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Optimal Threshold for Locating Targets Within a Surveillance Region Using a Binary Sensor Network Zhenxing Luo and Thomas C. Jannett Department of Electrical and Computer Engineering The University of Alabama at Birmingham Birmingham, AL 35294 Abstract - This paper considers the design of the optimal threshold for locating targets within a specified surveillance range using a network of binary sensors. The threshold was determined by minimizing a cost function representing the summation of the variances of estimation errors for the x and y coordinates of the target. The cost function was evaluated using multiple integration over the specified ranges for target power and target location. A geometrical interpretation of the optimal thresholds for targets with known power and location is presented to offer insight into the problem. The optimal threshold was validated in Monte Carlo simulations using a field of sensors having a uniform grid layout. In simulations, the summation of the variances of the estimation errors achieved using the optimal threshold approached the minimum of the cost function.

I. INTRODUCTION

T

localization is an important and challenging problem in sensor networks [1]-[4]. In a network or network region, sensors transmit data to a central processing node or fusion center. For the target localization problem, the processing node receives data from each sensor and estimates the target location. Some methods are based on time difference of arrival (TDOA) [5], [6]. However these methods require accurate timing, which is not feasible with inexpensive sensors. Energy-based methods, which use signal strength information, are more practical in real situations [7]-[10]. Much recent work has been focused on maximum likelihood target localization in networks of inexpensive sensors that have limited energy and communications resources [1], [9]. When resources are limited, it is desirable for the sensors to transmit quantized data to the processing node [1], [11]. A maximum likelihood estimator that uses quantized data and the Cramer-Rao lower bound (CRLB) for this estimator are derived in [1]. The CRLB is the theoretical performance bound for the variances of the estimation errors achieved using the maximum likelihood estimator. The CRLB has been used as the basis for several methods used to control the network localization performance, including designing detection thresholds, evaluating network layout, and reconfiguring a network after loss of resources [1], [12]. In this paper, we extend the work in [1] by developing an ARGET

optimal threshold for binary sensors detecting targets having a range of powers and locations within a specified surveillance region. The paper is organized as follows. Since this paper extends the work in [1], Section II closely follows the presentation and notation in [1]. Section II presents the sensor model, exemplary network layout, and maximum likelihood estimator for quantized data. Section II also reviews a method for designing optimal quantization thresholds for targets with known power and location. Section III provides a geometrical interpretation of the optimal thresholds for targets with known power and location. Section IV develops the optimal threshold for binary sensors locating targets having a range of powers and locations within a specified surveillance region. Methods for computing the optimal threshold are presented and demonstrated. The optimal threshold is validated and evaluated in Monte Carlo simulations. Discussion and conclusions are presented in Sections V and VI. II. PROBLEM FORMULATION We consider an exemplary field of N = 441 sensors uniformly deployed with sensors placed 9 m apart in each of the x and y directions (Fig. 1). Many ideas presented in this paper apply for other sensor deployments, including fields in which sensor are placed at random locations. The target emits a signal, with the signal intensity decreasing as the distance from the target increases. Following the presentation in [1], we consider a signal power model

ai2 =

Gi P0'

( di

d0 )

n

(1)

where ai is the signal amplitude at the ith sensor, Gi is the gain of the ith sensor, and P0' is the power emitted by the target measured at a reference distance d0 . The Euclidean distance between the target and the ith sensor is

d i = (xi - xt ) 2 + (yi - yt ) 2

(2)

2  0  1  Di =  M L − 2   L − 1

100 80 60

Y-coordinate (m)

40

−∞ < si < ηi1 ηi1 < si < ηi 2 M . ηi ( L − 2) < si < ηi ( L −1) ηi ( L−1) < si < ∞

(6)

20

The probability that Di takes specific value l is

0

ur ηi ( l +1) − ai η − ai pil (ηi , θ ) = Q ( il ) − Q( ) (0 ≤ l ≤ L − 1) (7)

-20

σ

-40 -60

where Q (⋅) is the complementary distribution function of the standard Gaussian distribution

-80 -100 -100

σ

-80

-60

-40

-20

0

20

40

60

80

100

Q ( x) = ∫

X-coordinate (m)



x

Fig. 1. Sensor field layout showing 441 sensors placed 9 m apart.

