Tracking Hidden Agents Through Shadow Information Spaces

Transcription

1 Trking Hidden Agents Through Shdow Informtion Spes Jingjin Yu Steven M. LVlle Deprtment of Computer Siene University of Illinois Urn, IL 601 USA Astrt This pper ddresses prolems of inferring the lotions of moving gents from omintoril dt extrted y roots tht rry sensors. The gents move unpreditly nd my e fully distinguishle, prtilly distinguishle, or ompletely indistinguishle. The key is to introdue informtion spes tht extrt nd mintin omintoril sensing informtion. This leds to monitoring the hnges in onneted omponents of the shdow region, whih is the set of points not visile to ny sensors t given time. When used in omintion with pth genertor for the roots, the pproh solves prolems suh s ounting the numer of gents, determining movements of tems of gents, nd solving pursuit-evsion prolems. An implementtion with exmples is presented. () I. INTRODUCTION Mny importnt tsks involve resoning out oservtions mde while detetle odies pss in nd out of the field-of-view of moving sensors. Figure shows severl senrios to whih the onepts in this pper pply. In suh senrios, generl prolems inlude: 1) ounting unpreditle people or roots tht move in omplited environment [1], [4], [], [1], ) pursuing n elusive moving trget y sweeping through omplited environment to gurntee detetion or pture [3], [], [8], [], 3) monitoring tem movement to ensure tht suffiient numers from eh tem re present in ritil prts of n environment, nd 4) trking movements of gents to determine their possile lotions [11], [], [16]. For eh of these nd other tsks, numerous senrios re possile depending on the types of roots, moving odies, sensors, environments, nd models tht re given. Our pper is motivted y the following question: Is there generl wy, regrdless of the senrio, to systemtilly proess ll umulted sensor oservtions nd mke inferenes out where the moving odies might e? Determining where odies might e is like plying omplited shell gme 1 : We wnt lgorithms tht effiiently mke perfet inferenes out unoservle moving odies. 1 This is gmling gme, usully involving deeption, in whih the plyer must trk the lotion of hidden ojet s the deler quikly shuffles severl potentil ontiners. () Fig. 1. ) Imgine three roots (white diss) tht rry visiility sensors to detet nd trk the movements of gents in n indoor (omplited orridors) or outdoor ( mpus of uildings) environment. There re seven onneted white regions in whih gents re out of view of sensors. ) Suppose there re holes in the overge of n tuted sensor network, nd we wnt to reson out possile movements of gents in the holes. ) In n urn setting, imgine trking people using heliopters, nd the field of view is frequently ostruted y uildings nd lndmrks. d) Consider using stellites to trk ots on the se while the lnd onstrins their motions nd moving louds ostrut prt of the view. This pper ddresses the generl prolem of inferring where unpreditle moving odies, lled gents, ould e s they pss in nd out of view. We only ddress the pssive prolem of mking onlusions sed on the motions of the sensors. This is useful for humns or roots, nd it is hoped tht the solution ids in the tive prolem, whih involves deiding how to move roots nd serh the informtion spe [6] to omplish given tsk. Regrdless of the prtiulr tsk or senrio, the pssive prolem mounts to proessing ritil hnges in the shdow region, whih is the portion of the environment tht is not visile to ny sensors t prtiulr (d)

