Mul i- a ge de ec ion and es ima ion wi h he use o massi e
independen , iden ical senso s
Tiancheng Li*a, Juan M. Co chadoa, Ja ie Bajob, Genshe Chenc
aFacul y o Science, Uni e si y o Salamanca, 37008 Salamanca, Spain; bSchool o Compu e
Science, Uni e sidad Poli écnica de Mad id, 28660 Boadilla del Mon e, Mad id, Spain; cIn elligen
Fusion Technology Inc., 20271 Golden od Ln S e., 2066 Ge man own, MD USA 20876-4104
* .c.li@{mail.nwpu.edu.cn, usal.es}; si es.google.com/si e/ ianchengli85/
ABSTRACT
This pape in es iga es he p oblem o using a la ge numbe o independen , iden ical senso s join ly o mul i-objec
de ec ion and es ima ion (MODE), namely massi e senso MODE. This is signi ican ly di e en o he gene al a ge
acking using ew senso s. The massi e senso da a allows e y accu a e es ima ion in heo y (bu may ins ead go
con e sely in ac ) bu will also cause a hea y compu a ional bu den o he adi ional il e -based acke . Ins ead, we
p opose a clus e ing me hod o use massi e senso da a in he same s a e space, which is shown o be able o il e
clu e and o es ima e s a es o he a ge s wi hou he use o any adi ional il e . This non-Bayesian solu ion as e e ed
o massi e senso obse a ion-only (O2) in e ence needs nei he o assume he a ge /clu e model no o know he
sys em noises. The e o e i can handle challenging scena ios wi h ew p io in o ma ion and do so e y as
compu a ionally. Simula ions wi h he use o massi e homogeneous (independen iden ical dis ibu ed) senso s ha e
demons a ed he alidi y and supe io i y o he p oposed app oach.
Keywo ds: Ta ge de ec ion, mul iple a ge acking, senso usion, non-Bayesian me hod
1. INTRODUCTION
Mul i-objec de ec ion and es ima ion (MODE), also e e ed o as mul i- a ge /objec acking, in ol es he join
es ima ion o he numbe o mul iple objec s and hei s a es in he p esence o clu e . The ealis ic MODE scena io can
be desc ibed as ollows: an unknown and ime- a ying numbe o objec s e ol e in e ms o spon aneous appea ance,
disappea ance (including me ging and spli ing) and andom mo emen (o possible high maneu e ) bo h in he s a e
space and in he ime, a ec ed by an unknown and ime- a ying le el o clu e and noise. Bo h he objec s and he
clu e can be de ec ed o missed by he senso s wi h an unknown p obabili y.
The gene al MODE solu ion, whe he o a single objec o mul iple objec s, is o es ablish a hidden Ma ko model
(HMM) whe e he sys em being modeled is assumed o be a Ma ko p ocess wi h hidden s a e. A Bayes il e o
app oxima ely, such as he Kalman il e , Gaussian sum il e , pa icle il e and hei ex ensions, can be employed o
ecu si e es ima ion o e ime. The Bayes il e ing o he p edic ion-co ec ion o ma has been well demons a ed in he
ield o decades. Howe e , a a ie y o models (and pa ame e s) a e in ol ed wi h he a ge s (including
appea ance/disappea ance/ de o ma ion and a ge e ol ing dynamics), senso s (e.g. miss-de ec ion), clu e and noises
(e.g. s a e p ocess noise, clu e densi y, obse a ion noise) e c. in he il e ing amewo k. In eal li e, hese models and
pa ame e s a e o en ime a ying and unknown, which poses c i ical challenges o any il e . The pe o mance o a
il e depends g ea ly on he coinciding deg ee be ween he eal sys em and he models assumed, see e.g. [1-5].
Howe e , i is no easy o iden i y hese models and pa ame e s in p ac ice bu ‘app oxima e’ assump ion has o be
conduc ed.
I has been well acknowledged ha all il e s su e om modelling e o s [6-8]. The impo ance o he model o he
Bayes il e canno be o e s a ed. Consequen ly, model assessmen [9], pe o mance assessmen [10], ou lie ea men
[11], adap i e and obus echnologies [12-15] o ime- a ying models ha e been de eloped. In he p esence o a high
maneu e ( he mo ion model o a ge s is unknown and highly ime a ying), he il e can easily ail e en in he case o
a single a ge . To maximally educe he misma ching be ween he model used and he eal one, conside able e o s
ha e been de o ed in o maneu e ing a ge acking o which he model and he s a es o a ge s need o be es ima ed
join ly. An e icien solu ion in his aspec is using mul iple models o ep esen he mo ion o a ge s and adap i ely
in e ac be ween hem, such as in e ac ing mul iple model (IMM) [16] and so on [4, 5]. The pa ame e can be ea ed as a
Senso s and Sys ems o Space Applica ions VIII, edi ed by Khanh D. Pham,
Genshe Chen, P oc. o SPIE Vol. 9469, 94690G · © 2015 SPIE
CCC code: 0277-786X/15/$18 · doi: 10.1117/12.2177973
P oc. o SPIE Vol. 9469 94690G-1
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms
componen o he s a e [13] o sepa a ely es ima ed based on he unde lying obse a ion [14]. Ne e heless, i is s ill
e y challenging o deal wi h he gene al MODE scene o e y high maneu e ing whe e bo h he sys em noises and he
models a e possibly highly ime a ying and comple ely unknown.