2

1 − t2 e dt . 2π

(8)

where (xi , yi ) and (xt , yt ) are the coordinates of sensor i and the target, respectively, and n is the power decay exponent.

After collecting the data, D , the processing node estimates the parameter vector θ = [ P0 xt yt ]T by maximizing the log-

We assume that Gi = G for i = 1,...,N , such that P0 = GP0' . In this paper, we use n = 2 and we assume that the target is at least d 0 = 1 meters away from any sensor at all times, resulting the simplified signal power model

likelihood function of D

ai2 =

N L-1 r lnp(D θ ) = ∑∑ δ(Di - l)ln [ pil (ηi ,θ)]

(9)

i=1 l=0

where

P0 . d i2

1, x = 0 δ(x) =  . 0, x = 0

(3)

The measured signal at the ith sensor is modeled as

In

(4)

si = ai + wi

summary,

the

maximum

(10)

likelihood

estimate

of

θ = [ P0 xt yt ]T is determined through the optimization problem:

for i = 1,...,N where wi is a Gaussian noise wi ~ N ( 0,σ

2

).

max lnp ( D θ ) .

(11)

θ

(5)

At each sensor, the measured signal si is quantized and transmitted to the processing node. As described in [1], the processing node receives quantized multi-bit data from each sensor. The received data are D = {Di : i = 1,...,N } , where Di can take on any discrete value from 0 to 2 M - 1 . Using the definition L = 2 M , the set of quantization thresholds for the ith sensor is r ηi = [ηi0 ,ηi1 ,...,ηiL ] , where ηi 0 = −∞ and ηiL = ∞ . The ith sensor supplies quantized data

The maximum likelihood estimate (11) and the corresponding Fisher Information matrix (FIM) and CRLB for the estimation error variance are derived in [1], which also addresses the design of sensor decision thresholds based on the CRLB using both optimization and heuristic methods. In this paper, we considered binary decisions made at each sensor that yield binary data 0 Di =  1

−∞ < si < η

η < si < ∞

.

(12)

3 A. Optimal Quantization Thresholds for Fixed Power and Fixed Target Location The CRLB for the ML estimate (11) is a function of the quantization thresholds and was used in [1] as the basis for the cost function r r J 11 η [J 33 η + J 22 r V( η ) = J

()

()

r

r

r

( rη ) ] - J ( η ) - J ( η ) (η) 2 13

2 12

(13)

which represents the summation of the variances of the location estimation errors in the x and y directions. In (13), J ij is

in a surveillance region and then examine the patterns of sensors that would fire if the optimal thresholds are applied. For P0 =10000 and binary decisions made at each sensor, optimal thresholds were determined over a subsection of the field (Fig. 1) represented by a 46 by 46 grid of target locations with x = 0.0, 0.2,…, 9.0 and y = 0.0, 0.2,…, 9.0. For this grid of target locations, the optimal threshold (14) varies symmetrically with target location from a minimum of 6.89 to a maximum of 11.24 (Fig. 2). For a target located far away from any sensor, the threshold is lower than for a target located near a sensor. The optimal threshold sometimes changes abruptly from one target location to another.

the element on the ith row and jth column of the FIM, J . duced here. The cost (13) is an obvious choice for designing a threshold that provides high accuracy in the estimation of the target location, although the resulting estimate of the power, P0 , may not be accurate. For a known P0 the optimal threshold for a target located at specific coordinates (xt , yt ) is the threshold that is the problem solution

r min r V η . η

()

Threshold

Expressions for J ij are presented in [1] and will not be repro-

10 8

5

5

(14)

0

0 -5

-5

III. GEOMETRICAL INTERPRETATION OF THE OPTIMAL THRESHOLDS FOR FIXED POWER AND TARGET LOCATION The optimal thresholds computed using (13) and (14) are valid only for a specified target location. In this section, we plot the optimal thresholds over a grid of target locations with-

X-coordinate (m)

a) Optimal thresholds over a field subsection having vertices [-9, -9], [9, -9], [-9, 9], and [9, 9].