2 1-4 S0 10 S1 S 1 d 1 v 1 7 v1 1 v v 6 8 v4 v3 v 4 3 Fig.. ) A D environment nd trjetory followed y root with omnidiretionl visiility. ) A grph tht enodes omintoril hnges in visiility. ) A iprtite grph, mde y ompressing informtion in. d) A resulting mximum-flow grph, used to nswer questions out trking or ounting gents. instnt. We onstrut n informtion spe whih umultes onstrints tht rise s the onneted omponents of the shdow region hnge. We introdue lgorithms tht enode the informtion stte s high-dimensionl polytope nd n effiiently nswer queries regrding possile lotions of gents y solving mximum flow prolem over n extended iprtite grph tht is derived inrementlly from the history of sensor oservtions. To the est of our knowledge, there hs een no ttempt to reson out the movements of trgets with the level of generlity onsidered in our pper. The min ssumption is tht the onneted omponents of the shdow region n e mintined. Beyond this, the environment my e two or three dimensionl, known or unknown, nd simply or multiply onneted. There my e one or more roots or humns rrying sensors. The sensor field-of-view my e limited y ny depth or ngle. The gents in the environment my e indistinguishle, distinguishle, or prtilly distinguishle (for exmple, there re severl tems). Figure illustrtes the high-level flow of onepts throughout the reminder of the pper. II. PROBLEM FORMULATION A. Roots nd visiility regions Suppose tht one or more roots move in world, W = R or W = R 3. To void onfigurtion-spe omplitions, ssume eh root is point. Suppose tht there re fixed ostles in W tht ostrut the roots. The ostle oundry is ssumed to e ounded v v v time nd pieewise-nlyti (to enle finite enodings). Let F W denote the free spe, whih is n open set of possile root positions. Let q denote the onfigurtion of the roots; if there re k roots, then q F k. The roots rry sensors tht illuminte suset of F. Let V (q) denote losed visile region, when the roots re in onfigurtion q. Let S(q) = F \ V (q) denote the shdow region. If the roots move long time-prmeterized pth q : [0,t] F k, then the shdow region itself eomes time-prmeterized: S( q(t )) is otined for every t [0,t]. It is ssumed tht V (q) nd S(q) ehve niely s q vries ontinuously; more preisely, ritrrily smll hnges in q led to smll hnges in V (q) nd S(q) (this n e formlized using the Husdorff metri). This enles the shdow omponents to e onsistently leled s q vries. B. Moving gents Let A denote set of n gents. Eh gent is point tht moves long ontinuous, not-neessrilyknown pth through F. There re no given ounds on the mximum speeds of the gents. Let r F n e n-dimensionl vetor tht speifies the gent positions. The vetors q nd r yield the positions of ll roots nd gents. Therefore, let x = (q,r) denote the stte, nd let X = F k+n e the stte spe. To set up different levels of distinguishility, it will e onvenient to ssign tem lels to the gents. Let T denote set of m tems, with m n. Let the mpping l : A T ssign tem to eh gent. It is ssumed tht gents n e distinguished only if they elong to different tems. At one extreme, we my hve T = 1, in whih se ll gents elong to the sme tem nd re indistinguishle. At the other extreme, we my hve T = n nd l is ijetion. In this se, the gents re ll distinguishle. A onvenient intermedite se is to ssign olors to the gents. For exmple, T = {red,green,lue}. There might, for exmple, e 30 gents, with 10 reds, 1 greens, nd lues. Wht does the sensor tully produe? Suppose tht for ny onfigurtion q, the onneted omponents of S(q) re leled with unique positive integers. As q hnges, the omponents of S(q) my undergo one of four ritil hnges: 1) two omponents merge, ) one omponent splits into two, 3) new omponent emerges, nd 4) omponent disppers. Eh ritil hnge is lled omponent event. These events re sed on infletion nd itngent rossings, nd re desried in more detil in [7], []. It is ssumed tht the omponent lels remin onstnt unless there is omponent event. This inludes generl position ssumption, to void tedious ses in whih three or more omponents re involved in hnge.