E en when he models a e es ablished co ec ly and he e is no sys em dis u bance, he co espondence be ween he
obse a ions and he a ge s in he clu e en i onmen is ambiguous and p e en s he di ec use o a il e . The way in
which his co espondence ( e e ed o as da a associa ion) is iden i ied dis inguishes wo main g oups o exis ing
solu ions. Fi s , he adi ional solu ions decompose he MODE p oblem in o mul iple sub-p oblems o single objec
de ec ion and es ima ion (SODE) based on da a associa ion [1, 8]. In his amewo k, he da a associa ion as well as he
de ec ion o he a ge s (de e mining he numbe o eal a ge s o e ime; an issue ha does no occu in SODE) is he
key o de e mining he il e ing esul . Secondly, s a e-o - he-a solu ions include inco po a ing ini e se s a is ics-based
poin p ocess modeling [17] o he (app oxima e) Bayesian il e ing amewo k. Signi ican ly di e en ly, ou app oach is
senso -o ien ed and is ee o Bayes il e ing.
Fu he mo e, wi h he apid de elopmen o senso s, massi e senso s a e becoming a ailable in p ac ice ha can p o ide
us a g ea e b ead h o obse a ion in o ma ion ega dless o he indi idual senso ailu e. While his will inc ease he
capabili y o he sys em o be e obse abili y, i can also cause hea y compu a ional bu den o he il e s. The opic o
mul i-senso da a usion has been in ensi ely in es iga ed [18]. Acco ding o ou knowledge, mos exis ing MODE
solu ions only use/deal wi h a ew senso s (e.g. lesse han 10 senso s), excep in he case o acking based on senso
ne wo k [19-20] whe e di e en combina ions o senso s a e used a di e en a eas (bu ew ha e o e lapping iew
ield). These implemen a ions ha may pe o m well in well-de ined en i onmen s a e s ill a om eaching he
ad anced MODE ealiza ion. This pape is pa icula ly conce ned wi h he challenging albei a o able use o a la ge
numbe o independen iden ical senso s in he unknown and ime- a ying en i onmen .
The p esen app oach does no need o make any assump ion abou he a ge ( ega ding o appea ance, disappea ance
and dynamics), clu e densi y and sys em noises, as i is ee o he use o any adi ional il e . The e o e, i is
insensi i e o he a ge /clu e model and is ex emely as compu a ionally. This howe e is based on he a o able case
o massi e senso s. The p esen solu ion can also be e y accu a e i he senso s a e o high quali y o i he numbe o
senso s is la ge. Al hough mul iple senso s ha e been in es iga ed in ensi ely wi hin he a ge - acking con en wi h he
use o il e s [18-20], i is o he i s ime exploi ed, join ly in a la ge numbe , o MODE wi hou he use o any il e .
Mo e comp ehensi e in o ma ion is a ailable in ou p ep in [21].
The pape is o ganized as ollows. Sec ions 2 and 3 p esen he b ie idea o he obse a ion-only (O2 o O2) in e ence
and massi e senso O2 in e ence espec i ely. Simula ion compa ison wi h he s a e o he a mul i- a ge acking
solu ion is gi en in Sec ion 4 be o e we conclude in Sec ion 5.
2. O2 INFERENCE: CONVERTINTG OBSERVATIONS INTO THE STATE SPACE
Loosely speaking, he g ea e he assump ion, he mo e un eliable he acke . In con as , a acke ha makes ewe
model assump ions will be be e able o achie e he desi ed esul s in p ac ice. The e o e, we de elop a senso -o ien ed
solu ion ha in e s he es ima e di ec ly om he obse a ions ecei ed by massi e independen homogeneous senso s
wi h no assump ion on he a ge /clu e model and sys em noises. In his pape , we illus a e he O2 in e ence wi h
ega d o pa icula senso s commonly used o a ge acking.
O2 in e ence
As senso s pe o m pe iodic scans, he obse a ion unc ion ℎ(∙) is gene ally o mula ed in disc e e- ime as
=ℎ(,) (1)
whe e indica es he disc e e ime-ins an (posi i e in ege ),  deno es he s a e, deno es he obse a ion namely
senso da a,  deno es he obse a ion noise.