Threshold

For a specified target location and power, this method gives the threshold that is optimal for use at all sensors in the field. However, the method does not address the practical problem of choosing the threshold needed to locate targets having unknown power and unknown positions within a surveillance region. Since the method for threshold optimization (14) depends on knowledge of the parameters θ = [ P0 xt yt ]T to be estimated, different thresholds would be required for different target locations within a surveillance range of interest, and the thresholds are optimal only for a specific P0 . However, the parameters are not known ahead of time, so the method cannot be used in practice. Despite its practical limitations, the optimal threshold (14) serves as a valuable benchmark for evaluating thresholds developed using other methods [1]. In this paper, the optimal threshold (14) is studied further. In the next section, we present a geometrical interpretation of the patterns of sensors that fire at the optimal thresholds for demonstrative target locations. The geometrical interpretation provides insight into the threshold design problem and the limitations of the performance of the optimal threshold (14).

Y-coordinate (m)

10 8 8 8 6

6 4

4 2

2 Y-coordinate (m)

X-coordinate (m) 0

0

b) Detailed view of optimal thresholds over a smaller field subsection having vertices [0, 0], [0, 9], [9, 0], and [9, 9]. Fig. 2. Optimal thresholds computed using (14) for the corresponding target locations over subsections of the field of Fig. 1 (P0=10000, binary decisions, σ = 1).

Next, we choose three demonstrative target locations for which the optimal thresholds used by sensors to generate bi-

4 nary decisions are at relatively low, medium, and high levels. We demonstrate the impact of the different thresholds by showing the patterns of sensors that fire (detect a target) for these thresholds at P0 =10000 (arbitrary units) in the noise free case (σ = 0). For a target located in the center of a grid of four sensors, the target is located as far away as possible from any of the four sensors, the threshold is low, and four sensors fire (Fig. 3a). For a target located an equal distance between two sensors, the optimal threshold is at an intermediate level and six sensors fire (Fig. 3b). For a target located close to a sensor, the threshold is relatively high and two sensors fire (Fig. 3c). Sensor Field Y-coordinate (m)

30 25 20 15 10 5 0

0 10 20 30 Sensor Field X-coordinate (m)

a) Target location (13.5,13.5) and optimal threshold 6.89. Sensor Field Y-coordinate (m)

30 25 20 15 10 5 0

0

10 20 30 Sensor Field X-coordinate (m)

b) Target location (18, 13.5) and optimal threshold 9.75. 30 Sensor Field Y-coordinate (m)

will not produce different firing patterns, resulting in poor identifiability (uniqueness of solution) and possibly, biased estimates. If the power is significantly lower than the power for which the threshold is optimal (power mismatch), few or no sensors may fire, resulting in large target location estimation errors [1]. Finally, we examine the consequences of applying suboptimal thresholds. If the applied threshold is higher than the optimal threshold, few or no sensors may fire, resulting in large estimation errors. If the applied threshold is lower than the optimal threshold, more sensors may fire than with the optimal threshold, perhaps with little consequence. For target locations near the border of the sensor field, a low threshold that results in a large pattern of sensors firing may produce an asymmetric firing pattern that introduces estimation errors. Further study of the log-likelihood function (9) and FIM or CRLB may provide insight into the relationship between performance, thresholds, and sensor firing patterns.

25 20 15 10 5 0

0 10 20 30 Sensor Field X-coordinate (m)

c) Target location (10, 18) and optimal threshold 11.38. Fig. 3. Demonstrative target locations (star) and corresponding patterns of fired sensors (solid circles) that occur for the optimal thresholds found using (14) if there is no measurement noise. For a), b) and c) the lower left sensor corresponds to location (0,0) in the field of Fig. 1. (P0=10000, binary decisions, σ = 0).