3 Let N e the set of nonnegtive integers. Let h : X Y e the sensor mpping, in whih Y = N m+1 is n oservtion spe. An oservtion y = h(x) Y is vetor of m + 1 integers. For i from 1 to m, the i th omponent of y indites the numer of gents from the i th tem tht lie in V (q). At the instnt in whih n gent enters or exits V (q), y m+1 gives the lel of the omponent of S(q) tht is entered or exited; otherwise, y m+1 = 0. The instnt t whih n gent enters or exits V (q) is lled field-of-view event. () C. Initil onditions An initil ondition is defined for eh shdow omponent. The totl numer of gents, n, my or my not e known to the root. For eh omponent, we llow lower nd upper ounds on ny susets T i of T. Let l i nd u i denote the lower nd upper ounds ssoited, respetively, with T i. Here re some exmples. We might know initilly tht one omponent ontins t lest nd t most red gents. In this se, T i = {red}, l i = nd u i =. Alterntively, we might know tht there re extly 7 gents, with unknown tems. In this se, T i = T nd l i = u i = 7. The ompletely unknown se eomes T i = T, l i = 0, nd u i =. Eh omponent my hve s mny s k pirs of ounds (k my e different for eh omponent). Let T 1,...,T k denote the susets of T hosen for onstrints; we ssume these susets re disjoint. An initil ondition for the omponent n e expressed s {(T 1,l 1,u 1 ),(T,l,u ),...,(T k,l k,u k )}, in whih the l i,u i ould e ny nonnegtive integer or positive infinity, nd l i u i. D. Possile Tsks Severl kinds of tsks n e defined sed on the formultion. Suppose tht the roots hve moved long some trjetory. Possile tsks re: Determine the minimum nd mximum numers of red (or ny other olor) gents in one of the shdow omponents. Count the totl numer of gents in F. Determine whether one finl shdow omponent ontins ny gents. Determine whether one finl shdow omponent hs more lue gents then red gents. Determine whether lue nd red gents re seprted into different shdow omponents. III. THE HISTORY INFORMATION SPACE To ddress the tsks desried in Setion II-D, we refully define nd mnipulte informtion spes (for generl introdution, see Chpter 11 of [6]). We will () Fig. 3. In this simple exmple, single root rries stndrd visiility sensor in polygonl free spe. The segment tht emntes from the root trjetory indites where omponent event ours (only few omponent events re shown). Agent trjetories re not shown. There re four possile types of omponent events: ) A shdow omponent splits into two. ) A shdow omponent ppers. ) Two shdow omponents merge. d) A shdow omponent disppers. use the revitions I-spe for informtion spe nd I-stte for informtion stte. The si ide will e to void preisely estimting the gent lotions, nd insted proess the sensor dt diretly to solve tsks. To omplish this, sustntil effort is required to identify ompt, omintoril representtions tht preserve ll informtion tht is pertinent to solving tsks. In this setion, we introdue the history I-spe, in whih n I-stte ontins initil onditions nd the entire timeprmeterized history of sensor oservtions. After time evolves from time 0 to some time t, perfet desription of everything tht ourred would e stte trjetory x t : [0,t] X. It is impossile to otin this, however, euse the gent positions re generlly unknown. Insted of the stte trjetory, we re offered the oservtion history, ỹ t : [0,t] Y, whih is time-prmeterized olletion of sensor oservtions. Suppose temporrily tht the root onfigurtions re lwys known (this ssumption is removed in Setion IV. Let η 0 denote the initil onditions from Setion II-C. The history I-stte t time t is η t = (η 0, q t,ỹ t ), whih represents ll informtion ville to the roots. The history I-spe I hist is the set of ll possile history I-sttes, whih is n unwieldy spe tht must e drmtilly redued if we expet to solve our tsks. (d) 3

