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Proceedings ArticleDOI

A particle filter to track multiple objects

08 Jul 2001-Current Opinion in Organ Transplantation (IEEE Computer Society)-pp 61-68
TL;DR: An extension of the classical particle filter where the stochastic vector of assignation is estimated by a Gibbs sampler is proposed and the merit of the method is assessed in bearings-only context and one application in image-based tracking is presented.
Abstract: We address the problem of tracking multiple objects encountered in many situations in signal or image processing. We consider stochastic dynamic systems nonlinearly and incompletely observed. The difficulty lies on the fact that the estimation of the states requires the assignation of the observations to the multiple targets. We propose an extension of the classical particle filter where the stochastic vector of assignation is estimated by a Gibbs sampler. The merit of the method is assessed in bearings-only context and we present one application in image-based tracking.

Summary (2 min read)

Introduction

  • Thedifficulty lies on the fact that theestimationof thestatesrequirestheassignationof theobservationsto themultipletargets.
  • The authors proposean extensionof the classicalparticle filter where the stochastic vectorof assignationis estimatedby a Gibbssampler.
  • The meritof themethodis assessedin bearings-onlycontextand wepresentoneapplicationin image-basedtracking.

1. Intr oduction

  • Multiple objecttracking(MTT) dealswith stateestimation of an unknown numberof moving targets.
  • As long asthe associationis consideredin a deterministicway, the hypothesisassociations must be exhaustively enumerated,which leadsto a NPhardproblem(asin JPDAF andMHT algorithms[3] for instance).
  • The above algorithmsdo not copewith nonlinearstateor measurementmodelsandnon Gaussianstateor measurementnoises.Undersuchassumptions(stochasticstateequationandnonlinearstateor measurementequation,nonGaussiannoises),particlefiltersare particularly adapted.
  • The authors proposeherea quitegeneralalgorithmfor multipleobjecttrackingapplicablebothin signalandimage analysis.
  • SectionIV beginswith a validationof their algorithmin thepassive sonar context, i.e. to estimatethetrajectoriesof multiple “small” targetsfrom their noisy bearings.

2. The basicparticle filter

  • For the sake of completeness,the basicparticlefilter is now briefly reviewed.
  • Unfortunatelythis modellingis notappropriatein many problemsin signaland imageprocessing,which rendersthecalculationsof theintegralsin (3) and(4) infeasible(no closed-form).
  • In the mostgeneralsetting[2], the displacementof particles is obtainedby samplingfrom an appropriatedensity > which might dependon the dataaswell.
  • After theserecalls,the authors proposean extensionof this algorithm to multiple-object tracking.

3.1. Notations

  • Let   bethenumberof objectsto track(first assumedto beknown andfixed).
  • The falsealarmsaresupposedto be uniformly distributed in the observationarea.
  • Of course,the authors do not associate any kinematicmodel to falsealarmsandthenno particles representtheir density.
  • This definition implicitly assumesthattheprobabilitiesº £ areindependent of themeasurementsastheir indexationis arbitrary.
  • Eachof theseconcatenatedvectors then gives jointly a representationof all objects.

3.2. The MOPF

  • The predictionis performedby samplingfrom someproposaldensity > .
  • To estimatethem the authors proposeto usea Gibbs samplerwhoseprinciplesare briefly recalled (see [1] or [9] for more details).
  • Now let ¦ be an integer in the secondproduct.

4. Application to bearings-onlyproblems

  • The authors first deal with the classical bearings-onlyproblem using synthetic data.
  • As soonasthedifference betweentwo bearingsissuedfrom two different targetsis lower thanthestandarddeviation of theobservationnoise, thetwo measurescannotbedistinguished,whichmakesthis scenariovery difficult.
  • The plot of the threeestimatedtrajectoriesshows that the dataassociationis overcome.
  • Anobject leaving thesurveillanceor vision areacanbedetectedby a dropof its º component, also known as Thevector º canthenin turnhelptheestimation.