Inspection of the sensor firing patterns for the optimal thresholds (Fig. 3) facilitates assessment of the impacts of noise and power mismatch. Noise may make more or fewer sensors fire, resulting in asymmetric sensor firing patterns that produce estimation errors. If the thresholds are the same over a range of target locations, then small changes in target location

IV. OPTIMAL THRESHOLDS FOR A RANGE OF POWERS OVER A SURVEILLANCE REGION The minimization in (14) provides a threshold that is optimal only for a known P0 and specified target location. However, our goal is to locate targets having unknown power and unknown positions within a surveillance region. In [1], the idea of integrating (13) over a surveillance region and range of powers was presented, but was considered impractical due to the prospect of a prohibitive computational load. In this paper, we use this idea for determining an optimal threshold for locate targets having unknown power and unknown positions within a surveillance region. First, we assume that the power is uniformly distributed within a specified interval, and we assume that the target locations of interest are uniformly distributed within a specified surveillance region. Next, we calculate an average cost function by employing multiple integration of (13) over the power interval and over the target locations within the surveillance region. In this approach, the elements of the FIM are understood to be a function of the threshold and the parameter vector θ = [ P0 xt yt ]T , and the cost r r r r r J 11 η ,θ [ J 33 η ,θ + J 22 η ,θ ] − J 132 η ,θ − J 122 η ,θ V (η ,θ ) = ∫∫∫ dxt dyt dP0 r J η ,θ r

( ) ( )

( ) ( )

( )

( )

(15) is computed by integrating over θ . The optimal threshold (14) designed for a specific power performs poorly for mismatched P0 values [1]. When the actual power is lower than the value of P0 used to calculate the optimal threshold, few or no sensors fire, and the location r variance V η computed using (13) becomes very high.

()

Therefore, we expect that the cost (15) will be dominated by the contributions of the low target powers. In a uniform sensor

5 field, for a fixed power, the optimal thresholds (Fig. 2) found using (14) for specific target locations and the corresponding r values of V η follow a symmetric pattern within the field.

()

An understanding of the sensitivity of the optimal threshold to mismatched low P0 values and recognition of the symmetry in r V η suggest that approximating the cost (15) as

()

r r J 11 η,θ [J 33 η,θ + J 22 r V ( η,θ) = ∑∑∑ xt yt p0 J

( )

( )

r

r

r

( η,θ ) ] - J ( η,θ ) - J ( η,θ ) r ( η,θ ) 2 13

2 12

(16)

may give useful results and avoid a prohibitive computational load in finding the optimal thresholds that are the solutions to r min r V η, θ . η

( )

(17)

A. Computation of the Optimal Thresholds Optimal thresholds were computed for exemplary cases that demonstrate the use of (16) and (17). First, the optimal threshold was found for targets having a fixed power and locations within a surveillance region having vertices [-36,-36], [36,-36], [-36,36], and [36,36] in the field of Fig. 1. Second, the optimal threshold was determined for targets that have P0 within a specified interval and locations within the surveillance region. Binary decisions were employed and σ = 1 in all cases. The optimal threshold for a fixed P0 = 8000 was determined for target locations within the surveillance region. The symmetry of the thresholds exhibited in Fig. 2 was exploited to approximately evaluate (16) for the surveillance region based on a grid of 81 target positions for x = 1,2,…9 and y = 1,2,…9. The minimization (17) was performed using exhaustive search. Optimal thresholds were also determined using the same method for P0 = 10000 and for P0 = 12000 for the same surveillance region (Table I). TABLE I OPTIMAL THRESHOLD AND COST VERSUS THE SET OF POWERS USED FOR OPTIMIZATION OVER A SURVEILLANCE REGION r Optimal Threshold, η Power, P0 Cost, V ( η,θ)

