Fast Sensor Deployment for Fusion-based Target Detection

Transcription

1 Fast Sesor Deploymet for Fusio-based Target Detectio Zhaohui Yua*, Rui Ta*, Guoliag Xig*, Cheyag Lu, Yixi Che *Departmet of Computer Sciece, City Uiversity of Hog Kog Departmet of Computer Sciece ad Egieerig, Washigto Uiversity i St. Louis Abstract Recetly, wireless sesor etworks (WSNs) have bee icreasigly available for missio-critical applicatios such as target detectio, object trackig, ad security surveillace. These applicatios ofte impose striget performace requiremets icludig high detectio probability ad low false alarm rate. A fudametal problem is how to deploy sesors such that the sesig performace of a etwork is maximized or the umber of required sesors is miimized due to the high cost of deploymet. The key challege of this problem is the computatioal complexity which is prohibitively high due to the eed to cosider the combiatio of sesig qualities from multiple sesors. I this paper, we develop fast sesor deploymet algorithms based o data fusio model which addresses the ucertaity of sesor measuremets ad takes advatage of collaboratio amog sesors. Numerical experimets ad simulatios o real data traces show that our algorithms effectively deploy sesors to meet the desired sesig performace. I. INTRODUCTION Deployig wireless sesor etworks (WSNs) for missiocritical applicatios (such as target detectio [1], object trackig [2], ad security surveillace [3]) ofte faces the fudametal challege of meetig striget performace requiremets imposed by users. For istace, a surveillace applicatio may require ay itruder to be detected with a high probability (e.g., > 9%) ad a low false alarm rate (e.g., < 1%). Due to the high cost of deploymet, it is importat to predict the sesig quality provided by a etwork before the deploymet. However, as the actual sesig quality achieved by a etwork is complex due to the ucertaity of sesor measuremets, it s challegig to fid a miimum deploymet that provides desired sesig performace. Most existig work o the sesor deploymet ad coverage maiteace are based o the ideal disc sesig model [4], [5], [6], [7], [8]. I particular, the sesig regio of a sesor is ofte modeled as a disc with a certai radius cetered at the positio of sesor. A sesor defiitely detects the targets (evets) withi its sesig regio. Although this disc model allows a geometric treatmet to the coverage provided by sesors, it fails to capture the stochastic ature of sesig. Moreover, the disc model does ot take advatage of possible collaboratio amog sesors. The aforemetioed complexities of sesig have bee the research focus of the data fusio commuity for over three decades [9]. A key advatage of data fusio is to improve the system detectio performace by joitly cosiderig the oisy measuremets of multiple sesors. I practice, may sesor etwork systems desiged for target detectio, trackig ad classificatio have employed some kid of data fusio schemes [1], [3], [1]. As data fusio explicitly cosiders the ucertaity of sesor measuremets, we ca predict the probabilistic detectio performace of a etwork before the fial deploymet. I this work, we focus o developig fast algorithms for sesor deploymet i data fusio based target detectio applicatios. I particular, we are aimig to effectively deploy sesors so that the umber of sesors is miimized while a certai level of sesig performace is guarateed. The key challege of the deploymet problem for fusio-based target detectio is the computatioal cost. I geeral, the computatioal cost for determiig the optimized detectio performace of a large umber of sesors is prohibitively high due to the eed to cosider the combiatio of sesig qualities from multiple sesors. Cosequetly, the computatioal complexity will sigificatly icrease with the umber of sesors to be deployed. We attempt to address the aforemetioed challeges ad desig fast sesor deploymet algorithms for fusio-based target detectio. This paper makes the followig cotributios: Based o realistic sesig characteristics of sesors, we formulate the sesor deploymet problem for fusiobased target detectio as a optimizatio problem. Based o the aalysis of the computatioal complexity of the problem, we develop several sesor deploymet algorithms icludig a global optimal algorithm, a fast divide-ad-coquer algorithm, ad a cluster-based dividead-coquer algorithm. The latter two algorithms fid approximate optimal solutios with acceptable computatioal complexities. We coduct extesive umerical experimets ad simulatios o real data traces collected i the DARPA SesIT vehicle detectio experimets [1] to evaluate the algorithms proposed. The results show that the cluster-based divide-ad-coquer algorithm ca effectively deploy sesors to provide a certai level of sesig performace. The rest of this paper is orgaized as follows. Sectio II reviews related work. Sectio III itroduces the backgroud of data fusio. I Sectio IV, we formally formulate our sesor deploymet problem for fusio-based target detectio. I Sectio V, we preset our sesor deploymet algorithms. I Sectio VI ad VII, we evaluate our algorithms via umerical experimets ad simulatios.

