Towards the Fusion of Distributed Binary Decision Tree Classifiers

Transcription

1 Towads th Fusion of Distibutd Binay Dcision T Cassifis Qian Zhang EECS Dpatmnt, Link Ha Syacus Univsity, Syacus, NY 344, USA Emai: qizhang@sy.du Abstact Mutip snso fusion and binay dcision t cassifis hav bn usd to succssfuy sov many a wod pobms. Ths topics a usuay studid spaaty. Fusion of binay dcision t cassifis in a mutip snso nvionmnt has civd vy itt attntion. In this pap, w fomuat th pobm, invstigat its scop, outin som issus associatd with dcision t cassifis and mutip snso fusion, and psnt som soution mthodoogis. Th suts a iustatd by mans of an xamp. Ky wods: snso fusion, binay dcision t. I. Intoduction Mutip snso dcision fusion is an impotant pobm with many pactica appications. This pobm has bn studid quit xtnsivy and many significant suts on this topic hav bn obtaind [- 3]. In most studis, on-stag dcision making pocdus a mpoyd at th snso as w as at th fusion cnt. By a on-stag pocdu, w man that a sing tst is mpoyd to distinguish btwn a th hypothss. Such on-stag dcision pocdus may bcom too compx and impactica in situations wh th a many vaiabs and many objct casss (hypothss). Ths typs of pobms ais in aas such as automatic tagt cognition (ATR), tam mdica diagnosis and tmdicin, and agsca suvianc systms. In ths situations, it may b dsiab to us mutistag dcision making pocdus at th snsos and/o at th fusion cnt. Th a sva appoachs fo impmnting mutistag dcision making stuctus. On popua appoach is by mans of a binay dcision t (BDT) [4]. Th basic ida is to tak a compx M-ay hypothsis tsting pobm, bak it into sva simp binay hypothsis tsting pobms that a oganizd in a hiachica t stuctu to mak dcisions gading th M hypothss o objct casss. H, w invstigat th us of BDTs in mutisnso fusion pobms. Th bnfits of using BDTs in mutisnso dcision fusion a mutifod:. It is w known that th dsign of distibutd dtction systms that mpoy on-stag dcision making fo binay hypothsis tsting pobms is Pamod K. Vashny EECS Dpatmnt, Link Ha Syacus Univsity, Syacus, NY 344, USA Emai: vashny@sy.du NP-had. This dsign fo M-ay hypothsis tsting is vn had. BDTs mak a squnc of binay dcisions in a hiachica mann that a asi to dsign, fficint and computationay simp to impmnt with simp dcision gions. Thus, th us of a BDT may mak th dcision making pocdu fasib fo pactica situations that hav tim o pocssing constaints. In addition, communication bandwidth fficincy may b achivd bcaus tansmission of binay dcisions instad of M-ay dcisions wi b quid.. Som of th avaiab snsos may not b capab of distinguishing a th objct casss in a paiwis mann. BDTs povid a famwok fo intgating th capabiity of a th snsos fo mutisnso dcision fusion and fo nhancd systm pfomanc. 3. Dcision making via a BDT has an inhnt fxibiity to dsign tsts at th intna nods. This fxibiity povids th abiity to hand snso dfcts, missing snso obsvations/dcisions, tc., thby nhancing systm obustnss. Aso, th fxibiity may hp in impoving systm pfomanc. Th dsign of a BDT basd mutisnso fusion systm invovs th dsign of th BDT, dsign of th tsts at th intna nods of snso BDTs, dsign of th fusion u, and dsign of th systm topoogy incuding communication stuctu of th mutisnso fusion systm. Goas of this dsign incud nhancd ova systm pfomanc (cognition abiity) and obustnss, using ast possib computation and communication. Som aspcts of this pobm