5. Application in image-basedtracking

  • The authors focuson a video sequencewherethreepersonsare moving accordingto unknown dynamics.
  • The modelizationof the trackingproblemconsistsin defininga statevector, its evolution modelandan observationmodel.
  • Suchadesignstageis quiteeasyin signal processingbecausethe modelsand the differentmeasuresarewell definedin theliterature.

5.1. Statespacemodel

  • In orderto keepreasonabledimensionof thestatespace, onehave to comeup with a compactrepresentationof the object silhouetteunlessstrong prior on the shapeof the object of interestis available.
  • An appealingand generic approachconsistsin usingFourierdescriptorsobtainedby inverseFourier transformof the truncateddiscreteFourier transformof theinitial objectcontour.
  • Theoutline of onemoving pedestrian(asobserved througha motion segmentationmap, seesection5.2) associatedto the five first Fouriercoefficientsis presentedin figure6.5.

5.2. Measurements

  • The authors usetwo typesof measurementsrelatedto position andvelocity of moving objectsin thescene.
  • Sucha segmentationprovidesinformation on thelocalizationof theobjectsin motion w.r.t. the camera.
  • The denominatorprevents a bias toward large contours.
  • The authors then estimatethe translationin the ß D-planeof each connectedcomponentbetweenthe frame at current time and the previous one.
  • As the velocity prior is very weak (uniform distribution, in (21)), the authors usethesemotion measurementsto constructan importancefunction that will performbetterthanstandard bootstrap-typechoicebasedonly on thedynamics,to guide particlestowardregionsof high likelihood.

5.3. Importance function

  • Givena partcile # , a translationestimationis considered available when only one of the object centersbelongsto the connectedcomponentunderconsideration.
  • Otherwise it meansthatseveralobjectsareobservedthroughthesame motion region and its global motion resultsin generalfrom anintricatemix of individualmotions.

6. Conclusion

  • Target statevectors andassociationprobabilitiesareestimatedjointly without enumeration,pruningor gating,by meansof particlesets representingthe joint a posteriori law of the target states.
  • The authors have demonstratedthe relevanceof the approachboth in bearings-onlytrackingandimage-basedtracking.

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A Particle Filter to Track Multiple Objects
Carine Hue Jean-Pierre Le Cadre
IRISA
Campus de Beaulieu
35042 Rennes Cedex, France
chue@irisa.fr lecadre@irisa.fr
Patrick P´erez
Microsoft Research
St George House, 1 Guildhall Street
Cambridge, CB2 3NH, UK
pperez@microsoft.com
Abstract
We address the problem of tracking multiple objects en-
countered in many situations in signal or image process-
ing. We consider stochastic dynamic systems nonlinearly
and uncompletely observed. The difficulty lies on the fact
that the estimation of the states requires the assignation of
the observations to the multiple targets. We propose an ex-
tension of the classical particle filter where the stochastic
vector of assignation is estimated by a Gibbs sampler. The
merit of the method is assessed in bearings-only context and
we present one application in image-based tracking.
1. Introduction
Multiple object tracking (MTT) deals with state estima-
tion of an unknown number of moving targets. Available
measurements may both arise from the targets if they are de-
tected, and from clutter. Clutter is generally considered as a
model describing false alarms. Its (spatio-temporal) statis-
tical properties are quite different from target ones, which
makes possible the extraction of target tracks from clut-
ter. To perform multiple object tracking the observer has
at his disposal a huge amount of data, possibly collected on
multiple receivers. In signal processing, elementary mea-
surements are receiver outputs, e.g., bearings, ranges, time-
delays, Dopplers, etc. In image-based tracking they have to
be computed from the images.
But the main difficulty comes from the assignment of a
given measurement to a target model. These assignments
are generally unknown, as are the true target models. This is
a neat departure from classical estimation problems. Thus,
two distinct problems have to be solved jointly: data asso-
ciation and estimation. As long as the association is con-
sidered in a deterministic way, the hypothesis associations
must be exhaustively enumerated, which leads to a NP-
hard problem (as in JPDAF and MHT algorithms [3] for in-
stance). As soon as the association variables are consid-
ered as stochastic variables and moreover statistically in-
dependent like in the Probabilistic MHT (PMHT), the com-
plexity is reduced. However, the above algorithms do not
cope with non linear state or measurement models and non
Gaussian state or measurement noises. Under such assump-
tions (stochastic state equation and non linear state or mea-
surement equation, non Gaussian noises), particle filters are
particularly adapted. They mainly consist in propagating
a weighted set of particles which approximates the proba-
bility density of the state conditionally to the observations.
Particle filtering can be applied under very weak hypothe-
ses, is able to cope with heavy clutter, and is very easy to
implement. Numerous versions have been used in various
contexts: the bootstrap filter for target tracking in [4], the
Condensation algorithm in image analysis [5] are two ex-
amples among others. In image analysis a probabilistic ex-
clusion principle has been developed in [7] to track multi-
ple objects but the algorithm is very dependent on the ob-
servation model and seems costly to extend for more than
two objects. We propose here a quite general algorithm for
multiple object tracking applicable both in signal and image
analysis.
This work is organised as follows. In section II, we briefly
recall the basic particle filter. Section III deals with our ex-
tension of the basic filter to multiple objects. Section IV be-
gins with a validation of our algorithm in the passive sonar
context, i.e. to estimate the trajectories of multiple “small”
targets from their noisy bearings. Then it is used to track
pedestrians in a video-sequence.
2. The basic particle filter
For the sake of completeness, the basic particle filter is
now briefly reviewed. We consider a dynamic system rep-
resented by the stochastic process

