Abstract This paper considers the problem of tracking objects with sparsely located binary sensors. Tracking with a sensor network is a

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1 Trackng on a Graph Songhwa Oh and Shankar Sasry Deparmen of Elecrcal Engneerng and Compuer Scences Unversy of Calforna, Berkeley, CA 9470 {sho,sasry}@eecs.berkeley.edu Absrac Ths paper consders he problem of rackng objecs wh sparsely locaed bnary sensors. Trackng wh a sensor nework s a challengng ask due o he naccuracy of sensors and dffcules n sensor nework localzaon. Based on he smples sensor model, n whch each sensor repors only a bnary value ndcang wheher an objec s presen near he sensor or no, we presen an opmal dsrbued rackng algorhm whch does no requre sensor nework localzaon. The rackng problem s formulaed as a hdden sae esmaon problem over he fne sae space of sensors. Then a dsrbued rackng algorhm s derved from he Verb algorhm. We also descrbe provably good prunng sraeges for scalably of he algorhm and show he condons under whch he algorhm s robus agans false deecons. The algorhm s also exended o handle non-dsjon sensng regons and o rack mulple movng objecs. Snce he compuaon and sorage of rack nformaon are done n a compleely dsrbued manner, he mehod s robus agans node falures and ransmsson falures. In addon, he use of bnary sensors makes he proposed algorhm suable for many sensor nework applcaons. I. INTRODUCTION In wreless ad-hoc sensor neworks, many nexpensve and small sensor-rch devces are deployed o monor and conrol our envronmen. I s envsoned ha sensor neworks wll connec us o he physcal world n a pervasve manner 1 3. Each devce, called a sensor node, has sensng, processng, and communcaon capables. Sensor nodes form a wreless ad-hoc nework for communcaon. The lmed supply of power and oher consrans, such as manufacurng coss and lmed package szes, lm he capables of each sensor node. For example, a ypcal sensor node has shor communcaon and sensng ranges, a lmed amoun of memory and lmed compuaonal power. However, he abundan number of spaally spread sensors wll enable us o monor changes n our envronmen accuraely despe he naccuracy of each sensor node. Targe rackng s a represenave applcaon of sensor neworks as exhbs dfferen aspecs of sensor neworks such as even deecon, sensor nformaon fuson, communcaon, sensor managemen, and decson makng. The applcaons of rackng usng sensor neworks nclude survellance, search and rescue, dsaser response sysem, pursu evason games 4, spaal-emporal daa collecon, and oher locaon-based servces 5. However, he ask of rackng wh sensor neworks s complcaed by wo problems: he naccuracy of sensors and dffcules n sensor nework localzaon. Each sensor node has a lmed supply of power and operaes n he low SNR regme, leadng o low deecon probably and hgh false deecon probables. The presence of false deecons and mssng observaons complcaes rack naon, manenance, and ermnaon. These mporan ssues are gnored by many rackng algorhms desgned for sensor neworks. Insead of usng an deal sensor model whch gnores he ssues lsed above, we ake he mnmals approach and use he bnary sensor model, n whch each sensor repors only a bnary value ndcang wheher an objec s presen near he sensor or no. Trackng algorhms desgned for sensor neworks so far rely on sensor nework localzaon Alhough many sensor nework localzaon algorhms have been proposed and esed, s sll a challengng problem n pracce 11, 1. Furhermore, even f he exac dsances beween pars of nodes are known, he localzaon n sparse neworks s shown o be NP-hard 13. Oher alernaves, such as he manual deploymen or he global posonng sysem (GPS), requre expensve hardware and are no scalable. Snce we canno avod localzaon errors, he localzaon-dependen rackng algorhms wll frequenly fal o rack a movng objec. In hs paper, we develop a dsrbued rackng algorhm based on he formulaon whch s ndependen of sensor nework localzaon. Hence, he performance of our algorhm s no affeced by he localzaon error. The connuous sae-space, dynamc-based rackng algorhms, such as he Kalman fler and a parcle fler, requre reasonably good measuremens