2 II. RELATED WORK A umber of prior research works o sesor deploymet promise miimizig the umber of sesors or maximizig the sesig quality provided by a etwork [4], [5], [6], [7], [8], [11]. Most previous work are based o the ideal disc sesig model [4], [5], [6], [7], [8]. Differet from them, we cosider realistic sesig characteristics of sesors ad adopt data fusio i our detectio model. There is vast literature o stochastic sigal detectio based o multi-sesor data fusio. Early work [9] focuses o aalyzig optimal fusio strategies for small-scale powerful sesor etworks (e.g., a hadful of radars). Recet work o data fusio [1], [1], [12] have cosidered the properties of wireless sesor etworks such as spatial ode distributio ad limited sesig/commuicatio capability. I practice, may sesor etwork systems desiged for target detectio, trackig ad classificatio [1], [3], [1] have icorporated some kid of data fusio schemes to improve the system performace. Clouqueur et al. [11] itroduce the path exposure to formulate the sesor deploymet problem, which icorporates a sigal decay model ad data fusio amog multiple sesors. However, the work assumes that the targets traverse i the moitorig field so that they ca be detected by the etwork. Differet from their work, our paper does ot assume the movemet of targets. Moreover, we are addressig fast sesor deploymet algorithms that guaratees desired sesig quality provided by the etwork. III. PRELIMINARIES I this sectio, we describe a sigle-sesor sesig model ad a multi-sesor data fusio model that are used i our solutios. A. Target ad Sesig Model For most physical sigals, e.g., acoustic, seismic, electromagetic ad etc., the sigal eergy atteuates with the distace from the eergy source. Sesors detect targets by measurig the eergy of sigals emitted by targets. Deote W (d) as the sigal eergy measured by a sesor which is d meters away from the target. We assume W (d) is a decreasig fuctio. Particularly, i the umerical examples ad simulatios of this paper, we employ a sigal atteuatio model as follows: { W (d/d W (d) = if d > d ) k (1) W if d d where W is the origial eergy emitted by the target, k is a decayig factor which is typically from 2 to 5, d is a costat determied by the size of the target ad the sesor. This sigal atteuatio model is widely adopted i the literature [13], [14], [15], [16]. We ote that the approaches ad algorithms proposed i this paper do ot deped o the specific form of W (d). Figure 1 plots the sigal eergy measuremets of a WINS acoustic sesor [17] i the DARPA SesIT vehicle detectio experimets [1], [18]. From the figure, we ca see Eergy measuremet (W ) The iverse of distace square (1/d 2 ) Fig. 1. Eergy measuremet v.s. distace. The x-axis is 1/d 2 where d is the distace betwee the sesor ad the target, which rages from 1m to 3m. that the eergy measuremet icreases liearly with 1/d 2, which matches the atteuatio model i (1) ad idicates k = 2. The sigal stregth measuremets of a sesor are corrupted by oise. Deote the oise stregth measured by sesor i is N i, which follows a zero-mea ormal distributio with variace of σ 2, e.g., N i N (, σ 2 ). Suppose sesor i is d i meters away from the target, the total eergy it measures is give by U i = W (d i ) + N 2 i (2) I practice, the parameters of target ad oise models are ofte estimated usig a traiig dataset before deploymet. B. Multi-sesor Fusio Model Data fusio [16], [11], [19] is a widely adopted techique for improvig the performace of detectio systems. A sesor etwork that employs data fusio is ofte orgaized ito clusters. Each cluster head is resposible for makig a fial decisio regardig the presece of target by fusig the iformatio gathered by member sesors i the cluster. There exist two basic data fusio schemes, amely, value fusio ad decisio fusio [11], [19]. I a value fusio scheme, each sesor seds its eergy measuremets to the cluster head, which makes a decisio based o the eergy measuremets received from all member sesors. I a decisio fusio scheme, each sesor makes a local decisio ( or 1) based o its measuremets ad seds its decisio to the cluster head, which makes a fial decisio accordig to the local decisios. Value fusio ofte yields more accurate detectio decisios tha decisio fusio as all the iformatio gathered by odes is cosidered i the decisio makig [19]. I this paper, we adopt a value fusio scheme as follows. Sesors sed their eergy measuremets to the cluster head, which i tur compares the average of all measuremets to a threshold η. If the average is greater tha η, the cluster head decides that a target is preset. Otherwise, it decides there is o target. The threshold η is referred to as the detectio threshold hereafter. The performace of a detectio system is characterized by false alarm rate ad detectio probability. False alarm rate (deoted by P F ) is the probability that a target is regarded to be preset whe the target is actually abset. Detectio

3 probability (deoted by P D ) is the probability that a target is correctly detected. Suppose sesors take part i the data fusio. Uder the aforemetioed value fusio scheme, the false alarm rate is give by: ( ) 1 P F = Pr > η = 1 Pr N 2 i ( ( Ni σ ) ) 2 η σ 2 As N i /σ follows the stadard ormal distributio, (N i/σ) 2 follows the Chi-square distributio with degrees of freedom whose Cumulative Distributio Fuctio (CDF) is deoted as X ( ). So (3) becomes: ( η ) P F = 1 X σ 2 (4) The detectio probability is give by: ( ) 1 ( P D = Pr W (di ) + N 2 ) i > η ( ( ) 2 Ni = 1 Pr η ) W (d i) σ σ 2 ( η = 1 X W (d ) i) σ 2 (5) where d i is the distace betwee sesor i ad the target. IV. SENSOR DEPLOYMENT PROBLEM FOR FUSION-BASED TARGET DETECTION I this sectio, we formulate the sesor deploymet problem for fusio-based target detectio. I Sectio IV-A, we itroduce the etwork model ad assumptios. I Sectio IV-B, we formally formulate the sesor deploymet problem. The problem is reduced i Sectio IV-C. A. Network Model ad Assumptios I a surveillace field, targets appear at a set of kow physical locatios referred to as surveillace spots, or spots for brief. We are oly cocered with the sesor deploymet for surveillace spots. Surveillace spots are ofte chose accordig to i situ surveys before deploymet. The choices of surveillace spots are applicatio specific. For istace, i fire detectio applicatios, the surveillace spots ca be chose at the veues with iflammables. I itruder detectio applicatios, as the detectio system is expected to provide spatially dese surveillace, the spots ca be chose desely ad evely i the surveillace field, e.g., at regular grid poits. As the sesors which are far away from the target ad hece experiece low Sigal-to-Noise Ratio (SNR) will ot cotribute to the target detectio, oly the sesors close to a surveillace spot take part i the data fusio. For ay surveillace spot, we defie the fusio regio as the disc of radius R cetered at the surveillace spot. The radius R is refereed to as fusio radius hereafter. Sesors withi the fusio regio of a surveillace spot forms a