whose tem-
poral evolution is given by the state equation:
 
(1)

!
Initialization:
"$#
%'&)(
*
%
+
%
-,/. 0 1
2,3445 607
!
For
8
-,454 69;:
<
Proposal: sample
=
#
from
>
* ? @A
#
@
CBDEGFH
for
1
2,454560I
<
Weighting:
J
K L
Compute un-normalised weights:
=
+
+
@NMOP
QHRST QHRSUNVWYX
S
O[Z
S\
P
QRS]W
^
O_P
Q
R
ST Q
R
SUNVC`
Z
S
W
for
1
-,544460I
Normalise weights:
+
P
a
RS
bc
R5d
V
P
a
R
S
for
1
2,454560I
<
Return
e
fhg
*i;jlk
]m
+
g
=
#

<
Calculate
n
0po
^q^
b
c
R5d
V
O
a
R
S3W*r
.
<
Resampling: if
n
0o
^/^s
0tuqvCo
Q
u/w
X[x
:
"
#
&
j
ky
m
+
y
{z
P
Q|
S
+
2,/. 0
1
-,545 60
, else
#
=
#
for
1
2,3445 607
Figure 1. Basic particle filter with adaptive resampling .
It is observed at discrete times via realizations of the
stochastic process
B
7-
]}
governed by the measure-
ment model:
B
l~
*
C

(2)
The two processes
_D)]
and
_)]
in (1)
and (2) are only supposed to be independent white noises.
Moreover, it is to be noted that no linearity hypothesis on
and
~
is done. We will denote by
B
% 
the sequence of the
random variables
B
%
54546B
and by
F
%
one realization of
this sequence.
Our problem consists in computing at each time
8
the con-
ditional density
of the state

given all the observations
accumulated up to
8
, i.e.,

(
* ? B
%
lF
%
545 6BlF]
and also in estimating any functional of the state
g
H
by
the expectation
f
g
45? B
% 

. The Recursive Bayesian fil-
ter, also named Optimal Filter, resolves exactly this problem
in two steps at each time
8
[4].
Suppose we know

. The prediction step is done ac-
cording to the following equation:
(
*lD4? B
%
@AlF
% 
@ E
4
R
(
h4? AG
]3
(3)
The observation
F]
enables us to correct this prediction us-
ing the Bayes’s rule:
DE
(
BEGF] ? D
(
*lD4? F
% 
@
R
(
_B
lF
? 
(

l? F
%

3
(4)
These equations provide a closed-form recursion if we as-
sume restrictive hypothesis such as Kalman Filter’s ones.
The functions
and
are then supposed to be linear and
the noises

and

to be Gaussian. Unfortunately this
modelling is not appropriate in many problems in signal and
image processing, which renders the calculations of the in-
tegrals in (3) and (4) infeasible (no closed-form).
The original particle filter, named bootstrap filter [4],
proposes to approximate the densities

by a finite
weighted sum of
0
Dirac densities centred on elements of

, named particles. The application of the bootstrap filter
requires that one knows how:
!
to sample from initial prior marginal
(