of he saes of a movng objec. Bu, due o he naccuracy of sensors and dffcules n sensor nework localzaon, s dffcul o fulfll such requremens wh currenly avalable sensor neworks. In addon, when an objec moves wh exremely complex dynamcs or when we are rackng wh sparsely locaed sensors, s dffcul o use a connuous sae-space, dynamc-based rackng algorhm. For example, he ask of rackng people ndoor requres a complex dynamcs model and he dynamc-based rackng algorhms canno be easly appled. We address hs ssue by developng a rackng algorhm over he fne sae space of sensors. Under hs new formulaon, we can rack objecs whou requrng an deal sensor model and sensor nework localzaon. In hs paper, he rackng problem s formulaed as a hdden sae esmaon problem n a hdden Markov model over he fne sae space of sensors. Then an opmal dsrbued rackng algorhm s derved from he Verb algorhm 14. We hen show provably good prunng sraeges for scalably and he condons under whch he algorhm s robus agans false deecons. The algorhm s also exended o handle non-dsjon sensng regons and o rack mulple movng objecs. Snce he compuaon and sorage of rack nformaon are done n a compleely dsrbued manner, he mehod s robus agans node falures and ransmsson falures. In addon, he use of bnary sensors makes he proposed algorhm suable for many sensor nework applcaons such as locaon-based servces. The algorhm presened n hs paper s suable for ndoor rackng problems where he movemen of an objec canno be easly modeled and sensor nework self-localzaon s dffcul. The algorhm can be appled o oudoor rackng problems f he maxmum degree of he passage connecvy graph of a sensor nework (see Secon II for s defnon) s no oo large. For arbrary oudoor rackng scenaros, a herarchcal mulple-arge rackng algorhm for sensor nework s developed n 15. The remander of hs paper s srucured as follows. The problem of rackng on a graph s descrbed n Secon II. The opmal dsrbued rackng algorhm s presened n Secon III and provably good prunng sraeges for scalably s gven n Secon IV. We sudy he robusness of he algorhm n he presence of false deecons n Secon V. The algorhm s exended o handle non-

2 dsjon sensng regons and o rack mulple objecs n Secon VI and VII. II. PROBLEM FORMULATION Suppose here are N sensor nodes, s 1,..., s N. We denoe he sensng regon of sensor node s by C and assume ha he sensng regon of each sensor node s dsjon from he sensng regons of oher sensor nodes. A sensor node s s passage conneced o a sensor node s j f here s a pah from he sensng regon of s o he sensng regon of s j such ha an objec can raverse. A sensor node s s drec passage conneced o a sensor node s j f s s passage conneced o s j and here s a pah from s o s j whch does no nersec sensng regons of he remanng sensors. We assume ha he passage connecvy (drec passage connecvy) s symmerc,.e., f s s passage conneced (drec passage conneced) o s j, hen s j s passage conneced (drec passage conneced) o s. Noe ha when here s no confuson, we wll denoe sensor s by s ndex. Defnon 1: A passage connecvy graph of sensor nodes s 1,..., s N s a graph G (V, E), where V {1,..., N} and (u, v) E f and only f u and v are drec passage conneced. Le G (V, E) be he passage connecvy graph of sensor nodes s 1,..., s N. An example of a passage connecvy graph s shown n Fgure 1. Noce ha snce u V s drec passage conneced o u, (u, u) E. Throughou hs paper, whou loss of generaly, we assume ha G s a conneced graph, snce, f G s dsconneced, we can consder each paron of G separaely. Le X {1,..., N} be he locaon of an objec on G a me for {1,..., T }, where T s he me horzon. An objec appears a V wh he nal sae probably π P (X 1 ). If X, an objec s locaed n C a me. Le NB {j V : (, j) E} be he neghborhood of. We assume ha he evoluon of an objec on G s Markovan such ha P (X +1 j X, X 1,..., X 1) P (X +1 j X ) : p j for all. Le P p j be he ranson probably marx. For each, j NB p j 1 and p j 0 for j NB. Noce ha our framework can handle more complex dynamcs of a movng objec by usng a k-h order Markov chan, for k. For each sensor node s, f an objec s n C, s deeced wh probably η. Le Y {0, 1} be he observaon made by s a