I is necessa y o poin ou ha a mo e gene al si ua ion would include an unknown obse a ion unc ion. This (unknown
senso y case) howe e is a e in he a ge acking con en and is omi ed he e. The obse a ion unc ion is a guably an
indispensable equi emen o he u iliza ion o he obse a ions in any es ima o and has o be iden i ied be o e a ge -
s a e es ima ion. We do no include his case o a oiding dis ac ion om he key con ibu ion o he pape .
A s aigh o wa d way o es ima e he s a e is o in e di ec ly om i s obse a ion, ega dless o he unobse ed s a e
P oc. o SPIE Vol. 9469 94690G-2
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms
model and clu e , namely he O2/O2 in e ence. I can be concep ually w i en as ollows (as long as i is in e ible):
=ℎ(,) (2)
whe eℎ is he in e se unc ion o ℎ in eal a iable space.
This simply maps/con e s he obse a ions in o he s a e space, wi h a de e minis ic accu acy ha is consis en o he
quali y o he senso s. Howe e , he in e sing will o en in oduce biases (i.e. he expec a ion o he es ima e is no equal
o he ue s a e) i ℎ(∙)is nonlinea ; he highe he nonlinea i y, he la ge he bias/e o . Simply, a nonlinea con e sion
o a Gaussian dis ibu ion is no mo e Gaussian and he e o e he si ua ion can be e y complica ed. This has been
ecognized when con e ing pola /sphe ical measu emen s o Ca esian coo dina es o he use o il e s, see e.g. [22,
23]. To a deg ee, he con e ing bias/e o can be emo ed explici ly o simple in e sing unc ion and noises (such as
Gaussian noises). Howe e , when he noise is unknown, a na u al solu ion is se ing i o be ze o. Then, Eq. (2)
educes o =ℎ(,) (3)
I he obse a ion noise is known, we p opose o use a sampling me hod o emo e he in e sing bias/e o as ollows.
The idea is sampling a g oup o ( andom o de e minis ic) samples om he noise dis ibu ion ()~(),=1,2,…
and use hem as noises sepa a ely in he in e sing calcula ion o (2) as
()=ℎ,(),=1,2,… (4)
Then, we ha e he unbiased (debiased) es ima e as he mean o hese sample-es ima es as
=∑()
 (5)
Ob iously, he sampling is unbiased and will a oid he bias caused by he nonlinea in e sing, ega dless he ype o
noises and he obse a ion unc ion. This is much simple and mo e e icien han he analy ical app oxima e me hods
gi en in [22, 23] and he e e ences he ein.
I is wo h no ing ha he obse a ion unc ion may be i e e sible, p e en ing he di ec in e sing calcula ion; u he
discussion is gi en in [21]. Ins ead o a comp ehensi e discussion on his undamen al ma hema ical p oblem, his pape
ocuses on he obse a ion unc ions used in he a ge acking con en only. Because o his, he in e sing issue is
limi ed o a ew ypical models o which obse a ion in e sing is ela i ely simple.
The O2 in e ence using ange-bea ing senso
Range-bea ing senso (e.g. ac i e ada ): he obse a ion is a noisy ange and bea ing ec o , gi en by
=󰇣
󰇤=,−,+,−,
a c an,,
,,+ (6)
whe e [,,,] and [,,,] a e he − posi ion o he a ge and he senso in he Ca esian coo dina e sys em
espec i ely, and  is he obse a ion noise.
To implemen he O2 in e ence, one es ima e can be in e ed by in e sing Eq. (2) a e aking o , lea ing
󰇣,
,󰇤=+/−



an()






+,
, (7)
As shown, in e sing he a c an unc ion in ol es a sign p oblem. O en, he s a e is bounded in a posi i e/nega i e s a e
space (since he coo dina e sys em is buil by he use wi h espec o he senso s used, hus allowing he use a oid his
p oblem easily) as shown in ou simula ion, so he sign is known. O he wise a leas wo ac i e senso s dis ibu ed a
di e en posi ions will be needed, and he sign o he es ima e can be de e mined by he iangula ela ionship among
he a ge and wo senso s. One pulsed ada and one o wa d-looking in a ed imaging senso can wo k oge he o
measu e he ange and bea ing in o ma ion sepa a ely, equi alen o one ac i e ada .
P oc. o SPIE Vol. 9469 94690G-3
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms
We omi he e mo e o he ypes o senso s such as bea ing-only senso , came a and Dopple senso , whose obse a ion
can be unde -de e mined (one bea ing-only senso is no enough o de e mine he posi ion o a ge s) o i e e sible.
Fu he discussion o his can be pa ly ound in [21] and will appea in ou u u e wo k. One ac i e ange-bea ing senso
is adequa e o conduc O2 in e ence, while a leas wo passi e bea ing-only senso s loca ed a di e en posi ions a e
equi ed o se e as one adequa e senso y uni o es ima ing. We no e ha , he es ima es ob ained by nonlinea
in e sing as shown in (7) is biased, al hough gene ally he bias is insigni ican when he a ge is a om he senso and
when massi e senso s a e used.