4 IV. THE SHADOW-SEQUENCE INFORMATION SPACE In this setion, we onstrut derived I-spe I ss whih is otined y mpping from I hist tht disrds informtion tht is irrelevnt to solving our tsks. Consider the informtion ontined in η t = (η 0, q t,ỹ t ). In I ss, we retin η 0, ut q t nd ỹ t re redued in some wy. The following redutions re mde: 1) The onfigurtion trjetory q t will e repled y grph struture tht simply lels the onneted omponents of S(q) nd trks how they evolve. This informtion n e determined ompletely from the onfigurtions, whih re given y q t. The prtiulr onfigurtions do not even need to e mesured, provided tht there is some lterntive wy to determine the shdow omponents (for exmple, they e inferred diretly from depth disontinuities in plnr, simply onneted free spe []). The grph struture is updted only when omponent event ours, whih mens omponents split, merge, pper, or dispper (see Figure 3). ) The oservtion history ỹ t is ompressed so tht only hnges in the oservtion need to e reorded. Every suh hnge orresponds to field-of-view event. The result is sequene of distint oservtions. 3) Preise timing informtion is disrded, exept the order in whih events our. The omponent events nd field-of-view events re interleved ording to their originl time ordering. We mke generl position ssumption tht no two events our simultneously. The result is n I-stte tht lives in derived I-spe, I ss, whih we ll the shdow-sequene I-spe. One ft tht should eome ler during the presenttion is tht ll of the onepts nd methods from Setions III to V n e pplied to the set of onstrints on eh T i (rell from Setion II-C) without onsidertion of onstrints with respet to other susets of T. We therefore present the onepts nd lgorithms for the se of single tem (i.e., T = 1). The pproh then extends nturlly to hndle eh T i for i from 1 to k (we eventully otin k independent mx-flow prolems, insted of one). By onsidering one tem only, it is simple to hrterize the possile onstrints for every initil shdow omponent nd every shdow omponent tht rises from n pperne omponent event, s shown in Figure 3. For the i th omponent, we llow lower nd upper ounds, represented s v i = (l i,u i ), in whih l i nd u i re nonnegtive integers (u i my e infinite) suh tht l i u i. For n initil shdow omponent, ny possile v i is llowle; however, it seems tht for omponent v 1 7 v1 1 v v 6 8 v4 v Fig. 4. A grph tht ounts for the evolution of shdow omponents over time. tht rises from n pperne, it is only possile tht v i = (0,0) ( new shdow omponent must e ler of gents). We will divide field-of-view events into two lsses to simplify the disussion. If k field-of-view events our in whih k gents enter shdow omponent, sy s i, tht ws just formed y n pperne omponent event, then we ssoite v i = (k,k) with s i. In the opposite diretion, if k gents leve shdow omponent s i nd in the next event s i disppers, then v i = (k,k) is ssoited with s i, nd it is reorded tht s i disppered. The remining presenttion up to Setion V-C ssumes tht only these kinds of field-of-view events n our (otherwise, our exmples eome luttered). Setion V-C then desries how the remining field-of-view events re hndled. The shdow-sequene I-spe I ss will now e desried using the exmple from Figure 3. A prtiulr I-stte in I ss is illustrted in Figure 4. The squre oxes indite the upper nd lower ounds v i = (l i,u i ), whih re ssoited with eh shdow omponent tht either: 1) existed initilly (the squre oxes t the top), ) ppers t omponent event, or 3) disppers t omponent event. The irles indite prtiulr shdow omponents. The I-stte n e onsidered s direted grph in whih the verties re the irles nd ll edges flow downwrd, whih orresponds to the progression of time. The oxes simply show the informtion ssoited with some verties. The edges indite how the shdow omponents were involved in split or merge omponent event. This ssoition is importnt euse we must ount for possile movements of gents. For exmple, Figure shows how n gent n move from one former shdow omponent to nother fter merge omponent event. Bsed on the informtion shown in the oxes in omintion with the grph struture, the possile numers of gents ehind eh finl shdow omponent n e v v v3 v time 4