me. When an objec s n C a me, he objec s deeced, Y 1, wh probably η, and he objec s no deeced, Y 0, wh probably 1 η. For now we assume here are no false deecons. We show he robusness of our algorhm n he presence of false deecons n Secon V. The model parameers, π, P, and η, can be learned from hsorc daa usng Baum-Welch mehod 14. So we assume ha hey are known n hs paper. We also assume ha node u can communcae wh node v relably n mely manner f (u, v) E. Snce he sensng regons are dsjon, we can assume ha each sensor makes an ndependen observaon a each me. Le Y (Y 1,..., Y N The jon dsrbuon s where ) T, Y 1:T {Y 1,..., Y T } and X 1:T {X 1,..., X T }. T T P (X 1:T, Y 1:T ) P (X 1) P (X X 1) P (Y X ), (1) P (Y X ) (η X ) Y X 1 (1 η X ) 1 Y X. () Now he rackng problem on a graph s o fnd he mos probable rajecory of a movng objec on he graph gven observaons y 1:T, Fg. 1. (a) An example of ndoor sensor nework deploymen (sold crcles are sensors). (b) The passage connecvy graph of he sensor nework shown n (a)..e., fndng he maxmum a poseror (MAP) soluon x 1:T : x 1:T arg max x 1:T P (x 1:T y 1:T ) arg max x 1:T P (y 1:T x 1:T )P (x 1:T ) P (y 1:T ) arg max x 1:T P (y 1:T x 1:T )P (x 1:T ) arg max x 1:T P (x 1:T, y 1:T ). (3) The Bayes rule s used for he second equaly and he normalzaon consan P (y 1:T ) s dropped n he hrd equaly snce does no change he maxmzaon problem. III. OPTIMAL DISTRIBUTED TRACKING ALGORITHM The mos probable sequence of hdden saes x 1:T gven observaons y 1:T can be effcenly found by he Verb algorhm 14. We frs defne δ () max P (x x 1,...,x 1,..., x 1, x, y 1:), (4) 1 for 1,..., N and 1,..., T. δ () s he maxmum probably of a sngle pah endng n sae a me, gven observaons y 1;. Once we have all δ T (), we can race backward from he sae whch maxmzes δ T () o fnd he sequence x 1:T. For more deal abou he Verb algorhm, see 14. We compacly denoe he even {x } by z. Noce ha δ () max x 1: 1 P (x 1: 1, z, y 1:) max P (z x x 1)P (y z)p (x 1: 1, y 1: 1) 1: 1 max P (z x x 1)P (x 1: 1, y 1: 1) P (y z) 1: 1 max P j {1,...,N} (z z 1) j max P (x x 1:, z j 1: 1, y 1: 1) P (y z) max P (z z 1)δ j 1(j) P (y z), (5) j NB where he doman of he maxmzaon s reduced from {1,..., N} o NB n he las equaly snce P (z z j 1 ) 0 for j NB ( NB j by symmery). Hence, n order o compue δ (), we only need {δ 1(j) : j NB } from he neghbors of s, ranson probables and he lkelhood P (y z).

3 Fg.. A graphcal llusraon of rackng. (a) Node receves δ 1 from s neghbors (sep a). (b) Node ransms δ () o s neghbors (sep c) Snce P (y 1 z) η and P (y 0 z) 1 η, (5) can be furher smplfed as δ () max P (z z 1)δ j 1(j) η y (1 j NB η)1 y. (6) Thus, we can compue δ () wh only local nformaon. The dsrbued mplemenaon o fnd he mos probable rajecory of a movng objec s composed of wo seps: rackng and backrackng. A. Trackng We nroduce varables ψ () o keep he bes prevous locaon of an objec for he backrackng sep. For each sensor node and 1,,..., T, 1) If 1, a) compue δ 1() π η y 1 (1 η ) 1 y 1 ψ 1() 0; b) ransm δ 1() o NB. ) If > 1, a) receve {δ 1(j) : j NB }; b) compue δ () max P (z z 1)δ j 1(j) j NB ψ () arg max j NB P (z z j 1)δ 1(j); η y (1 η)1 y c) ransm δ () o NB. The rackng sep of he algorhm s graphcally llusraed n Fgure. In Fgure (a), node receves δ 1 from s neghbors. Afer compung δ () and ψ (), node ransms δ () o s neghbors (Fgure (b)). B. Backrackng Suppose ha he sensor node s b requess a rajecory of he movng objec a me T. Snce rajecory nformaon ψ () s sored n a dsrbued manner, we backrack o recover he mos probable rajecory by collecng verces along he pah dreced by ψ (). For each sensor node and T, T 1,..., 1, 1) If T a) compue φ T () δ T (); b) ransm (φ T (), x T () {}) o ψ T (). ) If 1 < T a) receve D {(φ +1(j), x +1:T (j)) : j J }, where J {j NB : ψ +1(j) }; Fg. 3. A graphcal llusraon of backrackng. (a) Node receves (φ +1, x +1:T ) from s neghbors (sep a). (b) Node ransms (φ (), x :T ()) o ψ() (sep (b)) b) f D, ) compue j arg max φ +1(k) k D φ () φ +1(j) x +1:T () x +1:T (j) x () ; ) f > 1, ransm (φ (), x :T ()) o ψ (); ) f 1, ransm x 1:T () o s b. The rajecory x :T () s smply concaenaed a each sep and a