I is necessa y o no e ha he O2 app oach only es ima es he dimensions o he s a e ha ha e been obse ed, while he
unobse ed dimensions will be u he in e ed h ough he obse ed dimensions based on hei physical ela ionship.
Typically, he di e en ia ion o he posi ion is he eloci y, and he di e en ia ion o he eloci y is he accele a ion. I
only he posi ion o an objec is obse ed (e.g. ange and bea ing obse a ions), he O2 in e ence can only di ec ly
p o ide he posi ion es ima e; he same occu s when only he eloci y is obse ed (e.g. Dopple obse a ion). No e ha i
is he same s o y in il e s whe e he unobse ed dimensions o he s a e a e also in e ed om he obse ed dimensions.
3. MASSIVE SENSOR O2 INFERENCE
In addi ion o he use o massi e independen iden ical senso s wi h known obse a ion unc ion, we conside he
ollowing e y gene al assump ions (A.1-3):
(A.1) Each a ge gene a es obse a ions independen ly o o he s and one a ge gene a es no mo e han one
obse a ion a each scan o each senso (ex ended a ge is no in ol ed);
(A.2) The clu e dis ibu ion is independen o he a ge and shall no concen a e locally mo e signi ican ly han he
obse a ions o a ge s;
(A.3) The a ge de ec ion p obabili y gi en by senso s is no oo low.
Clu e il e ing based on unsupe ised clus e ing
As add essed so a , he di ec applica ion o he O2 in e ence on he senso da a o massi e senso s will gene a e a huge
numbe o obse a ions ha include he undis inguished obse a ions o eal a ge s and alse ala ms because o he
clu e . The key ask equi ed o MODE as compa ed wi h SODE o dis inguish he eal obse a ions o each indi idual
a ge om each o he and om he alse ala ms, o which we will de elop an unsupe ised online clus e ing p ocedu e.
Since he obse a ions o he same a ge gi en by di e en senso s will concen a e a he same a ea while alse ala ms
will no , we ha e he ollowing c i e ion o dis inguish he obse a ions o eal a ge s om alse ala ms:
C i e ion 1 The obse a ions a e o high densi y in he a ea con aining a ge s and o low densi y in o he a eas.
The e o e, we can con i m a ge exis ence in he a ea o high densi y o obse a ions. The obse a ions om di e en
senso s lying in he same high-densi y a ea a e mo e likely om he same o close a ge s. Fo illus a ion pu poses, an
example is gi en in Fig.1 and 2. Fig.1 gi es he obse a ions (g een ‘x’) epo ed in six independen iden ical ac i e
ada s and Fig.2 gi es all he obse a ions epo ed om en ac i e ada s in he same s a e space. In ui i ely, he eal
obse a ions can be dis inguished om alse ala ms based on he spa ial dis ibu ion o he da a poin s. To deal wi h his,
we a e p oposing he e a clus e ing me hod based on unsupe ised lea ning o he senso da a as ollows (mo e de ail o
he mul i-sou ce da a clus e ing is sepa a ely gi en in [24]).
Algo i hm 1 Clu e il e ing based on clus e ing
1) Apply he O2 in e ence on all obse a ions as add essed in Sec ion 2, ob aining undis inguished da a-poin s
(including s a e-es ima es o a ge s and alse ala ms) in he same space.
2) Calcula e he dis ances be ween any wo da a-poin s om di e en senso s. Da a-poin s om di e en senso s will
be iden i ied as connec ed i hei dis ance is smalle han a h eshold =× (whe e  es ima es he s anda d
de ia ion o he obse a ion noise in he s a e space, and we use a scaling pa ame e ∈[1,4]).
3) Since da a-poin s in he same clus e can be om a single a ge o mul iple close a ge s, a de ec ion o he
numbe o da a-poin s in each clus e shall be applied o dis inguish isola ed a ge s om close- a ge s. He e, ano he
P oc. o SPIE Vol. 9469 94690G-4
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms
h eshold is needed o gi e he a e age numbe o da a-poin s in a single clus e ha con ains one a ge . I shall be
designed wi h espec oe.g.=0.8×in ou simula ion. The s a ic pa ame e 0.8is scalable o ine adjus men .
-3.1) I a da a-poin has been connec ed wi h mo e han bu smalle han2× o he da a-poin s, he da a-poin and
i s connec ions will be iden i ied om a single a ge , o ming a sub-clus e o ex ac one es ima e.
-3.2) I a da a-poin has been connec ed wi h mo e han × (bu smalle han (+1)× whe e≥2) o he da a-
poin s, ha da a-poin and i s connec ions a e iden i ied om mul iple a ge s. Then, hese connec ed da a-poin s will be
pa i ioned in o +1 g oups based on hei p oximi ies in he s a e space, each o which has app oxima ely  bu no
mo e han da a-poin s ha shall all come om di e en senso s; See Rema k 2.