detectio cluster (3) t 1 s 1 s 6 R t 3 s 5 s 2 s 3 t 2 s4 Fig. 2. Network model s 9 s 8 t 5 s 1 t 4 s 7 surveillace spot sesor to detect whether a target is preset at the surveillace spot by comparig the average of all eergy measuremets with a detectio threshold, as described i Sectio III-B. A cluster head is selected for each detectio cluster to perform data fusio, e.g., the sesor closest to the surveillace spot. Figure 2 illustrates the etwork model. I the figure, the dotted circles are the fusio regio of differet surveillace spots represeted by solid dots. For example, sesors s 1, s 2 ad s 3 fuse their measuremets to detect whether a target is preset at spot t 1. How to choose a proper fusio radius is discussed i Sectio V-E. We ote that such etwork model is also adopted i [2]. We have the followig two defiitios: Defiitio 1 (Shared sesor): A sesor is a shared sesor if it is withi the fusio regio of at least two surveillace spots. Defiitio 2 (Dedicated sesor): A sesor is a dedicated sesor if it is oly withi the fusio regio of a surveillace spot. For example, i Figure 2, s 1, s 3, s 5, s 7, s 8 are shared sesors, ad s 2, s 4, s 6, s 9, s 1 are dedicated sesors. As a shared sesor belogs to multiple detectio clusters, it will report its measuremet to multiple cluster heads. For example, suppose s 2 ad s 4 are the cluster heads correspodig to t 1 ad t 2, respectively. Sesor s 3 should report its measuremet to s 2 ad s 4 i each detectio sessio. We make the followig assumptios before we formally formulate the problem. First, all sesors have sychroized clock. Secod, the commuicatio rage of sesors is greater tha fusio radius R, which esures that the member sesors ca reach the cluster head i oe hop. This assumptio ca be relaxed if a routig algorithm is employed. B. Problem Formulatio We defie the followig otatios. 1) A is the surveillace field. 2) Total K surveillace spots are chose ad deote T = {t j 1 j K} as the set of surveillace spots, where t j = (x j, y j ) A is the coordiates of the j th surveillace spot. 3) Deote C j as the fusio regio of t j. 4) Deote S = {s i 1 i N} as the sesor deploymet, where s i = (x i, y i ) A is the coordiates of the i th

4 sesor ad N is the umber of sesors. We deote S as the cardiality of S, i.e., S = N. 5) j is the umber of sesors withi fusio regio C j, i.e., the size of the detectio cluster correspodig to t j. 6) η j is the detectio threshold for the surveillace spot t j. 7) P Fj ad P Dj are the false alarm rate ad detectio probability of surveillace spot t j, which ca be calculated by (4) ad (5), respectively. We defie the coverage of a surveillace spot i terms of false alarm rate ad detectio probability as follows. Defiitio 3 ((α,β)-coverage): Give two real umbers, α (, 1) ad β (, 1), the surveillace spot t j is (α,β)- covered if the false alarm rate P Fj ad detectio probability P Dj satisfy: P Fj α, P Dj β The (α,β)-coverage actually defies the sesig quality provided by the sesor etwork at a surveillace spot. Our problem is to deploy the fewest sesors to provide (α,β)- coverage at all surveillace spots. We formally formulate our problem as follows: Give a surveillace field A ad a set of surveillace spots T, our objective is to fid a list of detectio thresholds {η j 1 j K} ad a sesor deploymet S such that the umber of sesors S is miimized subject to the costrait that each surveillace spot is (α,β)-covered. I this problem, the variables to be determied are the umber of sesors N, the sesor deploymet S = {(x i, y i ) 1 i N} ad the detectio thresholds {η j 1 j K}. Hece, the umber of variables is 1 + 2N + K. Oe may come up with a straightforward solutio as follows. Begi with N = 1, check whether N sesors are eough to cover all surveillace spots. If yes, we fid the optimal solutio; otherwise, we icrease N by oe ad cotiue to check. However, for a certai N, it s hard to kow whether N sesors are eough to cover all surveillace spots due to the complex oliear relatioship betwee the detectio performace (i.e., P F ad P D ) ad the combiatio of multiple sesor positios. The exhaustive search of positios of sesors will cause expoetial complexity. C. Problem Reductio The umber of variables ca be reduced to 1 + 2N by exploitig the optimal detectio thresholds. For a certai sesor deploymet, P Fj α is a ecessary coditio of (α,β)-coverage of surveillace spot t j. Accordig to (4), this ecessary coditio becomes: η j σ2 X 1 (1 α) j where X 1 ( ) is the iverse fuctio of X ( ). Furthermore, accordig to (5), P Dj decreases with η j. Therefore, i order to obtai the maximized detectio probability subject to a bouded false alarm rate for a certai deploymet, the detectio threshold should be: η j = σ2 X 1 (1 α) (6) j Executio time (secods) Fig e.34n The umber of required sesors (N) The executio time v.s. the umber of required sesors We compute the optimal detectio threshold of each surveillace spot by (6). Usig the optimal detectio thresholds, each surveillace spot is (α,β)-covered if ad oly if the detectio probability for each spot is greater tha β, or equivaletly, mi 1 j K {P Dj } β. Hece, there are 1 + 2N variables left to be determied, i.e., N itself ad S = {(x i, y i ) 1 i N}. The problem formulated i Sectio IV-B is reduced to: Give a surveillace field A ad a set of surveillace spots T, our objective is to fid a sesor deploymet S such that the umber of sesors S is miimized subject to the followig costrait: mi {P D j } β (7) 1 j K V. SENSOR DEPLOYMENT ALGORITHMS I Sectio V-A, we preset a sesor deploymet algorithm that obtais optimal solutio of the problem formulated i Sectio IV. However, the global optimal algorithm ca ot hadle large-scale problems due to prohibitive time complexity. I Sectio V-B, we propose a divide-ad-coquer approach to hadle large-scale problems. The divide-ad-coquer approach is implemeted as a deploymet algorithm i Sectio V-C. I Sectio V-D, we propose to itegrate a surveillace spots clusterig algorithm, which makes the divide-ad-coquer algorithm more time efficiet. I Sectio V-E, we aalyze the optimal fusio radius. A. Global Optimal Deploymet A straightforward approach fidig the optimal solutio of the problem formulated i Sectio IV is to search the sesor deploymet with the miimum sesor umber ad meawhile the costrait (7) is satisfied. Algorithm 1 is the procedure fidig the global optimal solutio. Algorithm 1 begis with N = 1 ad iterates for icremetal N. For each N, we maximize the miimum of detectio probabilities amog all surveillace spots, i.e., mi 1 j K {P Dj }. Oce the costrait (7) is satisfied, the global optimal solutio is foud. I this work, the optimizatio that maximizes the miimum of detectio probabilities (Lie 3) is doe by a oliear programmig solver based o Costraied Simulated Aealig (CSA) algorithm [21]. CSA exteds covetioal Simulated Aealig to look for the global optimal solutio of a costraied optimizatio problem with discrete variables. CSA allows the objective fuctio ad costrait fuctios to be specified i a procedure istead of i a closed form.