evaluate (16) for the surveillance region based on the grid of 81 target positions described earlier (Table I). The minimization (17) was carried out using exhaustive search. In consideration of Table II in [1], the cost was expected to be dominated by the contributions from the low powers within the P0 interval. The cost was evaluated for P0 within the interval from 8000 to 12000 using a P0 set having only three elements because using more elements would be expected to increase the computational load without much improvement in the solution. In addition, for P0 within the interval from 9000 to 11000, the cost (16) was minimized using the same method for a P0 set [9000 10000 11000] (Table I). B. Evaluation of Thresholds in Monte Carlo Simulations The optimal thresholds were evaluated in Monte Carlo simulations (1000 runs, binary decisions, n=2, σ = 1) in which targets were placed at random locations uniformly distributed over the surveillance region represented by the square having vertices [-36,-36], [36, -36], [-36, 36], and [36, 36]. Simulations were initially performed for fixed P0 values of 8000, 10000, and 12000 with the thresholds shown in Table II. Optimum thresholds matching the power or power intervals were studied. We also studied a threshold mismatch in which a suboptimal threshold was applied. The suboptimal threshold was the lowest of the optimal threshold values found for any target location within the surveillance region. TABLE II RESULTS OF MONTE CARLO SIMULATIONS EVALUATING THRESHOLDS AT DIFFERENT POWER LEVELS OVER A SURVEILLANCE REGION Power, P0

Cost at Optimal Optimal Applied Threshold, Threshold, r Threshold η V ( η,θ)

Sum of Location Estimation Error Variances Using Applied Threshold

8000

7.11

7.11

4.38

4.84

10000

7.91

7.91

4.05

4.21

12000

8.66

8.66

3.85

4.18

Random (8000-12000)

-

7.481

-

4.92

10000

7.91

6.892

4.05

4.73

1

8000

7.11

4.38

10000

7.91

4.05

12000

8.66

3.85

[9000 10000 11000]

7.80

4.14

[8000 10000 12000]

7.48

4.30

The optimal threshold for P0 within the interval from 8000 to 12000 was determined for target locations within the surveillance region. The cost (16) was evaluated over a P0 set [8000 10000 12000] by exploiting symmetry to approximately

7.48 is the optimal threshold found for P0 set [8000 10000 12000] in (16). 6.89 is the lowest of the optimal threshold values found for any target location within the surveillance region (Fig. 2a).

2

Additional simulations were performed for random P0 uniformly distributed within the interval 8000 to 12000. The sum of the experimental location estimation error variances for estimates of the target position (xt , yt ) was computed and compared to the minimum cost (16) at the optimal threshold r min (Table II). r V η, θ η

( )

6 V. DISCUSSION The optimal thresholds determined for fixed P0 levels of 8000, 1000, and 12000 increase with P0 (Table I). The optimal threshold determined for the P0 set [8000 10000 12000] was less than the threshold for the set midpoint P0 = 10000, but was closer to the optimal threshold for the set midpoint P0 = 10000 than to the optimal threshold for P0 = 8000. The optimal threshold determined for the P0 set [9000 10000 11000] was also less than the threshold for the set midpoint P0 = 10000. The costs at the optimal thresholds for fixed P0 levels of 8000, 10000, and 12000 decrease as the power level used for optimization increases (Table I). The cost at the optimal threshold determined for the P0 set [8000 10000 12000] was greater than the cost at the threshold for P0 = 10000, and was closer to the cost for the optimal threshold for P0 = 8000 than to the cost at the optimal threshold for P0 = 10000. The cost at the optimal threshold determined for the P0 set [9000 10000 11000] was greater than the cost at the threshold for P0 = 10000. These results confirm that the costs, and corresponding optimal thresholds, were dominated by the contributions from the lower P0 values. Despite the approximations and the coarseness of the representation of the surveillance region and power interval used in evaluating (16) in the minimization used to find the optimal thresholds, the optimal thresholds gave good performance (Table II). In simulations, the sum of the error variances achieved using the optimal thresholds approached the minimum cost for all P0 values considered. When a threshold mismatch, in which the applied threshold was not the optimal threshold, was studied for P0 = 10000, the sum of the error variances achieved (Table II, line 5) slightly exceeded the sum of the error variances achieved using the optimal threshold (Table II, line 2). VI. CONCLUSION This paper addresses the important problem of decision thresholds for target localization. In this paper, the work of [1] was extended by developing and demonstrating a method for finding the optimal threshold used to make binary sensor decisions for targets having power and location within specified intervals. Symmetry was exploited to find the optimal thresholds using minimization performed over a subset of the surveillance region, which reduced the computational effort expended in finding the optimal thresholds. The optimal threshold resulted in target localization performance approaching the minimum cost. The new method for designing the optimal threshold for targets having power and location within specified intervals could be useful in designing a sensor field to meet target localization performance requirements. A geometrical interpretation of the optimal thresholds that depend on knowledge of the target location and power was presented for a sensor field with sensors placed in a uniform