Rema k 2. The s a e-es ima es o close-dis ibu ed a ge s, in e ed om he same senso shall be clus e ed in o di e en
g oups e en hey a e closely dis ibu ed in he space, namely canno link (CL) cons ain on he es ima es om he same
senso (see [24]). The e o e, he e shall be only one es ima e in each sub-clus e ha is om he same senso . The
ob ained sub-clus e s will ha e app oxima ely equi alen numbe o da a-poin s. In ou cu en applica ion, he e a e ew
a ge s (≤3) mo ing closely, he e o e making he pa i ioning o he clus e ela i ely easy. Close- a ge is also a
e y challenging p oblem o he il e -based mul i- a ge acke [25].
Gi en ha he independen iden ical es ima es om eal a ge s a e dis inguished om alse ala ms in he abo e
clus e ing p ocess, he nex is o use hese es ima es o each indi idual a ge as one single inal es ima e. Based on he
independen iden ical p ope y o hese senso s, he inal posi ion es ima e o each a ge can jus be gi en as he mean o
he co esponding es ima es. By clus e ing, he p oposed massi e senso O2 in e ence o MODE wi hou he use o any
il e is able o handle miss-de ec ion and clu e ( alse ala ms). In addi ion, he O2 in e ence almos su ely ha e a as e
compu a ional speed han any il e , which is highly p e e able in p ac ice [21]. Howe e , we do no emphasize his in
ou simula ion s udy bu he il e and he O2 in e ence use he same amoun o obse a ion in o ma ion.
In he p esen clus e ing me hod, we ocus on he applica ion o homogeneous independen senso s and hei iew ields
co e all a ge s, ega dless o complex issues such as senso co ela ion, he e ogenei y and inconsis ency e c. Fu u e
wo k is desi ed o dealing wi h hese p ac ical, complica ed issues.
Fig. 1 Obse a ions (g een ‘x’) o six ac i e ada s
Fig.2 Obse a ions (g een ‘x’) o en ac i e ada s,
mapped in he same s a e space
4. SIMULATIONS
We compa e he mul i-senso O2 in e ence wi h he s a e-o - he-a SMC-PHD il e (i.e. he sequen ial Mon e
Ca lo/pa icle il e implemen a ion o he PHD il e ). To ex ac he es ima es om he join PHD o mul iple a ge s,
addi ional mul i-es ima e ex ac ion is u he equi ed. In ou simula ions we use he mul i-expec ed a pos e io (MEAP)
es ima o [25], which has been demons a ed o be eliable, accu a e and compu a ionally as . As s a ed, he bigges
1000
2000
30
210
60
240
90
270
120
300
150
330
180 0
Senso 1
1000
2000
30
210
60
240
90
270
120
300
150
330
180 0
Senso 2
1000
2000
30
210
60
240
90
270
120
300
150
330
180 0
Senso 3
1000
2000
30
210
60
240
90
270
120
300
150
330
180 0
Senso 4
1000
2000
30
60
90
120
150
180 0
Senso 5
1000
2000
30
60
90
120
150
180 0
Senso 6
500
1000
1500
2000
30
60
90
120
150
8
0 0
P oc. o SPIE Vol. 9469 94690G-5
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms

challenge o he il e is modeling e o . Howe e , in ou simula ions we will only employ pe ec ly co ec models o
he il e s o allow hem o achie e he bes possible pe o mance. This is he mos a o able si ua ion o il e s
(o he wise i he il e s use models ha a e di e en om he ue model, hei pe o mances will be highly educed).
Tha is o say, he pe o mance o he il e -based solu ions is as good as possibly.
The g ound u h o he ajec o ies o a ge s a e plo ed in Fig.3 whe e he colo dis inguishes di e en bi h models and
each ajec o y s a s a ‘∆’ and ends a ‘□’. As shown, new a ge s appea om ou di e en a eas ollowing Poisson
RFS wi h di e en in ensi y=∑,(∙;,)
 , whe e new a ge bi h wi h ini ial s a es=[−1500,0,250,0,0],
=[−250,0,1000,0,0], =[250,0,750,0,0], =[1000,0,1500,0,0], =diag ([50,50,50,50,6∗/180]) and
p obabili ies ,=0.02, ,=0.02, ,=0.03,,=0.03. The a ge s a e a iable =[,] consis s o he
plana posi ion and eloci y =[,,󰇗,,,,󰇗,] and he u n a e. Each a ge ei he con inues o exis a ime 
wi h su i al p obabili y =0.99 and o mo e o a new s a e wi h a nea ly cons an u n- a e (NCT) s a e ansi ion
model o disappea wi h p obabili y0.01. The NCT s a e ansi ion model can be w i en as
=()+, =+△ (8)
whe e
()=




1∆
0−
∆

0∆0 −∆
00∆

∆ 10∆

∆




,=




∆
0
∆0
00∆

∆




~(∙;0,),~(∙;0,),∆=1s,σ=15/ and σ=/180 ad/s.
