Multisensor Particle Filter Cloud Fusion for Multitarget Tracking

Transcription

1 utenor Partce Fter Coud Fuon for uttarget Trackng Dane Danu, Tha Krubaraan Deartment of Eectrca and Comuter Engneerng cater Unverty, Hamton, Canada Abtract Wthn the area of target trackng artce fter are the ubect of content reearch and contnuou mrovement. The uroe of th aer to reent a nove method of fung the nformaton from mute artce fter trackng n a mutenor muttarget cenaro. Data condered for fuon under the form of abeed artce coud, obtaned n the muaton from two robabty hyothe denty artce fter. Dfferent way of data aocaton and fuon are reented, deendng on the tye of artce ued (e.g. before reamng, reamed, of equa or of dfferent cardnate). A muaton reented at the end, whch how the mrovement obe by ung more than one artce fter on a gven cenaro. Keyword: Trackng, artce fterng, data aocaton, fnte random et. Introducton Wthn the at decade and a haf, artce fter have been the ubect of content reearch and mrovement, wth eca emha on target trackng [2]-[0]. The eenta work n [2] take on a nge target and, ung the boottra artce fter, exand trackng erformance beyond the caca Extended Kaman Fter (EKF) for nonnear tate dynamc and meaurement. In [3] evera veron of amng and reamng deveoed n the mean-tme are reented, whe n [4] the author reent an extenon of the artce fter to the muttarget cae. The fundamenta work n [5] on muttarget trackng baed on random fnte et (RFS) and fnte et tattc (FISST) derve equaton for equenta etmaton of the frt order muttarget moment aka robabty hyothe denty (PHD). Baed on the FISST and PHD aroache, new artce fter deved for muttarget cenaro, wth the abty to hande target brth, death and awnng were deveoed by author n [6]-[8]. The PHD artce fter wa mroved n [9]-[0] wth track dentte, or track abe, ue of whch made n the current work durng the fuon roce. The uroe of th aer to reent a nove method of fung the nformaton from mute uch artce fter trackng n a mutenor muttarget cenaro. Secfc of the artce fter ued Thoma Lang Genera Dynamc, Ottawa, Canada Tom.Lang@gdcanada.com chae cdonad DRDC Ottawa, Canada ke.cdonad@drdc-rddc.gc.ca n oca etmator are detaed n ecton 2. The agorthm for aocatng and fung artce coud (where a coud a cuter of artce abeed a ertanng to the ame target) generated by two dfferent artce fter are reented n ecton 3. Ground target trackng muaton and reut are gven n ecton 4, whe concuon are drawn n ecton 5. 2 Partce Fter Becaue of t abty to hande target brth (or death), track ntaton and awnng n a muttarget cenaro, the PHD mementaton of a artce fter (PHD-PF) wa condered for generatng artce coud at oca enor eve. Further n th aer enor and oca etmator have a mar meanng, both denotng a oca enor-etmator entty that end t outut data for fuon. 2. PHD Partce Fter The PHD defnton baed on the theory of RFS, whoe tattc a.k.a. FISST defne the target n a cenaro a a RFS (meta-target) and the et of obervaton a another RFS (meta-obervaton) [5]. The PHD wa ntroduced n [5] a the frt order muttarget moment the denty functon whoe ntegra over a regon the exected number of target n that regon. Beng defned n ngetarget tate ace, t vaue at a gven ont the robabty denty functon of target reence, therefore rovde target ocazaton. In the ame work [5] the author derve, wthn the Bayean framework of FISST, the equenta etmaton of the frt order muttarget moment, further known wthn the target trackng communty a the PHD fter. The PHD fter ha the abty to ntate track of newy born target, awnng target, a we a to termnate track for dead target n a muttarget cenaro of dynamc target cardnaty, whe ao abe to account for cutter wthn the meaurement et. A equenta onte Caro mementaton of the PHD fter, known a the PHD artce fter, wa derved n [6]- [8]. 2.2 Track abeng One man te of the data fuon roce n target trackng data aocaton, n whch data (et) from dfferent 9