hav bn addssd in th itatu. Dmibas [5] poposd a non-paamtic cntaizd objct cognition schm basd on a BDT. Each snso pocsss its data and xtacts som fatus that a tansmittd to th fusion cnt. Objct cognition is caid out using a BDT gnatd fom a taining st. Dasaathy [6] concntatd on th achitctua aspcts of a systm that fuss binay dcisions into a sing M- ay dcision. Th main goa was to dsign achitctus that satisfy pocssing tim constaints. Zhu t a. [7] aso considd th pobm of M-ay hypothsis tsting using a paa fusion topoogy wh oca dtctos tansmit binay dcisions. Thy focussd on th dsign of dcision us and on systm

2 pfomanc. Th main goa of this pap is to xamin th ova pobm of BDT basd mutisnso dcision fusion pobm, idntify th issus that nd to b addssd futh and popos som soution mthodoogis. In Sction II, w fomuat th pobm. In Sction III, w consid a spcific BDT basd dcision fusion systm. In Sction IV, w giv an xamp to iustat th suts of Sction III. In Sction V, w mak som concuding maks. II. Pobm Ovviw W focus ou attntion on th paa achitctu fo BDT basd mutisnso fusion systms in this pap. Th bock diagam of a BDT basd paa dcision fusion systm is shown in Figu. Th systm consists of K snsos that obsv a common phnomnon in paa. Th goa is to cogniz a givn unknown objct that bongs to th st of objcts {O, O,, O M. Lt X k b th obsvation vcto of th kth snso. Each snso uss a BDT to mak its dcisions. Lt T k dnot th BDT usd by th kth snso and U k th dcision mad by th kth snso. Ths dcisions a tansmittd to th fusion cnt that combins thm to yid U, th goba dcision. Th fusion u Γ (.) is a function that maps oca dcisions u,, u K into u and maks a dcision gading th unknown objct. X Snso S BDT T U U =Γ (U,,U K ) U U k X K Snso S K BDT T K Figu : BDT basd paa dcision fusion systm W assum that th ad is famiia with th notion of t and associatd tminoogy. Fo dtais, th ad may f to [4]. A gna BDT is shown in Figu. X dnots th fatu vcto. U dnots th dcision mad by th BDT at tmina nods. Sinc ach vau u of U cosponds to a uniqu path fom th oot nod to a tmina nod, u can b ncodd as th squnc of binay dcisions mad by a th nods in th cosponding path. At nod t, Φ(t) dnots th st of fatus usd by th BDT, and Γ(t) dnots th dcision u, which is a function that maps th spac of fatus spcifid by Φ(t) into st {,. nod t Φ (t), Γ(t) X oot O a U=u a O b U=u b Figu : A gna BDT Using th basic systm achitctu fo th dcision fusion systm shown in Figu, sva dsign appoachs and mods of opation can b nvisagd. H w catgoiz BDT basd paa dcision fusion systms in fou typs:. Each snso mpoys a BDT fo dcision making. Ths snso BDTs a assumd avaiab and a dsignd indpndnty of ach oth. Snso dcisions a snt to th fusion cnt. Th fusion cnt ith uss a on-stag pocdu o a BDT to dtmin th fina dcision. Th main issu h is th dsign of an optimum fusion pocdu. This pobm is anaogous to th dsign of an optimum fusion u in distibutd dtction systms [9]. This pobm is appicab to th fusion of BDT cassifis that may hav bn dsignd indpndnty.. BDTs at th individua snsos a dsignd jointy by mpoying coupd cost functions. Dcisions a avaiab ocay so that appopiat action can b takn at th snso. Dcision fusion is not mpoyd to combin snso dcisions. This pobm is anaogous to th on considd by Tnny and Sand [] in a distibutd dtction contxt. 3. Dcisions mad by snso BDTs a convyd to th fusion cnt that combins thm to yid th fina dcision. No oth communication is aowd. Snso BDTs and th fusion u a dsignd. This systm is a gnaization of distibutd dtction systms wh ony on-stag dcision pocdus a aowd. 4. In this cas, th systm is th sam as systm 3 xcpt that two-way communication btwn th snsos and th fusion cnt is aowd. Th