The homogeneous senso s a e o he same obse a ion unc ion and a e loca ed a he same planne posi ion [0, 0]; he
ange-bea ing obse a ion egion is he hal disc o adius 2000m. This co esponds o he scena io in which senso s a e
a anged in he same planne posi ion bu a di e en al i udes. This is no manda o y as one can place hese senso s a
di e en posi ions and hey will ha e a di e en iew ield. The a ge de ec ion p obabili y is he same,()=
0.95([,,,];0,6000)/(0;0,6000). The obse a ion is a noisy ange and bea ing ec o gi en as (6) o
[,,,]=[0, 0], whe e ~(∙;0,), wi h =diag([,]), σ=20m, σ=/90 ad/s. Since he acking
scena io is in he a ea o ,>0, he sign o he s a e in y-dimension is always posi i e o he O2 in e ence (7), while
in x-dimension i is he same wi h an(). The e o e, his in e se unc ion does no ha e a sign p oblem.
In his simula ion, di e en numbe s o senso s om 1 o 200 will be used. Clu e is uni o mly dis ibu ed o e he
egion wi h an a e age a e o =10 poin s pe scan. Fig.4 gi es he ange and bea ing obse a ions o e ime
sepa a ely, which also gi es he s a ing and ending ime o a ge s. 1000 pa icles pe expec ed a ge a e used and he
o al numbe o pa icles is ha d-limi ed o be no ewe han 600. The op imal sub-pa e n assignmen (OSPA) me ic
[26] is used o e alua e he mul i-objec es ima ion accu acy. Fo ini e subse s ={,,…,} and =
{,,…,} whe e,∈ℕ={0,1,2,…}, he OSPA me ic o o de  be ween and  is de ined as (i ≤)
()(,)=󰇡min∈∑(),()

 +(−)󰇢 (9)
whe e ()(,)=min(,(,)), he cu o alue >0 and (,) is he Eule dis ance. ()(,)=
()(,)i ≥ and()(,)=0 i ==0. The pa ame e s used o he OSPA a e=100,=2.
The e a e se e al ways o inco po a e di e en senso s o wo k o he SMC-PHD acke , di e ing om one ano he in
e ms o how o use he senso da a [2, 8, 27]. Fi s , we apply a nai e ack- o- ack (T2T) usion ha is o un he SMC-
PHD il e s sepa a ely o di e en senso s, and hen use hei es ima es a he end. The o e es ima ion o he numbe o
a ge s o one il e wi h ega d o he a e age o o he s will be simply elimina ed. Secondly, we apply a single “supe ”
senso based SMC-PHD il e whe e he single senso has a much lowe obse a ion noise ~(∙;0,󰆒), whe e
󰆒=diag([󰆒,󰆒])=/ , i.e. 󰆒=
√m, 󰆒=
√ ad/s. This co esponds o an obse a ion quali y ha is
equi alen o he Kalman usion o  senso s. This is e e ed o as obse a ion usion and acking (OFT).
P oc. o SPIE Vol. 9469 94690G-6
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms
Fi s , we se =10 o each o he h ee me hods, i.e. en senso s o he O2 in e ence and he T2T SMC-PHD il e .
The es ima es o six senso s o he same scena io has been gi en in Fig.1 o ime =40 in one ial o simula ion
(clu e a e=10). Co espondingly, all he es ima es om en senso s and he ue s a e posi ions a e gi en in Fig.5.
The a e age esul s o he es ima es o he numbe o a ge s and he mean OSPA o di e en il e s o e 100 Mon e
Ca lo uns a e gi en in Fig.6. The a e age esul s o e 100 s eps×100 MC uns a e summa ized in Table 1 whe e he
ime e e s o one i e a ion o he ull il e . Al hough he mul i-senso O2 me hod does no employ any knowledge abou
he a ge /clu e , i can s ill es ima e he numbe o a ge s wi h accep able accu acy and has p oduced an e en be e
es ima ion han he mul i-senso T2T SMC-PHD il e . To he bes o ou knowledge, MODE has been achie ed o he
i s ime in he clu e en i onmen wi hou any p io knowledge o he a ge /clu e model.