2 oca etmate are etabhed (groued) a ertanng to the ame target. The fued etmate reut from the combnaton of data contaned wthn uch an aready aocated grou (the two te mght overa or even a through evera teraton n ome nove aroache). Data aocaton a very comutatonay ntenve tak, n mot cae ncreang exonentay wth the number of enor and target. When fung artce coud, the robem comcated even more, by havng for each target a hgh number of rereentng artce. Th robem greaty aevated by abeng artce etmated a rereentng the ame target at the enor eve, ror to fuon. A PHD artce fter wth the artce abeng caabty wa deveoed n [0]. The man dea to run n arae another etmator n order to reerve track dentte (e.g. Kaman fter) and aocate PHD eak to current track dentte. A a reut, the PHD artce enterng the fuon roce, bede the tate and weght, have the track abe nformaton, whch bacay them n evera heren caed coud. For exame, after roceng the meaurement receved at a gven tme k, the outut of a gven artce fter (where =.. S and S the number of enor) the et of abeed artce correondng to the etmated PHD, D ˆ, kk :,, { ξ } Dˆ = w,, =.. Nˆ, =.. L (), kk k, k, kk, k, ș, Here, ξ k a artce of track abe and ndex, w k, t weght, N ˆ, kk the etmated number of target, ˆ =.. Nkk, are a target abe (numbered here from to N ˆ, kk, though th edom obe), and L, k the number of artce n the coud correondng to track abeed, a at tme k and at enor. In further notaton the tme ndex k droed for mfcaton, a fuon erformed tatcay (at a common ntance, thu tme beng rreevant for th aer uroe). One advantage of ung abeed artce n the artce fter equenta etmaton that new artce can be thrown nto roceng at any frame around every meaurement (true or fae) and no etmaton of fae meaurement ata dtrbuton needed for generatng thee new artce. The artce correondng to fae meaurement are emnated n the next frame, a they w not generate uuay confrmed track, and artce of non-confrmed track are not roagated for more than two or three frame. (Fae track, when generated, are uuay of ma duraton.) The advantage of ung abeed artce for fuon that artce of non-confrmed track do not enter the fuon roce, thu decreang a ot the data aocaton robem ze and communcaton. 3 Fuon of abeed artce coud The fuon art dvded nto two man tage, the frt beng the data aocaton and econd the etmaton from combned aocated data. Data to be fued here cont of abeed artce and ther correondng weght. The two tage are treated earatey next, n deveong the fuon method for two enor. 3. Aocaton of artce coud Ung notaton n () wth tme ndex droed, we denote 2 by Ξ a -abeed coud at enor one, and by Ξ 2 a 2 - abeed coud at enor two, wth abe =.. Nˆ and ˆ 2 =.. N2. (We aume that on both enor abe, 2 are contguou and tart from ony for the uroe of cot Ξ, Ξ 2 mfyng the notaton). We denote by ( 2) the cot of the hyothe that coud Ξ and Ξ 2 2 correond to the ame target. Smary to the cae of track-to-track aocaton, f we aume that the coud aocaton event among dfferent coud ar are ndeendent, the mot key coud-to-coud aocaton hyothe can be found, wthn the 2D agnment formuaton, by ovng the foowng contraned otmzaton: L L2 2 mn χ 2cot ( Ξ, Ξ2) (2) ubect to χ = 0 2= 0 L = 0 χ =, =.. L L2 χ 2 =, =.. L (3) 2 = 0 where χ 2 a bnary varabe. The abe =0 and 2 =0 n (2-3) are ued to denote the aocaton wth dummy (therefore no aocaton for the coud on the other enor). The mnmzaton n (2)-(3) can be eay oved ung for exame aucton agorthm [2]-[3]. It reman to determne the evauaton of the coud-to-coud aocaton cot, ( ) cot Ξ, Ξ 2 2, whch treated n the next ecton. In comutng the coud-to-coud aocaton cot, two tye of coud are condered (from the artce reamng vewont): coud of unreamed artce and coud of reamed artce. A gatng between coud ha to be ued frt, uch that ome coty mutagnment (defned next) are ked. 3.2 Cot of coud to coud aocaton A artce coud a fnte et of artce, therefore the cot of aocatng two coud can be een a the dtance The otmzaton robem generaze to N-D agnment when the number of enor S > 2. 92