%
;
!
to sample from
(
_
for all
8
;
!
to compute
(
_BDF] ? D
for all
8
through a
known function
such that
*Fi
(
_BDF@? 
where missing normalization must
not depend on
.
The algorithm then consists in making evolve the particle
set

#
+
]m
` `
k
, where
#
is the particle position
and
+
its weight, and to use it to estimate the density
by
the density

S
-j
k
]m
+
z
Q
R
S
. The weak convergence of
the probability density
S
towards
when
0
with
rate
,/. 0
can be proved. To avoid the degeneracy of the
particle set, i.e. only few particles with high weights and
the others with very small ones, a resampling is done in an
adaptive way when the number of effective particles, esti-
mated by
n
0po
^q^
, is under a given threshold [2]. Besides the
discretization of the filtering integrals, the use of such parti-
cles enables to voice many hypothesis on the position of the
object and to keep in the long term only the particles whose
position is likely given the sequence of observations. In
bootstap filter, the particles are “moved” by sampling from
the dynamics (1), and importance sampling theory shows
that the weighting is only based on likelihood evaluations.
In the most general setting [2], the displacement of parti-
cles is obtained by sampling from an appropriate density

>
which might depend on the data as well. The complete
procedure is summarized in figure 1. The reader will find
more details on the different filters in [4], [5] or [2] and on
adaptive resampling in [6] and [2]. After these recalls, we
propose an extension of this algorithm to multiple-object
tracking.
3. Multiple object particle filters
3.1. Notations
Let
be the number of objects to track (first assumed to
be known and fixed). The state vector we have to estimate
is made by concatenating the state vector of each object. At
time
8
,
*
44546¢¡
follows the state equation
,/
decomposed in
partial equations:
£
l¤£
£

t£
¥§¦,545 
(5)
The noises
¨
£
and
_
£ª©
are only supposed to be white
both temporally and spatially, independent for
¦¬«¦H®
.
The observation vector at time
8
is denoted by
F
*F
5444F{¯
S
. Following the seminal ideas of R. Streit
and T. Luginbuhl [10], we introduce the stochastic vector
°
A²±N,454 
´³
¯
S
such that
°Iµ
¦
if
F
µ
is issued from
the
¦
u
object. In this case,
F
µ
is a realization of the stochas-
tic process:
B
µ
~
£

£

µ
if
°¶µ
¦
(6)
Again, the noises
_
µ
and
_
µ
©
are only sup-
posed to be white noises, independent for
·
«
·
®
.
We assume that the functions
~
£
are such that they
can be associated to functional forms
£
defined by
£
FE
(
_B
µ
GF?
°¶µ
G¦
£

.
We make the assumption that one measurement can
originate from one object or from the clutter and that one
object can produce zero or several measurements at one
time. For that, we dedicate the model
¸
to false alarms.
The false alarms are supposed to be uniformly distributed
in the observation area. Their number is assumed to arise
from a Poisson density of parameter
¹
where
is the
volume of the surveillance area and
¹
the number of false
alarms by volume unity. Of course, we do not associate
any kinematic model to false alarms and then no particles
represent their density. Let
º
¼»
¸
@, ½*¡¾
defined by
º
£
¿
°¶µ
¦
for all
·
À,545ÁÂ
. This definition
implicitly assumes that the probabilities
º
£
are independent
of the measurements as their indexation is arbitrary. These
assumptions imply that
Á
may differ from
and that
the association is exclusive and exhaustive. In particular,
j
¡
£
m
%
º
£
Ã,
. Furthermore, it is assumed that the assign-
ment vector
°
has independent components (see[10]). To
estimate the density
Â
(
ÂÄ*
445 Å¡
5? F
% 