5 Algorithm 1 The procedure fidig global optimal solutio Iput: α, β, surveillace field A, a set of surveillace spots T Output: Sesor deploymet S where S is miimized 1: N = 1 2: repeat 3: deploy N sesors i A to maximize mi 1 j K {P Dj } 4: compute η j for each t j T by (6) 5: compute P Dj for each t j T by (5) 6: N = N + 1 7: util mi 1 j K {P Dj } β 8: retur S Fig. 5. s 1 R 2R t surveillace spot 2 s 2 s s 4 3 sesor t 1 t 3 fusio rage subregio The divide step: the subregio of a surveillace spot s 1 s 2 t 1 R (a) A sigle spot is covered by at least two sesors R s 2 t 1 s 1 t 2 s 3 R (b) Two spots are covered by at least three sesors Fig. 4. A example: shared sesors ca reduce the size of deploymet. Settigs: W =.65, d =1, k =2, σ 2 =.1, α=.1, β =.9, R=1.6. Theoretically, CSA is a global optimal algorithm that coverges asymptotically to a costraied global miimum with probability oe (Theorem 1 of [21]). I theory, the complexity of CSA, like other stochastic search algorithms, icreases expoetially with respect to the umber of variables [21]. Therefore, the oliear programmig solver has a expoetial time complexity with respect to the umber of sesors. So for a large-scale deploymet problem which meas a large umber of sesors are required to be deployed, the global optimal algorithm becomes time prohibitive. Figure 3 shows the executio time of the global optimal algorithm versus the umber of sesors required i the optimal solutio. We ca see that the executio time icreases rapidly with the umber of sesors. The dotted lie i the figure is the liear regressio of the executio time with respect to the umber of sesors. I practice, it s usual to deploy up to 1 sesors. If 1 sesors are available to deploy, the global optimal algorithm will cosume about e 36 secods accordig to the liear regressio, which is prohibitive. B. Divide-ad-Coquer Approach As we treat all the surveillace spots together at a time i the global optimal algorithm, the oliear optimizatio i Algorithm 1 becomes time prohibitive whe the umber of sesors becomes large. If we treat a sigle surveillace spot at a time to compute a local solutio usig the oliear programmig solver ad the combie all local solutios ito a global solutio, the time complexity will be polyomial as the executio time for each local solutio is bouded. The basic idea of our divide-ad-coquer approach is to treat the surveillace spots i a sequetial fashio, i.e., we deploy fewest sesors to cover the surveillace spots oe by oe. Before divig ito the details of our divide-ad-coquer approach, we first cosider a umerical example which provides several isights of the problem. I this example, the parameters of the sigal eergy model defied i (1) ad the oise model are: W =.65, d = 1, k = 2, σ 2 =.1. We wat to achieve the sesig quality of (.1,.9)-coverage, i.e., α =.1 ad β =.9. If there is oly oe surveillace spot t 1 to be covered, we eed at least two sesors, as show i Figure 4(a). If there are two surveillace spots which are far apart from each other so that they have o commo fusio regio, we eed to deploy total four sesors to cover them separately. If there is aother surveillace spot t 2 which is 1.2 meters away from t 1 so that t 1 ad t 2 have commo fusio regio, the global optimal solutio shows that three sesors are eough to cover t 1 ad t 2, as show i Figure 4(b). I the optimal deploymet, there are two dedicated sesors (s 2, s 3 ) ad oe shared sesor (s 1 ). This example shows that if there exist commo fusio regios amog the surveillace spots, the size of deploymet may be reduced by deployig shared sesors. I the divide-ad-coquer algorithm, we treat each surveillace spot oe by oe. Whe we compute a local solutio for a surveillace spot, we eed also to care about the eighbour spots which have commo fusio regios with the spot we wat to cover. For istace, i the previous umerical example show i Figure 4, whe we compute a local solutio for spot t 1, we eed to care about t 2 as a shared sesor of t 1 ad t 2 ca improve the global solutio. Based o this observatio, i the divide step, we defie a subregio A j for each surveillace spot t j, which is a disc cetered at t j as illustrated i Figure 5. We will oly care about the eighbour spots withi the subregio of a surveillace spot we wat to cover. To simplify the algorithm desig, we choose a uiform radius for all subregios. Due to the fusio radius of R, oly the sesors deployed withi the fusio regio of surveillace spot t j ca affect the detectio performace of t j. Moreover, the sesors deployed withi the fusio regio of t j caot affect ay spots which are 2R away from t j. So the radius of subregios should t be greater tha 2R. I the coquer step, for each surveillace spot t j, we compute a local solutio to cover t j ad all eighbour spots i the subregio of t j. Fially, all surveillace spots ca be covered. I the divide-ad-coquer approach, the local solutio computed for the j th surveillace spot may be affected by the local

6 Algorithm 2 The procedure of the coquer step Iput: α, β, all subregios {A j 1 j K} ad correspodig surveillace spots {T j 1 j K} Output: Local optimal sesor deploymet S 1: S = 2: for j = 1 to K do 3: = 4: repeat 5: deploy additioal sesors i C j (deoted by ) to maximize mi th T j {P Dh } uder deploymet S 6: compute η h for each t h T j uder deploymet S 7: compute P Dh for each t h T j uder deploymet S 8: = + 1 9: util mi th T j {P Dh } β 1: S = S 11: ed for 12: retur S solutios computed i the previous iteratios. For example, i Figure 5, we first compute a local solutio for the subregio of t 1 so that t 1 ad t 2 are covered. I the ext iteratio, we compute a local solutio for the subregio of t 2 to cover t 2 ad t 3. As t 2 is covered by the previous local solutio but t 3 is ot covered, we eed to deploy additioal sesors i the fusio regio of t 2. Obviously, the curret local solutio is affected by the previous local solutio. Due to the iterdepedece of the detectio performace betwee eighbour surveillace spots, we caot divide the problem ito mutually idepedet sub-problems. Thus, the divide-ad-coquer approach caot always fid the global optimal