complee rajecory s avalable when he backrackng s ermnaed. However, an alernave approach s o ransm x o s b a each and ransm only φ (j) o ψ () so ha he ransmsson packes are of he same sze. The backrackng sep of he algorhm s graphcally llusraed n Fgure 3. In Fgure 3 (a), node receves (φ +1, x +1:T ) from s neghbors. Afer he sep (b), node ransms (φ (), x :T ()) o ψ () (Fgure 3 (b)). C. Dscusson Noce ha n order o carry ou above calculaons, node needs o know only π, η, and P resrced o NB and he remanng varables are compued on he fly. Hence, he calculaons requred for rackng and backrackng can be done n a compleely dsrbued manner. The algorhm mplemens he Verb algorhm 14 and fnds he mos probable rajecory x 1:T. Snce he compuaon and sorage of rack nformaon are done n a compleely dsrbued manner, he mehod s robus agans node falures and ransmsson falures. If d s he maxmum degree of he graph G, he number of ransmssons a each me s bounded above by dn and does no depend on T. Bu he memory requred o keep rack nformaon ncreases as T ncreases. For a large graph, he number of sensors nvolved n rackng ncreases as T ncreases. Hence, n order o bound he memory requremen and he number of sensors nvolved n rackng and make he algorhm scalable, we need a mehod o prune away unlkely racks. For hs purpose, we presen provably good prunng sraeges n Secon IV. IV. PRUNING The man dea behnd he prunng sraeges dscussed n hs secon s o prune a rajecory hypohess f he number of deecons s low. Snce prunng s performed, based solely on local nformaon, s mporan o analyze he performance loss of he algorhm for he chosen prunng sraegy. We frs presen he updaed rackng and

4 backrackng seps and hen descrbe prunng sraeges such ha, for gven ɛ > 0, he algorhm fnds an opmal rajecory x 1:T wh probably a leas 1 ɛ. An effec of prunng s shown n Fgure 4 where he prunng sraegy gven n Theorem 1 s used. A. Trackng wh Prunng We nroduce a varable ϕ () o keep he number of deecons along he pah dreced by ψ (). A rack hypohess s pruned away f g(, ϕ ()) 1 and he choce of g(, ϕ ()) s dscussed n Secon IV-C. δ () and ψ () are all nalzed o zeros. 1) If 1, a) compue δ 1() π η y 1 (1 η ) 1 y 1 ψ 1() 0 ϕ 1() y 1; b) ransm (δ 1(), ϕ 1()) o NB. ) If > 1, a) receve D {δ 1(j), ϕ 1(j) : j NB }; b) f D, ) compue δ () max P (z z 1)δ j 1(j) η y (1 j NB η)1 y ψ () arg max j NB P (z z j 1)δ 1(j) ϕ () ϕ 1(ψ ()) + y ; ) f g(, ϕ ()) 0, ransm (δ (), ϕ ()) o NB. B. Backrackng wh Prunng Backrackng s he same as Secon III-B bu he sep 1.(b) s replaced by 1.(b ) f φ T () > 0, ransm (φ T (), x T () {}) o ψ T (). C. Prunng Sraeges Ths secon gves wo prunng sraeges such ha, for gven ɛ > 0, he algorhm fnds an opmal rajecory x 1:T wh probably a leas 1 ɛ. Le W be he even ha he algorhm ermnaed a me reurns an ncorrec soluon,.e., when a correc soluon s pruned away. The omed proofs appear n Appendx. Theorem 1 (Fne Horzon): Le G be a passage connecvy graph of sensor nodes s 1,..., s N. Suppose ha η > 0, for all 1,..., N, and le η mn η. For ɛ > 0 and T > 0, choose s N, T such ha exp ( ) η 0 ( ɛ < s T for some 0 1, and choose θ η 1 log ( ) ) T. If he followng prunng sraegy, η sɛ { 1 f ϕ() < θ for ks 0, k N g(, ϕ ()) (7) 0 oherwse, s used, hen P (W T ) < ɛ. The followng heorem shows an exsence of a prunng sraegy when T. Theorem (Infne Horzon): Le G be a passage connecvy graph of sensor nodes s 1,..., s N. Suppose ha η > 0, for all 1,..., N, and le η mn η. For ɛ > 0, choose s N, such ha s > log ( 1+ɛ η ɛ followng prunng sraegy ) (, and choose θ η 1 log ( 1+ɛ ηs ɛ { 1 f ϕ() < θ for ks s, k N g(, ϕ ()) 0 oherwse, s used, hen P (W ) < ɛ. ) ). If he (8) V. ROBUSTNESS In prevous secons, we have assumed ha here are no false deecons n our observaon model. In hs secon, we sudy he robusness of he algorhm n he presence of false deecons. We assume ha he algorhm s run whou prunng so f a prunng sraegy s used, unless saed oherwse, he argumens here should be undersood n a probablsc sense,.e., gven ɛ > 0 and a prunng sraegy gven n Secon IV, he argumens hold wh probably larger han 1 ɛ. Le be he false deecon probably for he sensor s. Thus, when an objec s no n C a me, he sensor