grid pattern. The optimal thresholds were found through minimization accomplished by approximate evaluation of (16) over a finite set of target locations (grid) within the surveillance region considered, and over a very coarse set of power values within the power interval considered. At the expense of a greater computational effort, better approximations and use of a finer grid might be employed to produce more accurate values for the optimal thresholds. Other future work could consider refinements in the methods used in computing (15)-(17), extension for multi-bit quantized data, detailed performance analysis, and random sensor layouts. REFERENCES [1]

R. X. Niu and P. K. Varshney, “Target location estimation in sensor networks with quantized data,” IEEE Trans. Signal Processing, vol. 54, pp. 4519-4528, Dec. 2006. [2] C. Cevher, F. D. Marco, and G. B. Richard, “Distributed target localization via spatial sparsity,” Proc.16th European Signal Processing Conference (EUSIPCO), Laussane, Switzerland, 2008. [3] Y. Zou and K. Chakrabarty, “Sensor deployment and target localization in distributed sensor networks,” ACM Trans. Embed. Comput. Systems, vol. 3, issue 1, pp. 61-91, Feb. 2004. [4] K. Chakrabarty, S. S. Iyengar, H. Qi, and E. Cho, “Grid coverage for surveillance and target location in distributed sensor networks,” IEEE Trans. Computers, vol.51, no.12, pp.1448-1453, Dec. 2002. [5] K. H. Yang, G. Wang, and Z.Q. Luo, "Efficient convex relaxation methods for robust target localization by a sensor network using time differences of arrivals," IEEE Trans. Signal Processing, vol. 57, pp. 27752784, July 2009. [6] E. Weinstein, "Optimal source localization and tracking from passive array measurements,” IEEE Trans. Acoust., Speech, Signal Process., vol.30, pp. 69-76, Feb. 1982. [7] D. Li, K. D. Wong, Y.H.Hu, and A. N. Sayeed, "Detection, classification, and tracking of targets," IEEE Signal Process. Mag., vol.19, no. 3, pp. 17-29, Mar. 2002. [8] D. Li and Y. H. Hu, "Energy based collaborative source localization using acoustic microsensor array," EURASIP J.Appl. Signal Process., no. 4, pp.321-337, 2003. [9] X. Sheng and Y. H. Hu, "Maximum likelihood multiple-source localization using acoustic energy measurements with wireless sensor networks," IEEE Trans. Signal Processing, vol.53, no.1, pp. 44-53, Jan. 2005. [10] N. Patwari and A. O. Hero, "Using proximity and quantized RSS for sensor localization in wireless Sensor Networks,” in Proc. 2nd Int. ACM Workshop on Wireless Sensor Networks and Applications, San Diego, CA, Sep. 2003, pp.20-29. [11] N. Katenka, E. Levina, and G. Michailidis, “Robust target localization from binary decisions in wireless sensor networks,” Technometrics, vol. 50, no. 4, pp. 448-461, Nov. 2008. [12] P. P. Joshi and T. C. Jannett, "Performance-guided reconfiguration of wireless sensor networks that use binary data for target localization," Third International Conference on Information Technology, New Generations, 2006, ITNG 2006, pp. 562-565.

Optimal Threshold for Locating Targets Within a ...

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