solutio. C. Divide-ad-Coquer Sesor Deploymet Algorithm I this sectio, we implemet the divide-ad-coquer approach preseted i Sectio V-B as a sesor deploymet algorithm. I the divide step, for each surveillace spot t j, we compute the set of eighbour spots withi the subregio of t j, which is deoted as T j. I the coquer step, we compute a approximate optimal local solutio for each surveillace spot t j. The pseudo code of the coquer step is give i Algorithm 2. I the j th iteratio (from Lie 3 to 9), the algorithm esures that spot t j ad all eighbour spots i the subregio of t j (i.e., T j ) are covered usig the fewest additioal sesors oly deployed i the fusio regio of t j. The optimizatio i Lie 5 is doe by the oliear programmig solver metioed i Sectio V-A. As we oly deploy additioal sesors i the fusio regio of t j (Lie 5), the detectio performace of other surveillace spots outside of A j will ot be affected. Algorithm 2 will deploy shared sesors which play a importat role i reducig the size of the deploymet. After K iteratios, all surveillace spots are covered by the sesor deploymet computed by Algorithm 2. We further propose a refiemet procedure to remove redudat sesors i the approximate optimal solutio computed by Algorithm 2. Basically, the refiemet algorithm removes all dedicated sesors i the approximate optimal solutio ad redeploys the fewest sesors ecessary to cover all spots. Algorithm 3 The procedure refiig the approximate optimal solutio Iput: α, β, all subregios {A j 1 j K} ad correspodig surveillace spots {T j 1 j K}, approximate optimal solutio S computed by Algorithm 2 Output: refied approximate optimal solutio 1: repeat 2: total sesor umber N = S 3: for j = 1 to K do 4: calculate dedicated sesors set D j of t j 5: if D j = the 6: skip this iteratio for t j ad cotiue 7: else 8: S = S D j /* remove dedicated sesors of t j */ 9: = 1: repeat 11: deploy additioal sesors i C j (deoted by ) to maximize mi th T j {P Dh } uder deploymet S 12: compute η h ad P Dh for each t h T j uder deploymet S 13: = : util mi th T j {P Dh } β 15: S = S 16: ed if 17: if S < S the 18: S = S 19: ed if 2: ed for 21: ew total sesor umber N = S 22: util N = N 23: retur S Algorithm 3 is the refiig procedure. I the j th iteratio (from Lie 4 to 19), we first remove all dedicated sesors of t j, ad compute a ew local deploymet S usig the oliear programmig solver metioed i Sectio V-A (from Lie 8 to 15). If the ew local deploymet S uses fewer sesors tha the origial deploymet S, we replace the origial by the ew oe (from Lie 17 to 19). I each roud (from Lie 2 to 21), all surveillace spots are treated oe by oe. The algorithm termiates if the curret roud does t reduce the umber of sesors aymore. Note that we do ot remove ay shared sesors i the approximate optimal solutio computed by Algorithm 2, because the removal of shared sesors will breach the coverage of eighbour surveillace spots. We ow discuss the covergece of Algorithm 3. As we oly accept ew deploymet with smaller size (from Lie 17 to 19) i each iteratio, obviously, the size of the itermediate ruig solutio i each roud is decreasig. Deote the size of the global optimal solutio (i.e., the output of Algorithm 1) as N, ad the size of the approximate optimal solutio (i.e., the output of Algorithm 2) as N. The upper boud of the umber of rouds i Algorithm 3 is N N. So Algorithm 3 coverges after fiite rouds. D. Cluster-based Divide-ad-Coquer I the divide-ad-coquer algorithm proposed i Sectio V- C, the oliear programmig solver is ivoked to do a local optimizatio for each surveillace spot may times. The code block from Lie 3 to Lie 9 i Algorithm 2 ad the code

7 block from Lie 9 to Lie 14 i Algorithm 3 have similar fuctio as the global optimal solver (i.e., Algorithm 1) but give a local approximate optimal solutio for a subregio. We refer these code blocks as to local solver. The local solver is ivoked for K times i Algorithm 2 ad for M K times i Algorithm 3, where M represets the umber of rouds before the termiatio of Algorithm 3. Accordigly, the local solver is ivoked for total (1 + M) K times. If a few of surveillace spots are to be moitored, the executio time of the algorithms i Sectio V-C is acceptable. However, for the applicatios that eed spatially dese moitorig, a large umber of surveillace spots will be chose, i.e., K is large. I such case, as the local solver is expesive, the executio time of the algorithms i Sectio V-C becomes uacceptable. For example, if there are three surveillace spots to be covered, as show i Figure 5, without surveillace spots clusterig we eed to ru the local solver for spot t 1, t 2 ad t 3 oe by oe. However, if we group the three spots ito oe cluster, we eed oly to ru the local solver for spot t 2 because t 1 ad t 3 will be covered by the local solutio. We further group the spots ito clusters, ad ru the local solver for a cluster of spots, ot per spot. We employ a greedy clusterig algorithm called Quality Threshold (QT) algorithm, which is proposed by Heyer et al.[22]. The basic idea of the QT clusterig is 1) to orgaize the surveillace spots ito clusters whose members are geographically close to each other ad 2) the cluster with more members is preferred. Specifically, i each iteratio of the QT clusterig, we build a cadidate cluster cetered at each uclustered spot t with the spots withi the fusio regio of t as members, the reder the cadidate with the most members as a real cluster. The clusterig procedure goes o util there is o uclustered spot left. More details of the QT clusterig algorithm ca be foud i [22]. We ote that the surveillace spots cluster is differet from the detectio cluster defied i Sectio IV. The detectio cluster is a group of sesors withi the fusio regio of a surveillace spot, while the surveillace spots cluster is a group of surveillace spots. Suppose the QT clusterig geerates L clusters. We refer the cetral spot i a spots cluster as to Spots Cluster Head (SCH). I the divide-ad-coquer approach, istead of defiig a subregio for each surveillace spot, we oly defie subregio for each SCH. Specifically, for each SCH t l, we defie