3 between the two et (of and 2 artce) defnng the coud. Pror to comutng the cot of aocatng two coud Ξ = { w, ξ}, =,2 (4) = on two enor (abeed coud, however, wth abe droed here for notaton mfcaton), both coud are normazed uch that ther weght um to one. Th done becaue we wh to re-nterret the artce coud a a dcretzed rereentaton of the target robabty denty functon n 2D-Cartean ace, of whch ntegra one: ( x, y ) dax (, y ) = (5) da R where R the regon over whch the denty etmated, the denty functon and da the eementary urface area (for etmaton n xy ane). A coud ent for fuon ony f t track wa confrmed, therefore the target amot urey reent and (5) hod over the etmaton urface R. For the normazed weght we have w = Δ ( ) = ( ) Δ ( ) = w A ξ ξ A ξ = = ΔA( ξ) = (6) From (6), uon normazaton each artce weght correond to the df of the ce urroundng the artce (of area Δ A( ξ ) around ξ ) muted by the area ce w = ( ξ) Δ A( ξ), rereentng the ce robabty ma. The Waerten dtance defne the dmarty between two robabty dente f, g, ung the dtance d (e.g. Eucdan for the cae of artce here) and the ont dtrbuton h(, xy ) whoe margna are the two dente h (, xy) d x= g () y and h(, xy) dy= f() x []: ( ) W d f, g = nf d( xy, ) h( xy, ) dxdy (7) h The ncuon of tate veocty comonent nto artce dtance coud be added (.e. a d = x-x2 + ε v-v2 ) wth ε havng a the hyca meanng ma tme nterva. In [] the muttarget m dtance ntroduced baed on the Waerten dtance a a mean to ae the muttarget trackng erformance (through rovdng a dtance between the et of etmated track and true target). Here we ue the adutment of Waerten dtance to fnte et a uggeted n [], the dfference beng that ntead of beng aed to et of target, t aed to et of weghted artce. We ue for functon f and g the um of Drac deta functon δ Ξ ( ξ ) and δ Ξ 2( ξ 2), defned on the urface ξ = {( x, y)}, covered by both enor =, 2 and wth vaue on the dcrete et correondng to the two artce coud 2 ξ,..., ξ Ξ = ξ,..., ξ Ξ = { }, 2 { 2 2 } δ () ξ = w δ ( ξ ), =,2 (8.) Ξ and h(x,y) defned a = 2 = = ξ h(, xy) wδ ξ () xδ ξ () y (8.2) = 2 where w are eement of the tranortaton matrx W known under th name from the near rogrammng eca cae of tranortaton robem. Introducng (8.) and (8.2) n (7), the Waerten dtance aed to two artce coud become: d 2 W (, 2) nf wd( ξξ, ) W = = Ξ Ξ = (9) wth = or = 2. Dfferent method for comutng the cot baed on (9) are derved further, deendng on the tye of artce ued (.e. reamed or before reamng) Ung coud of reamed artce Ung coud of (abeed) reamed artce at the artce fter eve roduce a fxed number of artce er coud, a havng ame weght (.e. / for artce er coud). The contant weght of a artce reut n the eca form of δ Ξ () ξ a δξ() ξ = δξ ( ξ), =,2 (0) = In th cae the tranortaton matrx W, ha the um on a row equa, and the ame hod for coumn: 2 w =, w = () = = 2 Equaton (9) and () rereent the contraned otmzaton to be oved n etmatng the dtance between the coud (obectve functon). Two ub-cae are n the comutaton of th cot: when Ξ = Ξ 2 (.e. coud are of equa cardnate) and when Ξ Ξ Reamed artce coud of equa cardnate For equa cardnate, = 2 = and a artce n both coud have dentca weght /. Conequenty the coud-to-coud aocaton cot can be mfed to the otma obectve functon of a me 2-D agnment (.e. W to have eement n {0, } ony). The mnmzaton (9) become W d (, 2) mn Ξ Ξ = d( ξξ, σ ) σ (2) = 93