s makes a false deecon, Y 1, wh probably and Y 0 wh probably 1. We need o updae he jon dsrbuon (1) as followng o allow false deecons. where T T P (X 1:T, Y 1:T ) P (X 1) P (X X 1) P (Y X ), (9) P (Y X ) N 1 P (Y X ) (η X ) Y X (1 η X ) 1 Y X 1 j X ( j) Y j (1 j) 1 Y j. Unforunaely, we can no longer compue he lkelhood P (Y X ) wh only local nformaon as shown n (). We need observaons from all sensors n order o compue he lkelhood and s dffcul o guaranee he opmaly of any dsrbued algorhm. We focus on he case, n whch he deecon probables and he false deecon probables are unform, for s smplcy and show he opmaly condons for he algorhm gven n Secon III n he presence of false deecons. Theorem 3: Le G be a passage connecvy graph of sensor nodes s 1,..., s N. Suppose ha η η.5 and.5, for all 1,..., N and m mn P (x 1:T ) > 0. Le q 1:T be he soluon compued by he algorhm gven n Secon III and le x 1:T be he opmal soluon, an MAP esmae o (9) gven y 1:T. Then he followng nequales hold P (q 1:T, y 1:T ) P (x 1:T, y 1:T ) αp (q 1:T, y 1:T ), (10) ( 1 ) log(p (q 1:T )/m) where α. In Theorem 3, α 1 snce.5. To ge a gher bound, we wan α o be as small as possble. Noce ha when P (q 1:T ) s small, α s small and we can be confden ha he esmae q 1:T from he algorhm gven n Secon III s close o he MAP esmae x 1:T. An exreme case s when.5. Then α 1 and q 1:T x 1:T. However, (10) s he wors-case bound. To ncrease he performance of he algorhm for an average case, we requre small. Hence, we assume ha s fxed a some small value and dscuss approaches o make he wors-case bound gh. Corollary 1: Le G be a passage connecvy graph of sensor nodes s 1,..., s N. Suppose ha G s a regular graph wh a unform nal sae dsrbuon and unform ranson probables. Then, under he condons specfed n Theorem 3, q 1:T x 1:T. Proof: G s a regular graph wh a unform nal sae dsrbuon and unform ranson probables so we have a unform pror. In parcular, P (q 1:T ) P (x 1:T ). Hence, α 1 and q 1:T x 1:T. Corollary : Assume he condons specfed n Theorem 3 and le r ( max( P (x 1:T ) )/ mn P (x 1:T )). For ɛ 1 > 0, le c exp log r log 1 / log(1 + ɛ 1). If η c, hen 1+c P (q 1:T, y 1:T ) P (x 1:T, y 1:T ) (1 + ɛ 1)P (q 1:T, y 1:T ).

5 Fg. 4. An effec of prunng: an objec moves horzonally along he cener row of sensor grd (η.7). A sensor node nvolved n rackng s paned wh darker color. (Top) Whou prunng. (Boom) Wh prunng ( 0 100, s 100 and ɛ.05). Proof: Snce η c, η c. So 1+c 1 η α ( 1 ( 1 ( 1 ) log(p (q 1:T )/m) ) log r ) log r log c ( ) 1 log(1+ɛ 1 log( ) 1 ) 1 + ɛ 1. Hence, an deal case s when r s small and η s bg (as specfed n Corollary ). For example, when r 1000 and.45, requres η.8808 for ɛ 1 1 and η.7793 for ɛ 1. Bu f r 1000 and.3, we need η.9998 for ɛ 1 1 and η.995 for ɛ 1. We noe ha, for bnary sensors, he deecon probably can be boosed o a desred value by deployng mulple sensors over he same survellance regon. VI. NON-DISJOINT SENSING REGIONS Unl now, we have assumed ha he sensng regons of sensors are dsjon, leadng o ndependen observaons. However, hs assumpon may no be sasfed n all cases. In some cases, we may wan o nroduce more sensors o mprove he deecon probably. In hs secon, we descrbe how o handle non-dsjon sensng regons. Suppose ha G s a passage connecvy graph. For he smplcy of exposon, we consder he case when he sensng regons of wo sensors A and B are no dsjon (see Fgure 5). However, he mehod can be appled o he case wh more han wo sensors. Consder all dsjon segmens of C A C B,.e., C A\B C A \ C B, C B\A C B \ C A and C A B C A C B (see Fgure 6). Recall ha C A s he sensng regon of sensor A. In hs example, an exra verex A B s added no G along wh new edges from A B (see Fgure 6). The sensor nodes A and B exchange observaons and he necessary compuaons a A B can be dsrbued beween A and Fg. 5. An example of non-dsjon sensng regons. Sensor nodes are shown n dos and dsks around sensors represen sensng regons B. The addonal parameers of hs updaed graph can be learned from hsorc daa usng Baum-Welch mehod as menoned earler. However, f he sensor model of each sensor node s known, we can compue he sensor models of each dsjon segmen. Le η A