subregio A l as the disc cetered at t l of radius equals to 2R. Let T l deote the set of surveillace spots i A l. The, we ru Algorithm 2 ad Algorithm 3 with {A l 1 l L} ad {T l 1 l L} as iputs. If the surveillace spots are chose desely, L will be much smaller tha K. The umber of the local solver ivoked is reduced to (1 + M) L. E. Optimal Upper Boud of Fusio Radius I this sectio, we aalyze the upper boud of fusio radius R, ad give the optimal value regardig sesor desity. Suppose a surveillace spot t is covered by sesors i the fusio regio of t. Accordig to (5), the detectio probability of t satisfies: P D = 1 X ( η W (d i) σ 2 ) β Replace η with the optimal detectio threshold give by (6) ad solve the sum of eergies W (d i) from the previous iequality, we get: W (d i ) σ 2 ( X 1 (1 α) X 1 (1 β) ) (8) Iequality (8) meas that give sesors, the sum of eergies received by sesors is lower bouded if the correspodig spot is covered. As the sigal eergy atteuates with the distace from the target, i the worst case, every sesor is placed at the edge of the fusio regio. So the miimum sum of eergies is { } mi W (d i ) = W (R) (9) Replace the sum of eergies i (8) with its miimum value give by (9) ad solve the fusio radius R, we get: ( ( σ R W 1 2 X 1 (1 α) X 1 (1 β) ) ) (1) where W 1 ( ) is the iverse fuctio of the sigal atteuatio model W (d). Deote the right had of (1) as R u, which is the upper boud of the fusio radius. The meaig of iequality (1) is: give sesors, if the fusio radius does t exceed R u, the surveillace spot t is always covered o matter how the sesors are placed i the fusio regio of t. If we set R = R u, the sesor desity is ρ = /(πru), 2 which is the miimum sesor desity that guaratees the coverage of the surveillace spot o matter how the sesors are deployed. Because both ρ ad R u deped o, it s hard to derive the margial relatioship betwee ρ ad R u with aalytic expressio. However, i practice, we ca fid the optimal R u that miimizes ρ by umerical approaches. For istace, Figure 6 shows several umerical results uder the sigal atteuatio model defied by (1) with differet settigs. From the figure, we ca see that the sesor desity ρ is a covex fuctio of R u uder differet settigs ad thus we ca fid the optimal R u, e.g., R u = 7.76 is the optimal value for the settig σ 2 = 1. Furthermore, for a certai fusio radius (e.g., R u = 1), the sesor desity for low SNR settig (e.g., σ 2 = 5) is greater tha the sesor desity for high SNR settig (e.g., σ 2 = 1). This is reasoable as more sesors are eeded for low SNR cases to achieve the same sesig quality. VI. NUMERICAL RESULTS I this sectio, we coduct umerical experimets to evaluate the performace of the sesor deploymet algorithms proposed i Sectio V. The impacts of surveillace spots clusterig ad fusio radius are evaluated i Sectio VI-A ad VI-B, respectively. We compare the divide-ad-coquer deploymet algorithm with the global optimal algorithm ad a greedy algorithm i Sectio VI-C.

8 Sesor desity (ρ).35.3 σ 2 = 1 σ 2 = 2.25 σ 2 = Upper boud of fusio radius (R u) Fig. 6. Sesor desity v.s. upper boud of fusio radius. Settigs: W = 4, d = 1, k = 2, α =.1, β =.9. I the umerical experimets, the parameters of the eergy model defied by (1) are set as follows: W = 4, d = 1, k = 2, σ 2 = 1. The surveillace field A is a 3 3m 2 square area. The surveillace spots are chose regularly (i.e., o regular grid poits) or irregularly (i.e., radomly). The maximum false alarm rate, α, is set to be 1%, ad the miimum detectio probability, β, is set to be 9%. A. Impact of Spots Clusterig I this sectio, we evaluate the impact of the QT surveillace spots clusterig algorithm, which is preseted i Sectio V-D. We ru the divide-ad-coquer deploymet algorithm with ad without QT clusterig for total 4 regular ad 4 irregular surveillace spots layouts, respectively. The results are show i Figure 7(a) ad Figure 7(b). Figure 7(a) plots the umber of deployed sesors versus the umber of surveillace spots regularly laid out at grid poits. The curve labeled clustered ad uclustered represets the results computed by the divide-ad-coquer deploymet algorithm with ad without QT clusterig, respectively. Figure 7(b) shows the results for irregular surveillace spots. We ca see that the cluster-based deploymet algorithm ca effectively reduce the umber of required sesors. For istace, i Figure 7(a), whe total = 225 surveillace spots regularly lay out at the grid poits, the divide-ad-coquer algorithm without clusterig eeds 172 sesors, however, the algorithm with clusterig eeds oly 13 sesors. So the spots clusterig ot oly makes the divide-ad-coquer deploymet algorithm more time efficiet, but also gives better solutios. Aother iterestig observatio is that the umber of sesors required does t icrease a lot whe the umber of surveillace spots icreases to a certai extet. For istace, i Figure 7(b), total 7 sesors are eough to cover 1 surveillace spots, ad total 11 sesors are eough to cover 2 surveillace spots. This observatio idicates that the whole field ca be fully (α, β)-covered by fiite sesors, which is extesively studied ad proved i [2]. B. Impact of Fusio Radius I this sectio, we evaluate the impact of fusio radius. Total 45 irregular surveillace spots scatters radomly i the field. The cluster-based divide-ad-coquer deploymet algorithm is used. We chage the fusio radius R from 3 to 12. Figure 8 The umber of sesors (N) clustered uclustered The umber of spots (K) Fig. 7. (a) Regular spots The umber of sesors (N) The umber of sesors (N) 5 45 clustered 4 uclustered The umber of spots (K) (b) Irregular spots The umber of sesors v.s. the umber of surveillace spots Fig Fusio rage radius (R) The umber of sesors v.s. fusio regio radius plots the umber of sesors required i the solutio computed by the deploymet algorithm versus the fusio radius. From the figure, we ca see that the umber of required sesors drops rapidly from 37 to 7 whe fusio radius R icreases from 3 to 7.6, ad gradually icreases to 17 whe R becomes larger. This simulatio result is cosistet with the umerical results show i Figure 6 i Sectio V-E. C. Sesor Deploymet