3 snsos navigat though thi BDTs und th supvision of th fusion cnt in a tuy coopativ mann. This is a nw systm achitctu that quis cos coodination btwn systm mnts. In th fou typs of systms dscibd abov, som o a of th foowing issus nd to b addssd. Dign of BDTs fo ach snso and/o th fusion cnt. Sction of snso fatus fo snso BDT dsign and dcision-making at intna nods of snso BDTs. Dcision-making us at intna nods of snso BDTs. Dcision-making us at th fusion cnt. Communication potoco usd by th snsos and th fusion cnt fo coodinatd navigation of thi BDTs. Evauation of systm pfomanc - cognition abiity, obustnss tc. Th tatmnt of th abov issus is guidd by many factos, such as snso chaactistics, avaiab pocssing soucs, opating nvionmnt and th natu of th objcts, tc. Du to th novty of ths pobms, much ffot is ndd to dvop a systmatic soution to th ova BDT basd mutisnso fusion pobm. In th foowing w mak som obsvations on th diffncs btwn how th abov issus a tatd in th convntiona sns and und th fomuation of BDT basd dcision fusion. Convntiona BDT dsign mthodoogis can b appid h, but with th goa of optimizing th ova systm pfomanc instad of just th pfomanc of individua snsos. A BDTs shoud b dsignd in such a way that thy coctivy mak th bst cassification of th unknown objct. A sing snso BDT may not b optima whn usd as a standaon cassifi. Snso fatus ought to b sctd in such a way that objcts a bst spaatd acoss th snso suit fo nhancd pfomanc of th ova systm. Th sctd fatus may not b th bst fo individua snsos. Th dcision us at intna nods of snso BDTs nd to b dsignd such that bst systm pfomanc is achivd. Ths us may not b th bst fo individua BDTs. III. Dsign of BDT Basd Intactiv Systms In this sction, w focus on BDT basd dcision fusion systms that invov two-way communication and wh a systm componnts a jointy dsignd. Th quaity of such a dcision fusion systm is givn by th cognition at of ach objct and th associatd avag numb of stps in a cognition opation. Th fist quantity svs as a masu of ffctivnss of th fusion systm in caying out dtction/cognition, whi th scond quantity indicats th fficincy in tms of computationa tim/ffot. W popos a mthod of dsigning BDTs, snso us and fusion us. W assum that th snso obsvations a conditionay indpndnt givn th objct cass. W aso assum that ach snso obsvation is chaactizd by a pobabiity distibution function givn ach objct cass and that it is known a pioi. Finay, ach snso uss a th avaiab fatus at ach nod of its BDT. Fist, w popos a way to coopativy us BDTs at th fusion cnt and th snsos. Th fusion cnt BDT is usd to cay out a squntia patition of th objct spac. At ach intna nod of its BDT, th fusion cnt tsts on subst of objcts against anoth subst of objcts. It cocts snso oca dcisions and uss thm to mak a goba dcision on which subst to tst futh, i.., it scts th path of th t to foow. Basd upon this goba dcision, it chooss an appopiat chid nod at which it tsts two nw substs of objcts. Th fusion cnt pats this pocdu ti it achs a tmina nod wh xacty on objct is ft, and thn dcas this objct to b th unknown objct. W notic that at ach nod of th fusion cnt BDT, th