4 wth =, 2, where σ a ermutaton of =,,. Equaton (2) rereent the 2-D agnment robem (wthout dummy aocaton aowed, a a artce houd be agned), whch can be oved by fndng otmum σ ermutaton ung aucton agorthm [2]- [3] Reamed artce coud of dfferent cardnate For coud of dfferent cardnate, 2, even though artce n each coud have equa weght (.e. / for = and / 2 for = 2), they are dfferent than artce weght n the other coud. In order to reduce th robem to a mar 2-D robem a n 3.2.., both coud need to be reduced to the ame number of artce and of equa weght (n order to aow one-to-one aocaton ony and therefore W wth eement n {0,}). Ung a method mar to the one n [], each artce =.. n the coud Ξ w be reamed nto * = 2/gcd(, 2) new artce, where gcd tand for the greatet common dvor, thu obtanng a coud of * Ξ = = 2/gcd(, 2) artce of weght w= gcd(, 2) /( 2). The ame done to the artce n the coud Ξ 2, n conequence obtanng * Ξ 2 = 22 = 2/gcd(, 2) artce of ame weght w. At th tage the cot can be comuted foowng the ame method n Ung coud of unreamed artce For two gven artce fter =, 2, wth, reectvey 2 tota artce er fter at a gven tme w, ξ have ntance, the unreamed artce { } = unequa weght. By t nature, for the PHD-PF, the tota um of artce weght equa to the etmated number of target (ratona number before roundng) ˆ w = N. = For the abeed artce fter, the number of target ao gven (nteger) by the number of abe, wth ony the abe of confrmed track counted. A abeed track condered confrmed ony f t ha meaurement aocated to t n evera conecutve frame. Ony thee artce, abeed a ertanng to confrmed track, are roagated for fuon. The cot of aocatng any two coud comuted a: ( ) 2 2 wd ξ ξ2 = = cot(, ) mn, atfyng the contrant Ξ Ξ = (3) w = w 2, =.. 2 (4) = 2 w = w, =.. (5) = Contrant (4) and (5) aure that each artce n both coud fuy agned (ung t fu weght) to artce from the other coud. The mnmzaton robem (3)-(5) a mutagnment robem, mar to the tranortaton robem, whch a eca cae of near rogrammng. 3.3 utagnment outon ung near rogrammng nteror ont method In order to tranate (3)-(5) n a caca near rogram mn dw T (6) ubect to Aw = b, w 0, w we ue the foowng for (7) 2 2 d R, w R, + 2, 2 2 A R, b R + : d = [ d d... d d d... d... d d... d ] rereent the vector of dtance d d( ξ, ξ2 ) =, w = [ w w... w w w... w... w w... w ] T the content of tranortaton matrx W n vector form, A = [ (8) ] and b = [ w w 2... w ] T w w w beng the vector of artce weght of both coud. A ame ackage ovng the above mnmzaton LIPSOL (Lnear-rogrammng nteror ont over) [4]. 3.4 utagnment outon a equenta 2D aocaton For coud wth very hgh artce count, a ubotma way of erformng the mutagnment between 94

5 unreamed coud of dfferent cardnate a a equence of 2D unweghted agnment. 3.5 Coud to coud fuon For two aocated artce coud (a correondng to ame target), Ξ and Ξ 2, of unreamed artce wth normazed weght (each coud weght um to one), three method for comutng the etmated fued tate are reented beow. ethod Derve from a drect combnaton of artce n both coud, a beow x 2 ˆ 0.5 = ξw + ξ2w 2 (9) = = ethod 2 Coud are weghted by ther ame covarance, foowng the nformaton fter aroach []: x 2 2 ˆ = ( P + P2 ) P ξw + P2 ξ2w 2 (20) = = Fgure Target evouton n tme n the x, y drecton. where P, P 2, are ame covarance of coud Ξ and Ξ 2, comuted through: ( )( ) = = P w ξ ξ ξ ξ, =,2 (2) T where ξ the ame weghted mean. In th method ndeendence of the etmaton error of both coud aumed. ethod 3 Fuon carred out makng ue of the etmated ame cro-covarance matrce between coud, P 2, P 2. Thee are etmated baed on the tranortaton matrx W between coud, derved n the aocaton te P w ( ξ ξ )( ξ ξ ) ( ξ ξ) ( ξ2 ξ2) w 2 T = = = 2 T = = = (22) and ued a P = P+ P2 P2 P2 (23) n the comutaton of fued etmate x N N2 ˆ = P P P ξ w + P P P ξ w (24) ( ) ( ) = = 4 Smuaton The muaton cenaro contan evera target whch occur and daear at dfferent moment, a hown through traectore n Fgure beow. Fgure 2 Target and meaurement n xy ane for the entre cenaro muaton. Fgure 2 how the true meaurement (crce) and fae aarm (croe). The two oca etmator ued at the two enor =, 2, are PHD artce fter wth