,
we propose to use particles whose dimension is the sum
of the ones of the individual state spaces corresponding to
each object, as in [7]. Each of these concatenated vectors
then gives jointly a representation of all objects. Let us
present the proposed multiple object particle filter (MOPF).
3.2. The MOPF
The initial particle set
%
¬
#
%
4,.0
]m
` `
k
is such
that each component
#
`
£
%
for
¦2,3454 
is sampled from
(

£
%
independently from the others. Assume we have ob-
tained
t$
#
@
+
@
]m
` `
k
with
jk
]m
+
@
Æ,
.
Each particle is a vector of dimension
j
¡
£
m
1
£Ç
where we
denote by
#
`
£
@
the
¦
u
component of
#

and where
1
£
Ç
designates the dimension of object
¦
.
The prediction is performed by sampling from some pro-
posal density
>
. In bootstrap filter case,
>
coincides with
the dynamics (5):
For
1
-,454 C0
=
#
ÈÉÊ
#
`

6Ë
`
.
.
.
t¡
#
`
¡

6Ë
`
¡
Ì4Í
Î
(7)
with
Ë
`
£
being realizations of
_
£
. Examine now the
computation of the likelihood of the observations condi-
tioned by the
1
u
particle. We can write for all
1
,544460
:
(
_BDEÏ*F
5444F{¯
S
4? 
=
#
E;Ð
¯
S
µ
m
(
*F
µ
?
=
#
Ð
¯
S
µ
m
»¨Ñ]Ò
S
ÓÏÔ
j
¡
£
m
£
*F
µ
=
#
`
£
º
£
½
(8)
It must be noted that first equality in (8) is true only un-
der the assumption of conditional independence of the mea-
sures, which we will make. Moreover, the normalization
factors between
£
and
(
_B
µ
GF?
°Iµ
¦
£

must be
the same for all
¦
to write the second equality in (8).
It remains to estimate the association probabilities
º
£
£
m
``
¡
which can be seen as the stochastic coeffi-
cients of the
Õ
component mixture. To estimate them
we propose to use a Gibbs sampler whose principles are
briefly recalled (see [1] or [9] for more details). For
Ö
*
°
6×
, it consists in generating a Markov chain
which converges to the stationary distribution
(
Ö
? B
%
which cannot be sampled directly. Given a partition
Ö
5454
Ö3Ø
of
Ö
, one samples alternatively from the condi-
tional posterior distribution of each component of the par-
tition. Assume the
Ù
first elements of the Markov chain
Ö
/5454
ÖÚ
have been drawn. We sample the
Û
compo-
nents of
ÖÚ
¾
as follows:
Draw
Ö
Ú
¾
from
(
Ö
? B
% 

ÖÜ
Ú
4454
ÖØ
Ú
Draw
ÖÜ
Ú
¾
from
(
ÖÜ
? B
% 

Ö
Ú
¾
ÖÝ
Ú
454 
ÖØ
Ú

!
Initialization:
"$#
%'&)(
*
%
+
%
-,/. 0 1
2,3445 607
!
For
8
-,454 69;:
<
Proposal: sample
=
#
from
>
* ? @A
#
@
CBDEGFH
for
1
2,454560I
<
Weighting:
,
Initialization of the Gibbs sampler:
Þ
º
£
`
%

Ñ]Ò
S
¡
¦-,454 
£
`
%
jk
]m
+
@
=
#
`
£
¦-,454 
ß
For
Ù
¸
454 
Ù
o
x
:
à
°Iµ
`
Ú
¾
& (
°¶µ
`
Ú
¾
l¦E
"
º
£
`
Ú
£
*F
µ