Performace I this sectio, we evaluate the effectiveess of our clusterbased divide-ad-coquer sesor deploymet algorithm. To evaluate the efficiecy of our divide-ad-coquer algorithm, we compare it with the global optimal algorithm proposed i Sectio V-A i terms of executio time ad the umber of sesors required. Figure 9 shows the executio time cosumed by differet deploymet algorithms versus the umber of sesors deployed for seve irregular surveillace spots layouts. From the figure, we ca see that our divide-ad-coquer algorithm cosumes much less time tha the global optimal algorithm. Figure 1 shows the umber of sesors required versus the area of surveillace field. From the figure, we coclude that our divide-ad-coquer algorithm gives approximate optimal deploymet. For further compariso, we employ a heuristic greedy algorithm as aother baselie. We ow itroduce the greedy deploymet algorithm briefly. I the greedy algorithm, sesors are also orgaized ito detectio clusters cetered at surveillace spots. As we ca compute the optimal detectio thresholds usig (6) for ay deploymet ad the false alarm rates of all surveillace spots are upper bouded by α with the optimal detectio thresholds, we eed oly cosider the detec-

9 4 35 optimal D&C The umber of sesors (N) Executio time (secods) Fig. 9. Executio time v.s. the umber of sesors deployed The umber of sesors (N) optimal D&C The area of surveillace field Fig. 1. The umber of sesors deployed v.s. the area of surveillace field The umber of sesors (N) D&C greedy The umber of spots (K) Fig. 11. (a) Regular spots The umber of sesors (N) D&C greedy The umber of spots (K) (b) Irregular spots The umber of sesors v.s. the umber of surveillace spots tio probabilities. I each iteratio of the greedy algorithm, we radomly place a sesor i the regio aroud the surveillace spot with the miimum detectio probability. The regio is a disc cetered at the spot of a relatively small radius, e.g., 2 meters i the followig simulatios. Oce every surveillace spot is (α,β)-covered, the greedy algorithm ceases. Similar greedy sesor deploymet algorithms are used i previous work [11], [23], [24]. I the first set of simulatios, total = 225 regular surveillace spots are chose at the grid poits, as show i Figure 12(a) ad Figure 12(b). Figure 12(a) shows the deploymet computed by our cluster-based divide-adcoquer algorithm, i which oly 13 sesors are deployed. Figure 12(b) shows the deploymet computed by the greedy algorithm, i which 15 sesors are eeded. I the secod set of simulatios, total 196 irregular surveillace spots radomly scatter i the field, as show i Figure 12(c) ad Figure 12(d). These two figures show the deploymet computed by our cluster-based divide-ad-coquer algorithm ad the greedy algorithm, i which 11 ad 15 sesors are deployed, respectively. We also coduct several simulatios with differet umber of regular ad irregular surveillace spots. The results are reported i Figure 11. I Figure 11(a), we compute the sesor deploymet by our cluster-based divide-ad-coquer algorithm ad the greedy algorithm for four regular surveillace spots layouts, i.e., spots chose at 4 4, 5 5, 1 1 ad grid poits. Figure 11(b) is the result for four irregular surveillace spots layouts, i.e., 2, 4, 1 ad 2 radomly scattered spots. We ca see from these two figures that our cluster-based divide-ad-coquer algorithm always deploy fewer sesors compared with the greedy algorithm. I average, our clusterbased divide-ad-coquer algorithm requires about 18% fewer sesors tha the greedy algorithm. VII. SIMULATIONS A. Simulatio Settigs ad Methodology We coduct extesive simulatios usig the real data traces collected i the DARPA SesIT vehicle detectio experimets [1]. I the experimets, 75 WINS NG 2. odes [17] are deployed to detect military vehicles drivig through several itersected roads. We refer to [1] for more detailed setup of the experimet. The dataset used i our simulatios icludes groud truth data ad the acoustic sigal eergy measuremets recorded by 17 odes at a samplig period of.75 secods, whe a Assault Amphibia Vehicle (AAV) drives through a road. The groud truth data iclude the positios of sesors ad the track of the AAV recorded by a GPS device every.75 secods. As i [18], we estimate the eergy atteuatio model usig a data trace as the traiig dataset. Our estimated parameters of the eergy model defied by (1) are: W =.51 (after ormalizatio), d = 2.6m, k = 2, σ 2 =.5. The surveillace field A is a 3 3 m 2 square area. The maximum false alarm rate, α, is set to be 1%, ad the miimum detectio probability, β, is set to be 9%. As the real data are collected by fixed sesors whe the target is movig, they ca ot be directly used i our simulatios. We geerate data for our simulatios as follows. For each eergy measuremet collected by a sesor, we compute the distace betwee the sesor ad the AAV from the groud truth data. Whe a sesor makes a measuremet i our simulatios, the eergy is set to be the real measuremet gathered at a similar distace to the AAV. Such a methodology accouts for realistic factors. For istace, there exists cosiderable deviatio betwee the measuremets of sesors i our simulatios ad the theoretic eergy atteuatio model. This deviatio is due to various reasos icludig the chagig oise levels caused by wid. B. Deploymet Validatio I this sectio, we validate the effectiveess of the solutios computed by the cluster-based divide-ad-coquer sesor deploymet algorithm usig the aforemetioed real data traces. We compute two sesor deploymets. The first deploymet uses 67 sesors to cover 196 regular surveillace spots scattered at grid poits, ad the secod deploymet uses 1 sesors to cover 25 irregular spots radomly scattered. I the simulatios, we evaluate the coverage of each surveillace spot oe by oe. For each surveillace spot, a target appears for a large umber of times (e.g., 1 i our simulatios) ad the detectio probability is calculated as the ratio of the umber of detectios to the umber of appearaces of the target. Figure 13(a) ad Figure 13(b) shows the CDF of the detectio probability uder the two sesor deploymets, respectively.