track-vaued (abeed) artce coud. The equaton and mementaton of the track-abeed PHD artce fter foow the one n [0], here extended to two-dmenona trackng, brefy decrbed beow, wth k denotng the tme ame, and fter ndex droed. Partce tate contan two-dmenona oton and veocty n Cartean coordnate, whe meaurement rovde ony oton nformaton. Intazaton For =,, L ame oton (x, y) of artce a ξ,( xy, ) N( Z, σ Z), where Z the et of a nta meaurement, and N tand for norma dtrbuton. The veocty comonent of each artce are ntazed unformy dtrbuted wthn maxmum veocte aowed for a target ξ,( vx, vy) U( vmax, vmax ), wth v max = 6 here. w = ς L n, where Inta weght are comuted a ( ) 95

6 ς the etmated number of target and n the number of meaurement. Predcton For =,, L k- (confrmed artce at tme k-), ξ q, where q the rooa denty, ame k ( ξk ) whch tranate nto T ξk = k k ξ 0 0 T + ν (25) wth ν k denotng the roce noe (v x,y =.0, v vx, vy = 0.5) and T = the amng tme te. Aumng no target awnng, weght are comuted a w k = w k e, where e = 0.95 the robabty of target urvva. For = L k- +,, L k- +J k newborn artce are amed baed on new meaurement n the et Z k. For th uroe Z k arttoned wth reect to ϒ k, the et of track at tme k-, nto Zkn, = { zk zkn vadaton regon of a track n ϒ k } and Zk, out = { zk zk n no vadaton regon of any track n ϒk } Partce oton comonent are amed for a meaurement a ξ k,( x, y) N( z, σ z), where σ z the tandard devaton of meaurement oton error. For meaurement n Z k,n artce veocty comonent are ˆ n ξ N x, σ, where x ˆ n the amed from k,( vx, vy) ( k v) veocty etmate of track n at tme k-, n whch regon the meaurement z wa acceted. For meaurement n Z k,out artce veocty comonent are amed unformy wthn [-vx max, vx max ] and [-vy max, vy max ]. The weght of a newborn artce are comuted a b ( k ξ k) w kk = (26) J ξ Z where ( k k) ( ) k k k k b ξ the ntenty functon of the ont roce (condered Poon here) generatng the target ξ Z the rooa denty. By brth and k( k k) conderng norma dente for b k and k a n [8],[0] the weght are comuted a I / J k, where the contant ntenty of the ont roce wa taken a I = 0.2 (e.g. one target brth exected every 5 th amng tme on average). Udate For =,, L k, where L k = L k- + J k, the udated weght are comuted a k w k = w k k β Phz D ( ξk k ) β2 Phz D ( ξk k ) PD + + z Zk, n λ c+ βck() z z Zk, out λ c+ β2ck() z (27) where P D the robabty of detecton (condered unty here), h the enor meaurement functon, condered a norma wth tandard devaton 2.5 for both enor, λ = 4 the average rate of cutter ont er can, c = the unform cutter denty for the whoe urveyed area, β =., β 2 = β /25 are degn arameter a n [0] and L k C () z = P h( z ) w (28) k D ξkk kk = Fgure 3 Etmaton of nge PHD-PF at end of cenaro wth hort fae track vbe. Coud at current (at) tme te: redcted of non-confrmed track bue, current meaurement yeow, confrmed track other coor. Fgure 4 Etmaton reut: red, voet oca fter (PHD-PF), bue true traectore, cyan fuon method, back fuon method 2. Fae track emnated after aocaton are not hown n ocay etmated track. Track abe are obtaned by ung hadow Kaman fter and the reouton ce technque [0]. The unreamed artce coud are condered for fuon, for whch aocaton cot are comuted a n (9) wth =. RSE 96

7 (Root ean Square Error) comuted over tme for each run and averaged over 00 C run are hown n tabe beow for nge fter and fued reut. Tabe RSE (x, vx, y, vy) for fter. Target Target 2 Target 3 Target 4 RSE x RSE vx RSE y RSE vy Tabe 2 RSE (x, vx, y, vy) for ethod of fuon Target Target 2 Target 3 Target 4 RSE x RSE vx RSE y RSE vy Tabe 3 RSE (x, vx, y, vy) for ethod 2 of fuon. Target Target 2 Target 3 Target 4 RSE x RSE vx RSE y RSE vy Fgure 5 Htogram of coud-to-coud dtance comuted : (a) a um of weghted dtance brown, and (b) ung equaton (9) bue for σ =.5, σ 2 =0.5. From reut above t can be een that oton etmate are mroved more than veocty etmate, and that the uage of ame covarance n ethod 2 doe not ncreae contenty the accuracy. 