5 v v Fig.. A omponent event tht enles n gent to relote. expressed s polytope s follows. Every vertex (irle in Figure 4) is n nonnegtive integer vrile x i, whih is ssoited to shdow omponent s i. Every v i = (l i,u i ) onstrint represents the ounds l i x i u i. Furthermore, every split nd merge omponent event orresponds to sum of vriles. For exmple, in Figure 4, when s 6 splits into s 8 nd s, we otin n ILP onstrint x 6 = x 8 +x. When s 4 nd s 10 merge into s 11, we otin x 11 = x 4 + x 10. Thus, pieewise-liner desription (polytope) of possile numers of gents is otined over the x i vriles. Answering prtiulr queries over the polytope eomes kind of integer liner progrmming (ILP) prolem. The generl ILP prolem is NP-hrd [10], whih might sound very disourging; however, our prtiulr ILPs hve speil struture tht enles them to e solved rther effiiently using mximum-flow lgorithms. To rrive t this, however, we need to further ompress the I-sttes. This rings us to the next setion, in whih iprtite grph struture enodes I-sttes, nd n I-spe results tht is even more ompt. V. COLLAPSING THE INFORMATION SPACE ONTO A BIPARTITE GRAPH A. Informtion sttes s n ugmented iprtite grph After refully studying the I-sttes in I ss, it eomes pprent tht the polytopes nd ssoited ILPs tht re otined hve some speil struture. Any piee of informtion tht origintes in n initil shdow omponent or shdow omponent tht is generted y n pperne omponent event must eventully ontriute to finl shdow omponent or shdow omponent tht disppered. Furthermore, the informtion umulted t ny finl omponent or omponent tht disppered is omposed entirely of informtion tht originted from the initil omponents nd ones tht ppered from omponent events. These ilterl dependenies suggest tht the informtion flow n e ptured in iprtite grph in whih the left side indites the soures of informtion nd the right side indites the destintions. Figure 6 illustrtes how to onstrut nd inrementlly mintin suh grph for eh of the four possile types of omponent events. The iprtite grph ontins weights on eh of the verties, whih result from ounds of the x y z x y z () () v (d) Fig. 6. Inrementlly omputing I-sttes in I ip : ) An pperne omponent event dds two verties nd n edge, with v ssoited with the left vertex. ) A split event splits vertex nd ll edges pointing to tht vertex. ) A merge event ollpses two verties into one nd ollpses their ingoing edges. d) A dispper event only ssoites v with the vertex on the right side. form v i = (l i,u i ). Initilly, the grph ontins left nd right vertex for every initil shdow omponent, nd n edge etween every orresponding pir. In ft, if there re k initil shdow omponents, then k opies of the onstrution shown in Figure 6() re mde. Coneptully, this is equivlent to introduing k shdow omponents vi pperne events. The four prts of Figure 6 show how the iprtite grph grows inrementlly while the roots move long onfigurtion trjetory. At ny time, the resulting iprtite grph, with ssoited weights on every vertex, is onsidered s n I-stte in new I-spe denoted s I ip. Figure 7() gives the iprtite grph for the exmple in Figure 3. Note tht in generl, prtiulrly if there re mny omponent events, the new grph is muh smller thn the kind onstruted in Figure 4. Note tht the onversion uses some informtion loss: All internl x y z x y z v

6 () S 0 10 S 1 S 1 Fig. 7. ) An I-stte in I ip, whih is iprtite grph. ) The orresponding mximum-flow grph for nswering questions out gent movements. All edges re direted from left to right. nodes from the I-sttes in I ss dispper s they re proessed to onstrut the iprtite grph. Furthermore, the time-ordering informtion is no longer preserved in the iprtite grph. The informtion spe I ip is more ompt thn I ss, ut it is nevertheless suffiient for solving our tsks. B. Trking gents s mximum flow prolem We lssify the iprtite grph verties into three types. Verties on the left re ll from initil or pper-event shdow omponents, nd re lled inoming verties. The verties on the right tht re from shdow omponents tht hve disppered re lled intive verties; ll of these hve fixed weights tht orrespond to the numer of field-of-view events tht ourred immeditely prior to dispperne. The remining right verties orrespond to shdow omponents tht hve not disppered, nd re therefore lled tive verties. One the iprtite is onstruted, the tsk of determining upper nd lower ounds on finl shdow omponents n e trnsformed into mximum flow prolem. A flow prolem usully hs soure nd sink verties, nd the edges hve pity nd flow vlues. For eh v i, ssume tht