10 surveillace spot sesor surveillace spot sesor surveillace spot sesor surveillace spot sesor (a) D&C: K = 225, N = 13 (b) Greedy: K = 225, N = 15 (c) D&C: K = 196, N = 11 (d) Greedy: K = 196, N = 15 Fig. 12. Sesor deploymets usig our cluster-based divide-ad-coquer (D&C) ad the greedy algorithm. The ceters of the dotted circles are SCHs. CDF Detectio probability (P D ) (a) 196 regular spots Fig. 13. CDF Detectio probability (P D ) (b) 25 irregular spots The CDF of the detectio probability From the figures, we ca see that over 9% surveillace spots are covered i both two deploymets. VIII. CONCLUSION I this paper, we develop sesor deploymet algorithms for the fusio-based target detectio. These algorithms are desiged based o a detectio model that cosiders oisy sesor measuremets ad data fusio amog multiple sesors. This distiguishes our algorithms from existig sesor deploymet algorithms that are based o ideal detectio models. Our umerical experimets ad simulatios o real data traces demostrate that our cluster-based divide-ad-coquer algorithms ca effectively deploy sesors to meet a desired detectio performace. REFERENCES [1] D. Li, K. Wog, Y. H. Hu, ad A. Sayeed, Detectio, classificatio ad trackig of targets i distributed sesor etworks, IEEE Sigal Processig Magazie, vol. 19 (2), 22. [2] F. Zhao, J. Shi, ad J. Reich, Iformatio-drive dyamic sesor collaboratio for trackig applicatios, IEEE Sigal Processig Magazie, March 22. [3] T. He, S. Krishamurthy, J. A. Stakovic, T. Abdelzaher, L. Luo, R. Stoleru, T. Ya, L. Gu, J. Hui, ad B. Krogh, Eergy-efficiet surveillace system usig wireless sesor etworks, i MobiSys, 24. [4] S. Dhillo, K. Chakrabarty, ad S. S. Iyegar, Sesor placemet for grid coverage uder imprecise, i FUSION, 22. [5] K. Chakrabarty, S. Iyegar, H. Qi, ad E. Cho, Grid Coverage for Surveillace ad Target Locatio i Distributed Sesor Networks, IEEE TRANSACTIONS ON COMPUTERS, pp , 22. [6] D. Tia ad N. D. Georgaas, A coverage-preserved ode schedulig scheme for large wireless sesor etworks, i Proceedigs of First Iteratioal Workshop o Wireless Sesor Networks ad Applicatios (WSNA 2), Atlata, USA, Sep 22, pp [7] X. Wag, G. Xig, Y. Zhag, C. Lu, R. Pless, ad C. D. Gill, Itegrated coverage ad coectivity cofiguratio i wireless sesor etworks, i Sesys, 23. [8] T. Ya, T. He, ad J. A. Stakovic, Differetiated surveillace for sesor etworks, i The First ACM Coferece o Embedded Networked Sesor Systems(Sesys 3), 23. [9] P. Varshey, Distributed Detectio ad Data Fusio. New York, NY: Spiger-Verlag, [1] M. F. Duarte ad Y. H. Hu, Vehicle classificatio i distributed sesor etworks, Joural of Parallel ad Distributed Computig, vol. 64, o. 7, 24. [11] T. Clouqueur, V. Phipataasuphor, P. Ramaatha, ad K. K. Saluja, Sesor deploymet strategy for target detectio, i WSNA, Sep 22. [12] M. Duarte ad Y.-H. Hu, Distace based decisio fusio i a distributed wireless sesor etwork, i The 2d Iteratioal Workshop o Iformatio Processig i Sesor Networks, Palo Alto,CA, April [13] M. Hata, Empirical formula for propagatio loss i lad mobile radio services, IEEE Tras. o Vehicular Tech., vol. 29, o. 3, 198. [14] D. Li ad Y. H. Hu, Eergy based collaborative source localizatio usig acoustic micro-sesor array, J. EUROSIP o Applied Sigal Processig, o. 4, 23. [15] M. F. Duarte ad Y.-H. Hu, Vehicle classficatio i distributed sesor etworks, Joural of Parallel ad Distributed Computig, vol. 64, o. 7, 24. [16] P. K. Varshey, Distributed Detectio ad Data Fusio. New York: Spiger-Verlag, [17] W. Merrill, K. Sohrabi, L. Girod, J. Elso, F. Newberg, ad W. Kaiser, Ope stadard developmet platforms for distributed sesor etworks, i Proceedigs of SPIE - Uatteded Groud Sesor Techologies ad Applicatios IV 4743, 22. [18] X. Sheg ad Y.-H. Hu, Eergy based acoustic source localizatio, i IPSN, April 23. [19] T. Clouqueur, K. K. Saluja, ad P. Ramaatha, Fault tolerace i collaborative sesor etworks for target detectio, IEEE Trasactios o Computers, vol. 53, o. 3, 24. [2] R. Ta, G. Xig, B. Liu, J. Wag, X. Jia, ad C.-W. Yi, Data fusio improves the coverage of wireless sesor etworks, City Uiversity of Hog Kog, Tech. Rep., 28. [21] B. W. Wah, Y. Che, ad T. Wag, Simulated aealig with asymptotic covergece for oliear discrete costraied global optimizatio, Joural of Global Optimizatio, vol. 39, pp. 1 37, 27. [22] L. Heyer, S. Kruglyak, ad S. Yooseph, Explorig expressio data: Idetificatio ad aalysis of coexpressed gees, Geome Research, vol. 9, o. 11, pp , [23] S. S. Dhillo ad K. Chakrabarty, Sesor placemet for effective coverage ad surveillace i distributed sesor etworks, i WCNC, 23. [24] L. F. M. Vieira, M. A. M. Vieira, L. R. Beatriz, A. A. F. Loureiro, D. C. da Silva Juior, ad A. O. Ferades, Efficiet icremetal sesor etwork deploymet algorithm, i Brazilia Symposium o Computer Networks, 24.