4. onte Caro muaton for Gauan dtrbuted coud In order to ae the rooed aocaton cot for coud of dfferent dtrbuton, onte Caro muaton were erformed for etmatng the aocaton cot of two coud, normay dtrbuted (correondng to the unreamed cae), wth dfferent tandard devaton. The number of artce condered wa = 200 on both coud. The cot dtrbuton obtaned n 50 run are comared wth the dtrbuton of the dtance between the coud center etmated ung the weghted um of artce 2 = ξ 2ξ2 = = (29) d w w From the htogram of the dtance comuted ung (9) and (29), comared n Fg. 5-9 for coud of dfferent dtrbuton, t can be een that the cot comuted baed on (29) nentve to the varance of the coud, a ony the frt order etmate enter (29), whe the one comuted baed on (9) ncreae wth the dfference between coud tandard devaton (varance). Whe thee muaton were ung Gauan coud, however the cot n (9) not makng any aumton on normaty, therefore dfference n hgher order moment of the coud are exected to be refected n the cot a we. Fgure 6 Same dtrbuton a n Fg. 5, for σ =.5, σ 2 =.0. Fgure 7 Same dtrbuton a n Fg. 5 for σ =.5, σ 2 =.5. 97

8 [3] Saneev. Aruamaam, S. ake, N. Gordon, and T. Ca, A tutora on artce fter for onne nonnear/non-gauan Bayean trackng, IEEE Tran. on Sgna Proceng, Vo 50, No. 2, , February [4] Carne Hue, J-P Le Cadre, Patrck Perez, Trackng ute Obect wth Partce Fterng, IEEE Tran. on AES, Vo. 38, No. 3, , Juy [5] Ronad P.S. aher, uttarget Baye fterng va frt-order muttarget moment, IEEE Tran. on AES, Vo. 39, No. 4, , October Fgure 8 Same dtrbuton a n Fgure 5-7, for σ =.5, σ 2 =2.5. [6] T. Zac and R. aher, A artce-ytem mementaton of the PHD muttarget trackng fter, Sgna Proc., Senor Fuon and Target Recognton XII, SPIE Proc. I. Kadar (ed.), Vo. 5096, , [7] Hedvg Sdenbadh, ut-target artce fterng for the Probabty Hyothe Denty, The 6 th Internatona Conference on Informaton Fuon, Carn, Autraa, Juy 2003, Proc. Vo.2, [8] Ba-Ngu Vo, Sumeeta Sngh, Arnaud Doucet, Sequenta onte Caro ethod for ut-target Fterng wth Random Fnte Set, IEEE Tran. on AES, Vo. 4, No. 4, , October Fgure 9 Same dtrbuton a n Fgure 5-7,, for σ =.5, σ 2 = Concuon A method of fung denty etmate of artce fter at the artce coud eve wa rooed. Coud-to-coud aocaton cot deendng on artce tye were deveoed together wth evera fuon method. Reut obtaned for a muated muttarget cenaro of dynamc cardnaty were reented. The fuon at artce eve reut n a fued etmate of target denty, not ony n a fued etmate of the target tate; the mrovement of the target denty etmate w be further nvetgated. Reference [] Y. Bar-Shaom, X.R. L, T. Krubaraan, Etmaton wth Acaton to Trackng and Navgaton: Theory, Agorthm and Software, Wey, New York, 200. [2] Ne. J. Gordon, D.J. Samond, and A.F.. Smth, Nove aroach to nonnear/non-gauan Bayean tate etmaton, IEE Proceedng-F Radar, Sonar and Navgaton, Vo. 40, No. 2,. 07-3, Ar 993. [9] K. Panta, B. Vo, S. Sngh, Imroved robabty hyothe denty (PHD) fter for muttarget trackng, The 3 rd Conf. on Integent Senng and Informaton Proceng, Bangaore, Inda, December 2005, Proc [0] Ln Ln, Yaakov Bar-Shaom, T. Krubaraan, Track abeng and PHD fter for muttarget trackng, IEEE Tran. on AES, Vo. 42, No. 3, , Juy [] John R. Hoffman and Ronad P. S. aher, uttarget Dtance va Otma Agnment, IEEE Tran. on Sytem, an and Cybernetc - Part A Sytem and Human, Vo. 34, No. 5, ay [2] Dmtr P. Berteka, and Davd A. Catañon, A forward/revere aucton agorthm for aymmetrc agnment robem, Technca Reort Ld-P-259, IT, 993. [3] R. Jonker and A. Vogenant, A hortet augmentng ath agorthm for dene and are near agnment robem, J. Comutng, Vo. 38, , 987. [4] Yn Zhang, Uer gude to LIPSOL nearrogrammng nteror ont over v0.4, Otmzaton ethod and Software, Vo. -2, ,