We i e a e ha such a p ope assump ion o he a ge and clu e model and ele an pa ame e s equi ed by he SMC-
PHD il e s ha is consis en ly he same as he u h is no a ailable in many eal-li e p oblems. Consequen ly, he il e s
will no gi e as good a esul o e en inapplicable esul s; howe e , he O2 in e ence will be he same since i does no
ely on he a ge /clu e model a all. In con as , i he obse a ion noise is exac ly known o he O2 in e ence,
debiasing can be applied o u he imp o e he es ima ion accu acy. Using co ec p io knowledge o he sys em, he
OFT SMC-PHD il e pe o ms be e han he T2T SMC-PHD il e , indica ing ha ou mul i-senso T2T solu ion (a
nai e e sion) implemen ed is no op imal. In he ollowing, we will only compa e he OFT SMC-PHD il e (using
MEAP) wi h he O2 in e ence unde di e en  (i.e. a di e en numbe o senso s used in he mul i-senso O2 app oach
and co espondingly a di e en obse a ion noise used in he OFT SMC-PHD il e ).
Fig. 7 gi es he obse a ions and es ima ion when 200 senso s a e used a ime=40. Fig. 8 gi es he mean OSPA
esul s ob ained by he O2 in e ence and he OFT SMC-PHD il e agains di e en . The esul s show ha wi h an
inc ease o , he O2 will su ely ge mo e eliable and accu a e esul s (up o a ela i ely s able le el), bu his is no
gua an eed wi h he SMC-PHD il e . The SMC-PHD il e ge s he bes accu acy when app oxima ely=10. An
obse a ion noise ha is oo la ge (small) is no good o all me hods while a e y small obse a ion noise co esponds
o a sha p likelihood dis ibu ion and he eby may cause signi ican sample degene acy o impo e ishmen [28] in
pa icle il e ( his is he p oblem o he pa icle il e ; u he esul s and discussion can be ound in [21]); bo h cases
will cause a educ ion in he pe o mance o he il e . This simply exhibi s he ad an age o he O2 me hod, which
gua an ees eliable pe o mance ha is consis en o he quali y o senso s and insensi i e o he p io knowledge. In
con as , adi ional il e based acke s may no bene i om mo e o accu a e senso s.
O e all, he esul has demons a ed ha he massi e senso O2 in e ence is quali ied o independen MODE which
enjoys s a is ically inc easing es ima ion accu acy wi h an inc easing numbe o senso s and can pe o m be e e en
han he pe ec ly modelled il e when he numbe o senso s inc eases. Mo e impo an ly, he massi e senso O2
in e ence can wo k e icien ly in he challenging unknown and ime- a ying scena io wi h e y ew backg ound
knowledge o which he adi ional il e -based acke is inapplicable. These indica e ha i may be no always
p e e able o employ a Bayesian il e o acking bu ins ead, in ce ain cases he non-Bayesian O2 app oach wi h he
use o p ope da a-lea ning echnologies may be mo e p omising.
Table 1. The pe o mance o di e en es ima o s when =10, N=10.
SMC-PHD (OFT) SMC-PHD (T2T) O2 In e ence (10 senso s)
OSPA 31.5482 40.3143 36.9991
Time (second) 1.8905 19.771 1.1224
P oc. o SPIE Vol. 9469 94690G-7
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms
Fig. 3 T ajec o ies o a ge s bo n in ou di e en a eas
Fig. 4 Range-bea ing obse a ions (black ‘o’) and he ue
ajec o ies o a ge s (blue line) when =10
Fig.5 Obse a ions (g een ‘x’) o 10 senso s mapped in
he same s a e space, ue s a es (black ‘o’) and he O2
es ima es ( ed ‘+’) a ime =40 when =10
Fig. 6 Mean es ima ed numbe o a ge s and mean OSPA
gi en by di e en es ima o s o e 100 Mon e Ca lo ials
Fig.7 Obse a ions (g een ‘x’) o 200 senso s mapped in
he same s a e space, ue s a es (black ‘o’) and O2
es ima es ( ed ‘+’) a ime =40 when=10
Fig. 8 Mean OSPA o 100 s eps×100 MC uns o he O2
in e ence and he OFT SMC-PHD il e o di e en .