and η B be he deecon probables of sensor A and sensor B, respecvely, and le A and B be he false deecon probables of sensor A and sensor B, respecvely. Le Y A and Y B be he observaons made by sensor A and sensor B, respecvely, and le X be he rue poson of he objec. Then he sensor models of dsjon segmens can be compued as shown n Table I. Now rackng can be done as dscussed n Secon III and IV. VII. MULTIPLE OBJECT TRACKING Suppose here are K ndependenly movng objecs and assume ha each sensor can denfy one objec from anoher. Le Y be an observaon made by sensor s a me, whch akes a value from a power se of {1,..., K}. Then we can express he sensor model as, for all k 1,..., K and 1,..., N, P (Y k X k ) η k (11) P (Y k X k ) 1 η k, where X k {1,..., N} s he locaon of he objec k a me and η k s he deecon probably a sensor for objec k. Then we can ndvdually rack each objec usng he algorhm gven n Secon III and IV. For he problem of rackng mulple objecs

6 TABLE I SENSOR MODEL P (Y A, Y B X) OF DISJOINT SEGMENTS IN FIGURE 6 Y A 0, Y B 0 Y A 0, Y B 1 Y A 1, Y B 0 Y A 1, Y B 1 X C A\B (1 η A )(1 B ) (1 η A ) B η A (1 B ) η A B X C B\A (1 A )(1 η B ) (1 A )η B A (1 η B ) A η B X C A B (1 η A )(1 η B ) (1 η A )η B η A (1 η B ) η A η B Fg. 6. A paron of sensng regons from he example shown n Fgure 5 VIII. CONCLUSION In hs paper, we have descrbed a novel rackng algorhm for sensor nework. The rackng problem s formulaed as a hdden sae esmaon problem n a hdden Markov model over he fne sae space of sensors. Then an opmal dsrbued rackng algorhm s derved from he Verb algorhm. The algorhm fnds an opmal soluon n a fully dsrbued manner. Furhermore, he algorhm s based on he smple bnary sensors and does no depend on sensor nework localzaon. We have descrbed provably good prunng sraeges for scalably of he algorhm and showed he condons under whch he algorhm s robus agans false deecons. We have also presened exensons of he algorhm o handle non-dsjon sensng regons and o rack mulple objecs. Snce he compuaon and sorage of rack nformaon are done n a compleely dsrbued manner, he mehod s robus agans node falures and ransmsson falures. ACKNOWLEDGMENT Ths maeral s based upon work suppored by he Naonal Scence Foundaon under Gran No. EIA Fg. 7. A orodal grd whou classfcaon nformaon, we refer readers o he herarchcal mulple-arge rackng algorhm for sensor neworks 15. A. Smulaon For smulaon, we consder a orodal grd passage connecvy graph G. An example of a orodal grd s shown n Fgure 7. We assume here are wo objecs and hey all have he unform nal sae dsrbuon whle p.1 and p j.9/ NB \ {} for j NB \ {}. The deecon probables are randomly chosen from.7, 1). However, we consder he suaon descrbed n 9, 16, n whch a sensor does no correcly classfy an objec when mulple objecs are presen n he same sensng regon. So when mulple objecs are presen n he same sensng regon, we assume ha he correspondng sensor does no repor deecon. We approxmae hs suaon by he sensng model (11). Two scenaros are consdered and hey are shown n Fgure 8 (a) and (c). The esmaed racks are shown n Fgure 8 (b) and (d). The esmaed racks are he same as he rue rajecores when objecs are deeced whle he esmaed racks follow he pah wh maxmum ranson probables and mnmum deecon probables when objecs are no deeced. A. Proof of Theorem 1 APPENDIX We frs need he followng lemma o prove Theorem 1 and Theorem. Lemma 1: Le Y 1,..., Y be ndependen random varables such ha P (Y 1) p and P (Y 0) 1 p, for arbrary 0 < p < 1. Le Z be a Bernoull process wh parameer p and, for all, p p. Le SY k1 Y k and SZ k1 Z k. Then, for all N and d, P (S Y < d) P (S Z < d). (1) Proof: We prove he clam by nducon. For 1, (1 p 1) (1 p). So P (Y 1 < 1) P (Z 1 < 1). Now suppose ha (1) s rue for. Then, for + 1 and any d + 1, we have, usng he nducon hypohess, P (S +1 Y < d) P (SY + Y +1 < d) P (SY + Y +1 < d Y +1 0)P (Y +1 0) + P (S Y + Y +1 < d Y +1 1)P (Y +1 1) P (S Y < d)p (Y +1 0) + P (S Y < d 1)P (Y +1 1) P (S Z < d)p (Y +1 0) + P (S Z < d 1)P (Y +1 1) P (S Z < d) P (S Z d 1)P (Y +1 1) P (S Z < d) P (S Z d 1)P (Z +1 1) P (S Z < d)p (Z +1 0) + P (SZ < d 1)P (Z +1 1) P (S +1 Z < d).