oca dcision fom a snso fcts to what dg this snso distinguishs th two substs of objcts that a und tst. To optimay utiiz th capabiity of this snso, it is ncssay that this snso tst th sam two substs of objcts as th fusion cnt bcaus adding/moving objcts to ths two substs woud mak th oca dcision of this snso ss vant to th cognition task at th fusion cnt. Basd on this fact, w t vy snso BDT patition th objct spac in th sam way as th fusion cnt BDT dos. Fo this ason, a th BDTs may b considd idntica with spct to th objct spac. Howv, sinc snso oca dcisions may not aways ag with th goba dcisions, som snsos may not choos th sam path as th fusion cnt if th snsos us thi own oca dcisions to sct chid nods. If so, th oca dcisions fom ths snsos a ss usfu to th fusion cnt. To mak su that a systm mnts foow th sam path of thi BDTs, a mchanism of coodination is ncssay. Such a mchanism is impmntd via a simp two-way communication potoco. Suppos th snsos and th fusion cnt aiv at a nod t, th snsos tansmit thi oca dcisions to th fusion cnt. Basd upon ths oca dcisions, th fusion cnt maks a goba dcision and snds it back to th snsos. Thn both

4 th fusion cnt and th snsos us this goba dcision to choos th sam chid nod. Athough a snso BDTs a th sam in th objct spac, thy a gnay diffnt in th fatu spac and snso dcision u spac. Futhmo, ach snso uss th past goba dcision to dtmin which fatus and what snso dcision u to us at a nod. Now t us invstigat how th stuctu of a BDT affcts th cognition at and th avag numb of stps. Th abiity of such a dcision fusion systm in achiving high cognition ats is mo on th actions at th high v nods of its BDTs. To s this, w not that jction of an objct at a nod pohibits any futh cassification and utimat cognition of this objct. Thfo, an o that occus at a high v nod is mo costy than an o that occus at a dscndnt nod. Fom this viwpoint, at ach nod of th BDT on nds to distinguish objcts that appa most dissimia to th snsos. Fo this pupos, on woud constuct a BDT that tsts and distinguishs th pai of most diffnt objcts at ach nod. Howv, th tota numb of intna nods of such a BDT gows astonomicay with th tota numb of objcts bcaus ony two objcts a distinguishd at ach intna nod. Thus, th avag numb of stps may bcom pohibitivy ag. To aviat this budn, on may choos to tst mo than two objcts at ach nod, but this consqunty dcass th dissimiaity btwn th objcts bing tstd and hnc th cognition ats. In th xtm cas, on may choos to tst a th maining objcts at a sing nod. This wi us th ast avag numb of stps but invitaby incas th pobabiity of o. Thfo, w nd to mak a compomis btwn th dissimiaity of objcts bing tstd and th numb of objcts to tst. In th foowing, w popos a BDT constuction mthod that maks such a compomis. Using this mthod, th mo distinct th objcts a, th ai th stag at which thy a tstd. Thus, th quaity of cassification monotonicay dgads aong a path. This povids a natua way of ncoding th goba dcisions into a squnc of binay bits with dcasing significanc. Such a fomat bcoms usfu whn it is dsid to dtmin ony th goup to which an objct bongs but not its xact idntity. Constuction of BDTs In ou mthod, w stat with th oot nod, and thn pat th foowing pocdu fo a nw nods as ong as thy contain mo than on objct. At ach nod w choos th two substs of objcts to distinguish, cat its chid nods and associat with thm th appopiat sts of objcts fo futh