l i = u i, whih orresponds to slr weight w i = l i = u i tht is ssoited with vrious verties. This ssumption will e lifted t the end of this setion. To trnsform our iprtite grph with vertex weights into flow grph, introdue soure vertex S 0, with edges pointing to ll inoming verties. Also introdue sink vertex, S 1, to whih ll intive verties point, nd seond sink vertex, S, to whih ll tive verties nd S 1 point (see Figure 7). Note tht shdow omponents from s 1 to s 4 re ll ollpsed into single vertex here euse we n equivlently ssume tht they re split from sme v i. After otining the extended grph, the weights on verties must e shifted to pities on edges. Let e(s,t) e n edge in the grph going from s to t, nd denote 1 the pity nd flow on tht edge s (s,t) nd f(s,t), respetively. Consider omputing the minimum numer of gents in finl shdow omponent, s m. In this se, we wnt to determine the minimum flow through the orresponding tive vertex in the grph. Every inoming vertex, i, now orresponds to fixed numer of gents, nd the numer is the pity for e(s 0,i). For every intive vertex j, the pity of e(j,s 1 ) is w j. By onservtion of flow, the pity of e(s 1,S ) is (S 1,S ) = (S0,i) (j,s 1 ). To otin the minimum for s m, we wnt the lest possile flow through e(m,s 1 ); therefore, (m,s 1 ) = 0. All other edges re ssigned infinite pity. After running the mximum-flow lgorithm, min(f(m,s 1 )) = f(s 0,i) f(j,s 1 ) f(s 1,S ), i (1) in whih f(j,s 1 ) = (j,s 1 ). Eqution (1) is orret euse minimum flow t one edge of the grph is equivlent to flowing the mximum mount possile through the rest of the grph. Note tht the prtiulr mximum-flow lgorithm is not ritil; this is lssil prolem nd mny effiient lterntives exist. To insted ompute the mximum numer of gents possile for finl shdow omponent, s m, let (m,s 1 ) = nd let (j,s 1 ) = 0 for ll other tive verties. Keep the rest of the grph the sme s for the minimum se. The resulting f(m,s 1 ) fter running mximum-flow lgorithm is the desired upper ound. The proedure n e repeted for every tive vertex. Note tht the lower (or upper) ounds on finl shdow omponents usully nnot e hieved simultneously y tul gents euse of the dependenies mong the omponents. Now we return to the more generl se in whih l i nd u i re not neessrily equl. In this se, we simply reple v i y l i for determining minim nd u i for determining mxim. C. Hndling the remining field-of-view events In Setion IV, we intentionlly left some field-of-view events out of the disussion to simplify the explntion. Those events ertinly need to e hndled euse gents my enter or exit the visiility region t ny time. To ount for those events, we n ugment the iprtite grph s it is omputed. For eh gent pperne fieldof-view event, we n tret it s if omponent splits into two nd then one of the two shdows disppered, reveling ertin numer of gents. We n tret eh gent dispperne field-of-view event s if new shdow omponent ppers nd merges into n existing one. Algorithmilly, this effetively trnsforms field-of-view events into omponent events. 6

7 VI. SOLVING A VARIETY OF TASKS The mximum-flow method explined in Setion V-B n e used to determine the minimum nd mximum numer of possile gents hiding in speified shdow omponent. In ddition to this, vriety of other tsks n e redily solved, nd re overed riefly in this setion. Refining initil ounds: Mximum flow omputtions n e used to refine the ounds given originlly on the initil shdow omponents, sed on informtion gined lter. For exmple, if s 1 originlly hs lower ound of 4, ut it is then oserved tht there re 6 gents ppering from s, then there must hve initilly een t lest 6 gents in s 1. Upper ounds n e refined similrly. To get etter lower ound for n initil shdow omponent, s m, let (S 0,m) = l m nd let the remining (S 0,i) ssume the weights of their inoming verties. The edges from intive verties weighted s efore, nd ll (j,s 1 ) re infinite. Let (S 1,S ) = (S 0,i) (j,s 1 ). The refined lower ound is given y l m = l m + j (j,s 1 ) j f(j,s 1 ) () in whih j rnges over the intive verties. To refine u m, let (S 0,m) = u m, (S 0,i) = l i for eh i m, nd the remining pities e the sme s in the lower-ound se. The refined upper