7 (a) (b) (c) (d) Fg. 8. (a) Scenaro 1 (10 10 orodal grd, rajecory of objec 1 n sold lne, rajecory of objec n dashed lne, observaons from objec 1 n crcles, observaons from objec n damonds). (b) Esmaed racks for scenaro 1 (rajecory of objec 1 n sold lne, rajecory of objec n dashed lne). (c) Scenaro (30 30 orodal grd). (d) Esmaed racks for scenaro. We now prove Theorem 1. Le Z be a Bernoull process wh parameer η and le SZ k1 Z k. Le X {1,..., N} be a Markov chan on G wh he nal sae dsrbuon and ranson probables gven n Secon II. Le Y be an observaon made a me. Y 1 wh probably η X and Y 0 wh probably (1 η X ). Le SY k1 Y k. Noce ha, for all, η X η. Gven a rack x 1:, he observaons are ndependen. By Lemma 1, for any, P (SY < d x 1:) P (SZ < d) for d. Furhermore, he nequaly holds for any x 1:. Hence, P (S Y < d) x 1: P (S Y < d x 1:)P (x 1:) ( x 1: P (x 1:) ) P (S Z < d) P (SZ < d). Le δ log ( ) T η sɛ. δ < 1 for all 0 snce s > T exp ( ) η 0 ɛ. Thus, P (W T ) < T 1 k1 T 1 k1 k1 k1 k1 k1 k1 I( ks)p (SY < θ ) I( ks)p (SZ < θ ) P (S ks Z < θ ks ) P (S ks Z < (1 δ ks )ηks) ( ) exp ηksδ ks ( exp ηks ηks log ɛ s T ɛ s T T s ɛ ( )) T sɛ where I s an ndcaor funcon and he Chernoff bound 17 s used n he fourh nequaly. B. Proof of Theorem Le Z be a Bernoull process wh parameer η and le SZ k1 Z k. Le SY k1 Y X k k. Then, as n he proof of Theorem 1, P (SY < d) P (SZ < d). Le δ log ( ) 1+ɛ ηs ɛ. Then δ < 1 by our choce of s. Thus, P (W ) < 1 k1 1 k1 k1 k1 I( ks)p (SY < θ ) I( ks)p (SZ < θ ) P (S ks Z < θ ks ) P (S ks Z < (1 δ)ηks) ) exp ( ηksδ ( exp ηks ( )) 1 + ɛ ηs log ɛ ( ) k ɛ ɛ, 1 + ɛ k1 k1 k1 where he Chernoff bound 17 s used n he fourh nequaly. C. Proof of Theorem 3 To avod confuson, le us denoe he jon dsrbuon whou false deecons (1) by Q(X 1:T, Y 1:T ) and he jon dsrbuon wh false deecons (9) by P (X 1:T, Y 1:T ). By he opmaly, for any feasble rack x 1:T,.e., P (x 1:T ) > 0, In parcular, we have P (x 1:T, y 1:T ) P (x 1:T, y 1:T ) Q(q 1:T, y 1:T ) Q(x 1:T, y 1:T ). P (x 1:T, y 1:T ) P (q 1:T, y 1:T ) (13) Q(q 1:T, y 1:T ) Q(x 1:T, y 1:T ). (14) From (13), we oban he frs nequaly n (10). Now consder he

8 followng jon dsrbuon rao R : P (q1:t, y1:t ) P (q1:t ) P (x 1:T, y1:t ) P (x 1:T ) P (q 1:T ) T 1 P (x 1:T ) T 1 P (q1:t ) P (x 1:T ) η yq T 1 ηyq T 1 P (y q) T 1 P (y x ) (1 η) 1 y q η yx (1 η) 1 yx (1 η) 1 y q T 1 ηyx (1 η) 1 yx Le S q T 1 yq and S x T ( 1 1 yx q y (1 ) 1 y x y (1 ) 1 y T 1 yx (1 ) 1 yx T 1 yq (1 ) 1 yq. Then R P (q1:t ) η Sq (1 η) T Sq Sx (1 ) T Sx P (x 1:T ) η Sx (1 η) T Sx Sq (1 ) T Sq ( ) Sq S η x ( ) Sx S q (15) P (q1:t ) P (x 1:T ) 1 η Q(q1:T, y1:t ) Q(x 1:T, y1:t ) ( 1 1 ) Sx S q. By (14), Q(q 1:T,y 1:T ) Q(x 1. Bu by (13), P (q 1:T,y 1:T ) 1:T,y 1:T ) P (x 1. Hence, 1:T,y 1:T ) ) Sx Sq ( ) 1. Snce.5, 1 and we mus have 1 S x S q. Bu S q canno be arbrarly small. Snce Q(q 1:T,y 1:T ) Q(x 1:T,y 1:T ) 1, ( ) Sq S P (q 1:T ) η x P (x 1:T ) 1 1 η S q S x log ( ) P (x 1:T ) P (q 1:T ( ) η 1 η ( ) log P (q1:t ) P (x 1:T ( ) log η 1 η log S x