET TRACKING

At sampling time t − 1, the selected cluster head CHt−1 executes the VF algorithm and provides the Gaussian predictive distribution N (xp (t); xt/t−1 , λt/t−1 ). The predicted position allows the selection of the cluster to be activated. Furthermore, this target position predictive distribution is employed by the +t minimizing CHt−1 to give the optimal quantization level, N the transmitting power under predicted Cram´er-Rao bound constraint. This optimal quantization level is then transmitted to the CHt before being diffused to the activated cluster sensors so that they use it to quantize their observations. These quantized observations are then used by the CHt to execute the VF algorithm at the sampling instant t.

F I(xt , si , Nti )

-

=

l,k

Ez i |x t

t

.

/ ∂ log(p(zti |xt )) ∂ log(p(zti |xt )) (10) ∂x(l,t) ∂x(k,t)

(l, k) ∈ {1, 2} × {1, 2}

where zti denotes the observation of the ith sensor at the sampling instant t, xt = [x1 , x2 ]T is the unknown 2×1 vector to be estimated, and Ezti |xt [.] denotes the expectation with respect the likelihood function p(zt |xt ), which is given by Nti −1

p(zti |xt )

=

0 j=0

)

*

)

p τj (t) < γti < τj+1 (t) N βti dj , σ"2

*

(11)

where )

*

p τj (t) < γti < τj+1 (t) =

%

τj+1 (t)

τj (t)

)

*

N ργti (xt ), σn2 dγti

(12) is derived according to the quantization rule defined in (1), in which ργti (xt ) = K!xt − si !η , (13) Then, the derivative of the log-likelihood function can be expressed as,

Ni

×

280

0t ,

k=1



exp  −

∂ log(p(zti |xt )) ηK = 1 (xl,t − sl,i )%xl,t − sl,i %η−2 2 ∂xl,t 2σn

   2 2 1 (τk − ργti (xt )) 1 (τk+1 − ργti (xt ))  − exp  −  2 2 2 σn 2 σn

  7 Ni τk − ργ i (xt ) 0t , 1 (zt (k) − dk )2 t  − × exp − / erfc 1 2 2 σ#2 2σn k=1   6 7 τk+1 − ργ i (xt ) 1 (zt (k) − dk )2 t   × exp − erfc 1 2 2 2 σ# 2σn 6

(14)

Substituting expression (14) in (10), the FI matrix is easily computed by integrating over the likelihood function p(zti |xt ) at the sampling instant t. It is worth noting that the expression of the FI given in (14) depends on the target position xt at the sampling instant t and on the quantization level Nti . However, as the target position is unknown, the FI is replaced by its expectation according to i the predictive distribution p(xt |z1:t−1 ) of the target position , i i < F I(xt , si , Nti ) >= Ep(xt |z1:t−1 ) F I(xt , Nt )

The overall proposed algorithm is summarized in Algorithm 1. Algorithm 1: Pseudo-code of the proposed algorithm — Initialization: 1) Select the detecting cluster. 2) Quantize using an uniform power. 3) Execute the VF algorithm. — Iterations: 1) Select the detecting cluster. 2) Compute the predicted Cram´er-Rao bound based on the predicted target position. 3) Compute the optimal level quantization by optimizing the transmitting power under predicted Cram´er-Rao bound using the equation (21). 4) Quantize using the optimal level quantization. 5) Execute the VF algorithm.

(15)

Computing the above expectation is analytically untractable. However, as the VF algorithm yields a Gaussian predictive distribution N (xt ; xt/t−1 , λt/t−1 ), expectation (15) can be efficiency approximated by a Monte Carlo scheme: < F I(xt , si , Nti ) >&

J 1 0 ˜ jt , si , Nti ), F I(x J j=1

˜ jt ∼ N (xp (t); xt/t−1 , λt/t−1 ) x

(16)

˜ jt is the j − th drawn sample at instant t, and J is where x ˜t. the total number of drawn vectors x B. Optimizing the quantization Contrary to the work proposed in [6], we assume here that the transmitting power could be controlled by the sensors. The objective is then the optimization of the transmitting power while ensuring a good tracking performance. The total amount of the required transmission power used by the ith sensor within a cluster [12] is proportional to: P i (t)

∝

riλ (Nti − 1).

(17)

The optimal resource allocation problem is expressed as an optimization of the quantization level Nti over the field Z+ subject to an average estimation error constraint M0 , 0 min P i (t)2 (18) Nti ∈Z+

i=1,...,Na

CRB(xt , Nti ) < M0

s.t

where Na is the number of activated sensors at instant t. The Lagrangian formulation of this constrained optimization yields the following expression to be minimized: 0 L = ri2λ (Nti − 1)2 + (µCRB(xt , Nti ) − M0 )(19) i

where µ is the Lagrange multiplier constant. + i is a solution of the following Then, the optimal value of N t equation: +ti ) = ∂L = (N +ti − 1) − g(N +ti ) = 0 f (N (20) i + ∂N

V. R ESULTS The performance of the tracking algorithm can be essentially quantized by the tracking accuracy, which is evaluated by the distance between the estimated trajectory and the true target trajectory. The system parameters considered in the following simulations are: η = 2 for free space environment, the constant characterizing the sensor range is fixed for simplicity to K = 1, the cluster head noise power σn2 = 0.5, the total number of sensors Ns = 100, the total sampling instants N = 100, and the sensor noise power σ"2 = 0.1. One can notice from Figure.2 and Figure.3, that the best tracking quality is ensured by the adaptive VF algorithm compared to binary variational filtering (BVF) and to the quantized proximity sensors with uniform power level using VF; (QVF-U). The method based on BSN neglected the information relevance of sensor measurements, thus resource consumption is restricted to the activated cluster, where intracluster communications are dramatically reduced. But the method with fixed level quantization resolved the problem of restriction of the consumption resource, while it neglected the change of quantization bits per observation. The percentage of saving is defined as (Punif orm − Popt )/Punif orm × 100. In Figure.4 a), we plot the optimal power scheduling and the uniform power, supposing that sensors have same observation noise variances. From the Figure.4 b), we can see that the amount of energy saving becomes significant in time.

t

i

+ti ) denotes − µ2λ ∂CRB(xit ,N+ t ) . where g(N + 2ri ∂N t Applying the Newton-Raphson procedure yields the following sequence converging to the solution of (20): +i +i +i + i (n + 1) = N + i (n) − f (Nt (n)) = N + i (n) − (Nt (n) − 1) − g(Nt (n)) N t t t i i $ $ + + f (Nt (n)) 1 − g (Nt (n))

VI. CONCLUSIONS

We show that the total energy-saving and energy-efficient WSN designs for information collection from sensors and the error estimation can be minimized by optimizing the quantization level. As for economical reasons, in the hardware layer, (21) the deployment of quantized sensors saves greatly energy; in

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MSE comparison

Sensor Field True Trajectory Sensor Nodes Estimated Trajectory using QVF−U algorithm Estimated Trajectory using AQVF algorithm

140

distribution of target position. We have also presented an approximate optimal solution to the energy minimization problem. It is shown by numerical simulations that energy saving up to 80% can be achieved compared to uniform quantization when each sensor sends the same number of bits.

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AQVF algorithm AQVF−mean QVF−U algorithm QVF−U−mean

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Figure 4. a) Optimal power allocation (red) and uniform bit allocation (blue). b) Percentage of energy saving.

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