cassification. Suppos w a daing with nod t wh a st Λ(t) of objcts main to b futh cassifid. Ou goa is to sct substs Λ and Λ of Λ(t) accoding to a cition that baancs th goa of objct dissimiaity and th numb of objcts to tst. Lt U dnot th cadinaity of which is th numb of objcts to distinguish. U, Lt D(Λ,Λ ) b th dissimiaity btwn substs Λ and Λ which is dfind as D, Λ = min d O,O ( ) ( ) O,O : O,O Λ wh d(, O ) O is an infomation distanc masu btwn objcts O and O. An ovviw of such distanc masus can b found in [8]. Lt D ( n ) = max (, Λ ) D UΛ = n wh D(n) is th bst possib dissimiaity that can b achivd whn n objcts a tstd. It is not difficut to s that D(n) is a monoton dcasing function of n. Duing th constuction of a BDT, it is dsiab to maximiz both D(n) and n. But thy a conficting objctivs and w nd to baanc ths two objctivs. H w maximiz n subjct to α n n D( n) + D( ) () wh α is a pscibd constant. This cition is basd upon th foowing obsvation. Suppos a objcts occu with th sam pobabiity π. In a cntaizd cognition schm, givn {Λ, Λ and U Λ n, th pobabiity of o P (n) can b = boundd by th foowing [8] P n O,O ( ) ( ) P O Λ O Λ wh P (, O ) O is th pobabiity of o whn ony O and O a tstd. Th ight hand sid can b boundd by a cass of infomation distanc masus btwn O and O [8] and w hav ( ) d ( O,O ) D( Λ, Λ ) P n c O Λ O Λ wh c is a constant. c O Λ O Λ Using th bst possib dissimiaity D(n) w hav P ( n) ( n) ( n) c D c n D O Λ O Λ Using th abov bound as an appoximation fo P (n), w hav

5 n ( ) = n D( n) + ( ) P ( n) P ( ) n D Th ft-hand sid fcts th pfomanc oss du to an incas in th numb of objcts to tst. Dnot th maximum toab pfomanc oss by α, w hav th cition (). Somtims, th a mutip pais of {Λ, Λ that cospond to th sam vau of D(n), i.., thy yid th bst possib dissimiaity. In such situations, on may compa two dsigns using th foowing mthod. Fo ach dsign, find th shotst int-objct distanc btwn Λ and Λ, and thn choos th dsign with th sma such distanc. If th is a ti, count th numb of a th objct pais btwn Λ and Λ that ba this distanc, choos th dsign with th sma count. If th is sti a ti, pat th pvious pocdu fo th nxt shotst distanc. If at th nd, th two dsigns a sti tid, andomy choos any on of thm. This mthod is basd on th fact that shot distancs contibut mo to th o than ong distancs. Finay w psnt ou agoithm fo th constuction of BDTs:. St α. Lt N dnot th st of nw nods. Put oot nod t into N. St t as th cunt nod t.. If th cunt nod t contains xacty on objct, mov t fom N, and go to stp 3; othwis us α to find th maxima n, D(n) and th cosponding {Λ, Λ. Cat ft chid nod t, t Λ( t )= Λ( t)- Λ. Cat ight chid nod t, t Λ( t )= Λ( t)-λ. Put nods t, t into N, mov t fom N. 3. If N is mpty, stop; othwis find a mmb of N, st it as th cunt nod t, and thn go to stp. Dsign of th fusion u and th snso us At ach nod t, th snsos mak oca dcisions gading {Λ, Λ. Basd on ths oca dcisions, th fusion cnt maks a goba dcision. Rca that a nod affcts th systm pfomanc mo than any of its dscndnt nods. Thfo at ach nod, w want to dsign th cosponding fusion u and snso dcision u in such a way that th pobabiity of miscassification at that nod is minimizd gadss of what happns at its dscndnt nods. This is basicay a gdy agoithm. Such us can b considd as th soution to a binay hypothsiststing pobm in which objcts in Λ a tstd against objcts in Λ. Sinc BDTs a usd at a th systm mnts, and th optima fusion