ound is u m = j f(m,j). (3) Counting: In this se, the totl numer n of gents is unknown. For determining n, the lower nd upper ounds on eh initil shdow omponent re speified s v i = (0, ). Using the lgorithm from Setion V-B, upper nd lower ounds n e otined on eh omponent. If l i = u i for eh of these, then n hs een determined. Otherwise, () nd (3) give lower nd upper ounds on n. Note tht if F is not ompletely explored, then the upper ound remins t infinity. Reognizing tsk ompletion: We might lso wnt to reognize the ompletion of ertin tsks. For exmple, in wild niml preserve, it my e required tht the totl numer of speies is verified periodilly. This redues to the prolem of eing given n nd wnting to ount for ll of them. To verify the ompletion of this tsk, we n keep trk of the lower ounds on the totl numer of gents, nd if the numer grees with n, then the tsk hs een omplished. Pursuit-evsion: Another tsk is pursuit-evsion, s defined in [3], [], [8], []. Suppose there is single evder nd the tsk is to determine where it might e. In this se, v i = (0,1) for eh initil shdow omponent. In the end, there re three possiilities for eh finl shdow omponent: 1) v i = (0,0) (the evder is not in s i ), ) v i = (1,1) (the evder is definitely in s i ), nd 3) v i = (0,1) (the evder my or my not e in s i ). Note tht this is pssive version of the pursuit-evsion prolem. We do not determine trjetory tht is gurnteed to detet the evder. In generl, this prolem is NP-hrd []. Nevertheless, the lultion method proposed in this pper n e used with heuristi serh tehniques (or even humn opertors) to orretly mintin the sttus of the pursuit. Multiple tems with ounds on single tems: Now suppose tht there re multiple tems, m > 1. There re two importnt ses. For the first one, suppose tht the ounds for eh initil shdow omponent refer to individul tems. Thus, eh T i (refer to Setion II-C) is just single tem. The tsk of determining lower nd upper ounds for m tems n then e solved y defining m distint single-tem trking prolems, one for eh tem. If the tsk is to find lower ound on ll gents in finl shdow omponent, we simply sum lower ounds of ll tems in tht omponent to otin the nswer. This pproh even works in the extreme se in whih ll gents re distinguishle. Hndling onstrints ross multiple tems: The seond se for multiple tems llows T i to ontin more thn one tem. For exmple, n initil shdow omponent my hve onstrint tht there re etween 3 nd 7 gents tht re red or lue. If we wnt to know the lower ound on red or lue gents, then we set red pity from S 0 to the orresponding vertex in the flow grph to zero, nd flow the reds through the grph to otin minimum of the red. We then do the sme for lue. Adding those two numers up gives the nswer. Upper ounds n e omputed similrly. VII. IMPLEMENTATION We implemented nd tested the lgorithms for single root tht moves in simply onneted polygonl region in R using n omnidiretionl visiility sensor. We hose this setting for simultion euse the itngents nd infletions n e lulted so we ould hve n orle for moving the gents round inside the free spe nd void simulting prtiulr gent movements. Rell tht the method works for ny numer of gents, ny sensor models, nd D or 3D worlds, s long s the shdow omponents n e mintined. We implemented the O(V E )-time Edmonds-Krp mx-flow lgorithm [], in whih V nd E re the numers of verties nd edges in the grph, respetively. The numer V is the totl numer of initil, ppering, disppering, nd finl shdow omponents. In the worse se, the iprtite grph my e omplete, ut in prtie there re fr fewer edges. 7