S q + Le b log(p (q 1:T )/P (x 1:T )). Then, we can bound S x as S q S x S q + b. ). Noce ha f P (q 1:T ) P (x 1:T ), S q S x and we ge an equaly S q S x. Hence, α 1 and q 1:T x 1:T. On he oher hand, we canno have P (q 1:T ) < P (x 1:T ), snce hs requres S q > S x, conradcng he opmaly of x 1:T. Now R akes he mnmum value f S x S q + b. So R P (q1:t ) ( ) b ( ) b η P (x 1:T ) 1 η 1 ( ) b 1 α. 1 ( 1 ) log(p (q 1:T )/m) Hence P (q 1:T, y 1:T )/P (x 1:T, y 1:T ) 1 α nequaly n (10). and we ge he second. REFERENCES 1 D. Culler, D. Esrn, and M. Srvasava, Overvew of sensor neworks, IEEE Compuer, Specal Issue n Sensor Neworks, Aug D. Culler and H. Mulder, Smar sensors o nework he world, Scenfc Amercan, June D. Esrn, D. Culler, K. Pser, and G. Sukhame, Connecng he physcal world wh pervasve neworks, IEEE Pervasve Compung, vol. 1, no. 1, pp , L. Schenao, S. Oh, and S. Sasry, Swarm coordnaon for pursu evason games usng sensor neworks, n Proc. of he Inernaonal Conference on Robocs and Auomaon, Barcelona, Span, J. Hghower and G. Borrello, Locaon sysems for ubquous compung, Compuer, vol. 34, no. 8, pp , Aug D. L, K. Wong, Y. H. Hu, and A. Sayeed, Deecon, classfcaon and rackng of arges, IEEE Sgnal Processng Magazne, vol. 17-9, March J. Aslam, Z. Buler, V. Cresp, G. Cybenko, and D. Rus, Trackng a movng objec wh a bnary sensor nework, n ACM Inernaonal Conference on Embedded Neworked Sensor Sysems, J. Lu, J. Lu, J. Rech, P. Cheung, and F. Zhao, Dsrbued group managemen for rack naon and manenance n arge localzaon applcaons, n Proc. of nd workshop on Informaon Processng n Sensor Neworks (IPSN), Aprl J. Lu, J. Lu, M. Chu, J. Rech, and F. Zhao, Dsrbued sae represenaon for rackng problems n sensor neworks, n Proc. of 3nd workshop on Informaon Processng n Sensor Neworks (IPSN), Aprl M. Coaes, Dsrbued parcle flers for sensor neworks, n Proc. of 3nd workshop on Informaon Processng n Sensor Neworks (IPSN), Aprl K. Whehouse, F. Jang, A. Woo, C. Karlof, and D. Culler, Sensor feld localzaon: a deploymen and emprcal analyss, Unv. of Calforna, Berkeley, Tech. Rep. UCB//CSD , Aprl X. Nguyen, M. I. Jordan, and B. Snopol, A kernel-based learnng approach o ad hoc sensor nework localzaon, n AAAI-004 Workshop on Sensor Neworks, July J. Aspnes, D. Goldenberg, and Y. Yang, On he compuaonal complexy of sensor nework localzaon, n Proc. of Frs Inernaonal Workshop on Algorhmc Aspecs of Wreless Sensor Neworks, Turku, Fnland, July L. Rabner, A uoral on hdden Markov models and seleced applcaons n speech recognon, Proceedngs of he IEEE, vol. 77, no., pp , S. Oh, L. Schenao, and S. Sasry, A herarchcal mulple-arge rackng algorhm for sensor neworks, n Proc. of he Inernaonal Conference on Robocs and Auomaon, Barcelona, Span, Aprl J. Shn, L. Gubas, and F. Zhao, A dsrbued algorhm for managng mul-arge denes n wreless ad-hoc sensor neworks, n Proc. of nd workshop on Informaon Processng n Sensor Neworks (IPSN), Aprl R. Mowan and P. Raghavan, Randomzed Algorhms. New York: Cambrdge Unversy Press, 1995.