u and th optima snso us fo th cunt nod dpnd upon past snso dcisions and past goba dcisions. This adds to th compxity of th pobm. It has bn shown [] that und a mid condition this pobm ducs to a convntiona binay dcision fusion pobm, and th optima fusion u and snso us can b dsignd basd on suts avaiab in []. IV. An Examp In this sction, w psnt an xamp to iustat th suts dvopd in th pvious sction. W us th communication potoco dvopd in th pvious sction to coodinat th snsos and th fusion cnt. W wi discuss constuction of th BDT, dsign of th snso us and th fusion u, and pfomanc vauation. W aso compa this dsign with th cntaizd schm, an ad-hoc M-ay dcision fusion schm and th optimum sing snso schm. Lt us consid a dcision-fusion systm consisting of th indpndnt idntica snsos and a fusion cnt. By idntica snsos w man that th snso obsvations hav th sam chaactistics and a th snsos us th sam BDT. This systm is usd to idntify fou quay iky objcts O, O, O 3 and O 4. Each snso obsvation is assumd to b a scaa. Th objcts a psntd by fou vny spacd points on th a in in th snso obsvation spac as shown in Figu 3. Th distanc btwn adjacnt points is assumd to b a. A snso obsvation is couptd by additiv whit Gaussian nois of zo man and unit vaianc. O 4 O 3 O O -.5a -.5a.5a.5a Figu 3: Objct constation Fo th pupos of compaing ou dsign to an ad-hoc M-ay dcision fusion schm that uss bits fo ach snso, w nd a two v BDT that uss on bit at ach v. Sinc th avag numb of stps is qua to in this pobm, w dsign a BDT and th cosponding snso us and th fusion u that maximizs th avag cognition at. T constuction W us th Kuback divgnc to comput th dissimiaity btwn substs of objcts. Sinc th th snsos a idntica and conditionay indpndnt, th dissimiaity is additiv and w hav d O,O = 3K O, O ( i j ) ( i j ) wh K(, O ) O is th Kuback divgnc btwn i j O i and O j using a sing snso obsvation. Futhmo, sinc th noiss a additiv whit Gaussian, th Kuback divgnc is a quadatic function of th Eucidan distanc btwn objcts x

6 K ( O,O ) i j = O O i Th int-objct distancs fo ou pobm a shown in Tab. Tab : Int-objct distancs d (.,.) O O O 3 O 4 j u = u k =d X k nod {Ο 3, Ο 4 vs. {O, O nod nod 3 u = u k =d O 3a a 7a O 3a 3a a O3 a 3a 3a {O 4 vs. {O 3 O 4 O 3 {O vs. {O O O O4 7a a 3a Sinc th dsid BDT has two vs, w nd to vny divid a objcts into two substs at th oot nod. H n=4, and it is not difficut to find out that D(4)=3a. Th a th candidat dsigns Dsign : Λ ={O, O, Λ ={O 3, O 4. Dsign : Λ ={O, O 4, Λ ={O, O 3. Dsign 3: Λ ={O, O 3, Λ ={O, O 4. Using th sction mthod that w dvopd in th pvious sction in cas of tis, w find that th Dsign is th bst. Hnc at th oot nod, w tst Λ ={O 3, O 4 against Λ ={O, O. Th dsign of th two scond-v nods is tivia, hnc is omittd. Th fusion cnt BDT is shown in Figu 4. U,U,U 3 {Ο 3, Ο 4 vs. {O, O nod nod nod 3 O 4 u = {O 4 vs. {O 3 O 3 u = O Figu 4: Fusion cnt BDT {O vs. {O O u = u = Th BDT usd by th kth snso is shown in Figu 5. In Figu 5, d is th binay dcision mad by th kth snso at stag. W not that th kth snso uss th goba dcision to navigat its BDT. u k =d u k =d u k =d u k =d Figu 5: Th kth Snso BDT Snso dcision u and fusion u Th systm taks two stags to idntify an unknown objct. At stag, it tsts {O, O against {O 3, O 4 and thn chooss on of thm say {O, O. At stag, it tsts O against O. Thus w nd to dsign snso us and th fusion u fo ach stag. At ach stag, w minimiz th