8 The progrm is written in Jv 1.. The yte ode ws run on Intel U00 1.GHz mhine with 1GB RAM. For the free spe in Figure 8(), the trjetory genertes 8 omponent events. We defined n orle tht rndomly distriuted 100 gents in the free spe s the omponent events our. This setting yields iprtite grph tht hs 41 verties nd 60 edges. Clulting the lower nd upper ounds for the finl shdow omponents for single tem took 0.1 seonds. The seond free spe, shown in Figure 8, hs 38 omponent events, 41 totl shdow omponents, 4 verties in the iprtite grph with 33 edges. The exmple involves million gents on tems tht intersperse. The ounds on finl shdow omponents for ll tems were omputed in. seonds. () Fig. 8. Complited exmples tht were used to test our pproh. The given root trjetories re shown. VIII. CONCLUSION We hve introdued nd solved prolem of trking unpreditle gents tht re mostly out of the rnge of sensors. The pproh uses initil hypotheses on possile gent lotions nd resons out how the shdow omponents evolve to mke finl onlusions. The resulting informtion spe is ollpsed into iprtite grph, nd queries re solved effiiently y solving speil mximum flow prolem. The onepts re very si nd generl. We therefore elieve there is gret potentil for pplying them to prtil prolems suh s serhnd-resue, ounting people, trking movements in uilding, nd nlyzing movement strtegies for tems of roots or humns. Mny interesting open prolems remin. The onepts pply to wide vriety of sensor models; however, the fesiility nd omplexity of mintining the shdow omponents vries nd should e studied in prtiulr ontexts. Also, the resulting informtion spes my further simplify in some ses. For exmple, it seems tht the derived I-spes, I ss nd I ip, for single root in simple polygon should e muh simpler thn tht for numerous roots in 3D free spe tht hs holes. Severl open questions remin regrding weker sensor models. For gents tht lie in V (q), we presently ssume their tem lels nd entering/exiting shdow omponents re ll oserved. Exmples of weker models inlude: 1) inry sensor tht indites only whether there is t lest one gent in V (q), ) ounter tht yields the totl numer of gents in V (q), 3) unertinty regrding whih shdow omponent n gent entered or exited. Aknowledgments: Yu is supported in prt y Sieel Fellowship. LVlle is supported in prt y the DARPA SToMP progrm (DSO HR ). The uthors sinerely thnk Chndr Chekuri for helpful disussion on ILPs nd mximum flow. REFERENCES [1] Y. Bryshnikov nd R. Ghrist. Trget enumertion vi Euler hrteristi integrls I: sensor fields. Preprint ville on Internet, Mrh 007. [] J. Edmonds nd R. M. Krp. Theoretil improvements in lgorithmi effiieny for network flow prolems. J. ACM, 1():48 64, 17. [3] B. Gerkey, S. Thrun, nd G. Gordon. Cler the uilding: Pursuitevsion with tems of roots. In Proeedings AAAI Ntionl Conferene on Artifiil Intelligene, 004. [4] B. Gfeller, M. Mihlk, S. Suri, E. Viri, nd P. Widmyer. Counting trgets with moile sensors in n unknown environment. In ALGOSENSORS, July 007. [] L. J. Guis, J.-C. Ltome, S. M. LVlle, D. Lin, nd R. Motwni. Visiility-sed pursuit-evsion in polygonl environment. Interntionl Journl of Computtionl Geometry nd Applitions, ():471 44, 1. [6] S. M. LVlle. Plnning Algorithms. Cmridge University Press, Cmridge, U.K., 006. Also ville t [7] S. M. LVlle nd J. Hinrihsen. Visiility-sed pursuit-evsion: The se of urved environments. IEEE Trnstions on Rootis nd Automtion, 17():16 01, April 001. [8] J.-H. Lee, S. Y. Shin, nd K.-Y. Chw. Visiility-sed pursuitevsions in polygonl room with door. In Proeedings ACM Symposium on Computtionl Geometry, 1. [] X. Liu, P. H. Tu, J. Rittsher, A. Perer, nd N. Krhnstoever. Deteting nd ounting people in surveillne pplitions. In Pro. IEEE Conferene on Advned Video nd Signl Bsed Surveillne, pges , 00. [10] G. L. Nemhuser nd L. A. Wolsey. Integer nd Comintoril Optimiztion. Wiley, New York, 188. [11] J. Singh, R. Kumr, U. Mdhow, S. Suri, nd R. Cgley. Trking multiple trgets using inry proximity sensors. In Pro. Informtion Proessing in Sensor Networks, 007. [] I. Suzuki nd M. Ymshit. Serhing for moile intruder in polygonl region. SIAM Journl on Computing, 1(): , Otoer 1. [] B. Tovr, R Murriet, nd S. M. LVlle. Distne-optiml nvigtion in n unknown environment without sensing distnes. IEEE Trnstions on Rootis, 3(3):06, June 007. [] C.-C. Wng, C. Thorpe, S. Thrun, M. Heert, nd H. Durrnt- Whyte. Simultneous loliztion, mpping nd moving ojet trking. Interntionl Journl of Rootis Reserh, 6():88 16, 007. [1] D. B. Yng, H. H. Gonzlez-Bnos, nd L. J. Guis. Counting people in rowds with rel-time network of simple imge sensors. In Pro. IEEE Interntionl Conferene on Computer Vision, volume 1, pges, 003. [16] F. Zho nd L. Guis. Sensor Networks: An Informtion Proessing Approh. Morgn Kufmnn, Sn Frniso, CA,