500
1000
1500
2000
30
60
90
120
150
180
010 20 30 40 50 60 70 80 90 100
0
500
1000
1500
2000
Time
Dis ance
010 20 30 40 50 60 70 80 90 100
0
1
2
3
Time
Bea ing
500
1000
1500
2000
30
60
90
120
150
180
T ue s a e o a ge s
Es ima e o O
2
020 40 60 80 100
0
50
100
Time
Mean OSPA
020 40 60 80 100
0
2
4
6
Time
Mean Numbe o a ge s
T ue numbe o a ge s
MEAP SMCPHD il e (OFT)
MEAP SMCPHD il e (T2T)
O2 in e ence by using en senso s
500
1000
1500
2000
30
60
90
120
150
180
0
T ue s a e o a ge s
Es ima e o O2
050 100 150 200
25
30
35
40
45
50
55
60
65
70
N
A e . mean OSPA
MEAP SMCPHD il e (OFT)
O2 in e ence
P oc. o SPIE Vol. 9469 94690G-8
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms
5. CONCLUSION
This pape conside s a pa icula class o MODE wi h wo special challenges: 1) li le/no p io in o ma ion is gi en
abou he backg ound; 2) massi e (independen iden ical) senso s a e a ailable. I is shown ha he use o massi e
senso s is ac ually a o able o ci cum en he poo backg ound/p io in o ma ion, al hough his may go con e sely in
adi ional il e s. Based on an unsupe ised clus e ing p ocess, he p esen massi e senso O2 in e ence needs nei he o
assume he a ge /clu e model no o know he sys em noises o MODE, while i is able o handle high maneu e
a ge s, a ge spli ing and me ging, unknown and ime a ying sys em noises and clu e densi y, e c., and can do so
compu a ionally as . I enjoys s a is ically inc easing es ima ion accu acy wi h an inc easing numbe o senso s and can
pe o m be e e en han he pe ec ly modelled il e . Simula ions based on massi e independen homogeneous senso s
ha e demons a ed he supe io i y o he p esen massi e-senso O2 in e ence. Mo e impo an ly, he massi e senso O2
in e ence can wo k e icien ly in he challenging unknown and ime- a ying scena io wi h e y ew backg ound
knowledge o which he adi ional il e -based acke is inapplicable. The p esen massi e senso O2 in e ence
iden i ies a benchma k o e alua e he e ec i eness o any o he mul i- a ge acke .
The abili y o deal wi h huge amoun s o da a aised wi h he use o massi e senso s is an eme ging end o ad anced
a ge acking, especially in challenging cases o which ew p io knowledge is a ailable. Ou u u e wo k will conside
complex senso issues such as senso egis e bias, senso da a co ela ion, he e ogenei y, inconsis ency and senso
ne wo k-communica ion e c.
ACKNOWLEDGMENT
Tiancheng Li’s wo k is pa ly suppo ed by he Excellen Doc o a e Founda ion o No hwes e n Poly echnical
Uni e si y and he Pos doc o al Fellowship o he Uni e si y o Salamanca.
REFERENCES
[1] Jazwinski, A. H., [S ochas ic P ocesses and Fil e ing Theo y], Academic P ess Inc. London, (1970).
[2] Blackman, S., and Popoli, R., [Design and Analysis o Mode n T acking Sys ems], A ech House, (1999).
[3] Li, X.R., and Jilko , V. P., “Su ey o Maneu e ing Ta ge T acking. Pa I: Dynamic Models,” IEEE T ans.
Ae osp. Elec on. Sys ., 39(4), 1333-1364, (2003).
[4] Li, X.R., and Jilko , V. P., “Su ey o maneu e ing a ge acking. Pa V: Mul iple-model me hods,” IEEE T ans.
Ae osp. Elec on. Sys ., 41 (4), 1255–1321, (2005).
[5] Yildi im, S., Jiang, L., Singh, S.S., and Dean, T.A., “Calib a ing he Gaussian mul i- a ge acking model,”
S a is ics and Compu ing. pp. 1-14, (2014).
[6] F iedland, B., “T ea men o bias in ecu si e il e ing,” IEEE T ans. Au om. Con . 14, 359–367, (1969).
[7] Li, T., Rod íguez, S., Bajo, J. Co chado, J. M., and Sun S., “On he bias o he SIR Fil e in pa ame e es ima ion o
he dynamics p ocess o s a e space models,” 12 h In e na ional Symposium on Dis ibu ed Compu ing and
A i icial In elligence, Salamanca, Spain, June 3-5, (2015).
[8] Ba -Shalom, Y., and Li, X.R. [Mul i a ge -mul isenso T acking: P inciples and Techniques], YBS Publishing,
S o s, CT (1995).
[9] Dju ić, P. M., and Miguez, J., “Assessmen o nonlinea dynamic models by Kolmogo o –Smi no s a is ics,” IEEE
T ans. Signal P ocessing, 58 (10), 5069–5079, (2010).
[10] Tulsyan, A., Huang, B., Gopaluni, R. B., and Fo bes, J.F., “Pe o mance assessmen , diagnosis, and op imal
selec ion o non-linea s a e il e s,” Jou nal o P ocess Con ol, 24, 460-478, (2014).
[11] Maíz, C.S., Míguez, J., Molanes-López, E. M., and Dju ić, P. M., “A obus i ied pa icle il e ing scheme o
p ocessing ime se ies co up ed by ou lie s,” IEEE T ansac ions on Signal P ocessing, 60(9), 4611-4627, (2012).
[12] Gus a sson, F., [Adap i e Fil e ing and Change De ec ion], Wiley, London, (2000).
[13] Cohn, S. E., “An in oduc ion o es ima ion heo y,” J. Me eo . Soc. Japan, 75, 257–288, (1997).
P oc. o SPIE Vol. 9469 94690G-9
Downloaded F om: h p://p oceedings.spiedigi allib a y.o g/ on 05/28/2015 Te ms o Use: h p://spiedl.o g/ e ms