tota pobabiity of miscassification. Stag At this stag, w hav a binay hypothsis-tsting pobm of {O,O vs. {O 3,O 4. Th snso dcisions u, u and u 3 a sing binay bits. Basd on ths snso dcisions, th fusion cnt maks a binay goba dcision. It is w known [9] that givn fixd snso us, and givn that th snsos a conditionay indpndnt givn th unknown objct, th optima fusion u is th MAP fusion u which can b xpssd as if p ( u ) p ( u ) i k j k i= k= j= 3 k= u = if p ( u ) < p ( u ) i k j k i= k= j= 3 k= wh p i (u k ) is th conditiona pobabiity of dcision u k of th kth snso whn th unknown objct is O i. It is aso w known [] that th ncssay condition fo a snso u to b optima und th conditiona indpndnc assumption is that it is a ikihood atio tst. Sinc ach snso obsvation X k is a Gaussian andom vaiab, a ikihood atio tst suts in a thshod tst and a binay patitioning of th a in x k. Rca that idntica snso us a usd, and on can asiy s that th dcision bounday is at th oigin x k =. With such snso dcision us, th optima fusion u futh simpifis to a majoity u if u + u + u 3 u = if u + u + u 3

7 Stag At this stag, th systm ith tsts O vs. O, o O 3 vs. O 4. Bcaus of th symmty of th objct constation, th snso dcision us and th fusion u at stag, th sut on tsting O vs. O is ssntiay th sam as that fo tsting O 3 vs. O 4. So without oss of gnaity, w ony consid tsting O vs. O. Now th snso dcisions u, u and u 3 a two-bits vctos with th fist bit psnting th snso dcision at stag and th scond bit psnting th snso dcision at stag. Basd on ths vctos, th fusion cnt maks a goba dcision. Simia to th sut at stag, th optima fusion u is 3 3 if p ( u ) p ( u ) k k k= k= u = 3 3 if p ( u ) < p ( u ) k k k= k= wh p i (u k ) is th conditiona pobabiity that th kth snso dcision is u k whn th unknown objct is O i. Again by th sam agumnt as usd fo stag [], th optima snso us a ikihood atio tsts. Such a tst cosponds to a binay patitioning by mans of a thshod ith in gion x k > if th fist bit of u k is, o in gion x k < if th fist bit of u k is. Ths dcision boundais a functions of th int-objct distanc a. W t a ang fom -5dB to +5dB with stp siz qua to.5db. Fo ach vau of a, w comput th optima snso dcision us, and thn xpss th MAP fusion u as a Booan function of th binay snso dcision vctos. This sut is givn in Tab. H u is th goba dcision that th fusion cnt maks at nod 3. Whn u =, O is dcad, and whn u =, O is dcad. u, u and u 3 a th incoming snso dcision vctos. Sinc th fusion cnt chooss {O, O at th oot nod using a majoity fusion u, th a at ast two s out of th fist bit of u, u and u 3. Diffnt pmutations of ths vctos sha th sam nty in this tab. On can s that th fusion u changs with th intobjct distanc a. Such phnomna a makd by adjacnt and fo u in Tab. Fo sma vaus of a (<-5dB), th fusion cnt pfs O () to O () unss th snsos hav stong suppot fo O. This is bcaus O is sandwichd btwn {O 3, O 4 and O, and th is itt oom fo O. Fo ag vaus of a (>4dB), th fusion cnt chooss O () ov O () unss th snsos stongy suppot O, In this situation, bcaus th signa is sufficinty stong, most of th tim th snso dcision vctos bong to th upp fou ntis in Tab. In such cass, th fusion u is ssntiay a majoity u. Fo intmdiat vaus of a (fom -5dB to 4dB), th fusion u xhibits a chang fom th wak signa fom to th stong signa fom. Tab : Stag fusion u: O (u =) vs. O (u =) a (db) (-5, -5) (-4.5, -3) (-.5, 3) 3.5 (4, 5) u u u 3 u Pfomanc vauation Th systm pfomanc is givn by th pobabiity of miscassification P mc. In Figu 6, th P mc of th BDT basd dcision fusion systm is pottd against th int-objct distanc a as a soid cuv. In th sam figu, w aso pot th pfomanc of th optima cntaizd cassifi and th pfomanc of th optima sing snso cassifi. Th optima cntaizd cassifi uss a th aw snso obsvations to mak a on-stag cassification of th unknown objct. It has th smast possib P mc that svs as a ow bound to th P mc of any oth schm. This P mc is pottd as a dottd cuv. Th optima sing snso cassifi uss on snso obsvation to cassify th unknown objct. Th cosponding P mc is pottd as squas in Figu 6. Th BDT basd dcision fusion systm outpfoms th optima sing snso cassifi. Aso it is sighty infio to th optima cntaizd cassifi. This is futh shown in Figu 7 wh th incas in P mc nomaizd by that of th cntaizd cassifi is pottd. Fo instanc, whn a=-db, fo th optima sing snso cassifi, P mc =.6558; fo th optima cntaizd cassifi, P mc =.588; fo th BDT basd systm, P mc = In Figu 7, th BDT basd dcision fusion systm is compad with th ad-hoc dcision fusion systm in which th th snsos a dsignd as idntica optima sing snso cassifis and th MAP fusion u is usd to fus thi dcisions. Fo ach systm, th incas in P mc with spct to th optima cntaizd cassifi is pottd. It is shown that th BDT basd dcision fusion systm is btt than th ad-hoc dcision fusion systm. Fo instanc, whn a=-db, fo th ad-hoc systm, P mc =.647. Th incas of P mc fo th BDT basd systm is -7dB, fo th ad-hoc systm is -5.5dB.

8 mthodoogy was iustatd by mans of an xamp. Many aspcts of this cass of pobms main unsovd and povid a fuitfu aa fo futu sach. Acknowdgmnt Rsach sponsod by Ai Foc Offic of Scintific Rsach, Ai Foc Systms Command, USAF, und Gant No. F Rfncs Figu 6: Pfomanc of th BDT basd dcision fusion systm Figu 7: Compaison of th BDT basd systm with an ad-hoc systm V. Summay W considd a nw cass of dcision fusion pobms that mpoys binay dcision ts at th snsos and/o at th fusion cnt. Such pobms w catgoizd into fou typs accoding to how th binay dcision ts a dsignd and usd by th fusion systm. Vaious aspcts of th dsign of such systms w discussd. W poposd a systmatic dsign mthodoogy fo on such systm. This [] Pamod K. Vashny, Distibutd Dtction and Data Fusion, Sping-Vag Nw Yok, Inc [] R. Viswanathan and P. K. Vashny, "Distibutd Dtction with Mutip Snsos-Pat I: Fundamntas," Poc. IEEE, vo. 85, No., pp , Jan [3] R. S. Bum, S.A. Kassam and H. V. Poo, "Distibutd Dtction with Mutip Snsos: Pat II Advancd Topics," Poc. IEEE, vo. 85, no., pp Jan [4] S. Rasou and D. Landgb, "A Suvy of Dcision T Cassifi Mthodoogy," IEEE Tans. AES, May 99, pp [5] K. Dmibas, Distibutd Snso Data Fusion with Binay Dcision Ts, IEEE Tans. AES, Vo. 33, No., pp. 3-34, Aug [6] B. Dasaathy, "Opationay Efficint Achitctus fo Fusion of Binay-Dcision Snsos in Mutidcision Envionmnt," Opt. Eng., 36(3), pp , Mach 997. [7] X. Zhu, M. Kam and C. Ros,"M-ay Hypothsis Tsting with Binay Loca Dcisions," Poc. CISS, Pincton, Mach 998. [8] M. Bassvi, Distanc Masus fo Signa Pocssing and Pattn Rcognition, Signa Pocssing, 8 (989) pp [9] Z. Chai and P.K. Vashny, "Optima Data Fusion in Mutip Snso Dtction Systms," IEEE Tans. AES, Vo. AES-, No., pp. 98-, Jan., 986. [] R.R. Tnny and N.R. Sand, "Dtction with Distibutd Snsos," IEEE Tans. AES, Vo. AES-7, pp. 98-, Juy 98. [] Q. Zhang, Ph.D. disstation in pogss.