£
`
Ú
if
¦h,3445 
º
%
.]
if
¦h
¸
á
º
¡
`
Ú
¾
&â
6,
Ô
1
£
°
`
Ú
¾

£
m
``
¡
ã
1
£
°
§älå±
·
:
°
µ
¦
³
.
æ
For each
¦
such that
ç·
4544
·
£
.
°IµCè
`
Ú
¾
¦C
é
J
K
Llê
Ú
¾
=
#
`
£
ë
Ú
¾
MO[Z ì
V
S
` `
Zì_í
SîT ï
í
S
mð
R
ñò
V
WqaR
S*UNV
bc
Rqd
V
MO[Zì
V
S
` `
Z ì
í
SóT ï
í
S
mð
R
ñCò
V
W/a
R
SUNV
1
2,3454 C0I
é
£
`
Ú
¾
&
jk
]m
ë
Ú
¾
z
ð
R
ñò
V
D
For each
¦
such that
ô·
.
°¶µ
¦
£
`
Ú
¾
&
j
k
m
+
z
P
Q
Rõ
í
S
ö
n
º
£
Ú÷*ø¨ù
Úø
R5ú
j
Ú6ø
Rqú
Ú
m
Ú÷ø_ù
º
£
`
Ú
¦,3454 
û
BÏ*F
44546FY¯
S
5?
=
#
i;Ð
¯
S
µ
m
»¨Ñ]Ò
S
ÓÏÔ
j
¡
£
m
£
F
µ
=
#
`
£
n
º
£
½
1
2,3454 C0I
ü
+
+
@YM/O_P
QHRS]WT QHRSUNVWYX
S
Oªý
S
T
P
QHRSW
^
O_P
Q
S
T Q
S*UNV
`
Z
S
W
1
2,3445 607
<
Return
e
fhg
*
i
jlk
]m
+
g
=
#

<
Calculate
n
0
o
^q^
b
c
R5d
V
O
a
R
S
W*r
.
<
Resampling: if
n
0
o
^/^
s
0
uqvCo
Q
u/w
X[x
:
"
#
&
j
k
y
m
+
y
z
P
Q|
S
+
-,/. 0
1
,545460
, else
#
=
#
for
1
,3445 60
Figure 2. MOPF: multiple object particle filter with adaptive resampling.
.
.
.
.
.
.
Draw
ÖØ
Ú
¾
from
(
ÖØ
? B
% 
Ö
Ú
¾
4544
Ö
Ø

Ú
¾
In our case, at a given instant
8
, the partitionning of
Ö
is:
J
K
L
Ö
µ
°Iµ
for
·
,3454 6ÁÂ
Ö
¯
S
¾
£
º
£
for
¦2,3454 
Ö
¯
S
¾þ¡p¾
£

for
¦2,3454 
(9)
We now detail the different steps of this Gibbs sampler.
,
The initialization of Gibbs sampler consists in assigning
uniform association probabilities, i.e.,
º
£
`
%

Ñ
Ò
¡
for all
¦tÿ,3454 
, and taking
`
%
jk
]m
+
@
=
#
, i.e., the
centroid of the predicted particle set. Then, suppose that at
instant
8
we have already simulated
Ö
`
]454 
Ö
`
Ú
.
ß
The
Ù
Ô
,
u
iteration is handled as follows.
à
As the
°
µ
µ
m
``
¯
S
are supposed to be independent,
their individual conditional density reads:
(
°
µ
? B
%

5
°
X
X

m
µ
C×
E
(
°
µ
? B
µ
6
C×
 
(10)
°Iµ
are discrete variables and we can write:
¿ã
°¶µ
¦ ? B
µ
lF
µ

6×
M/O ý
ì
S
m
Z
ì
S
T
ì
S
m
£
`
ï
S
`
S
W

O
ì
S
m
£
T ï
S
`
S
W
MOªýþì
S
m
Z ì
S5T ï
S
`
S
W
"
º
£
£
*F
µ

£
if
¦h2,3445 
º
%
.]
if
¦h
¸
(11)
The realizations
µ
`
Ú
¾
of the vector
°
`
Ú
¾
are then sam-
pled according to the weights
(
µ
`
%
`
Ú
¾
º
%
.]
, and
(
µ
`
£
`
Ú
¾
º
£
`
Ú
£
*F
µ

£
`
Ú
for
¦h2,3445 
.

á
Mixture proportion vector
×
¡
`
Ú
¾
is drawn from the con-
ditional density:
(
×
¡
?
°
`
Ú
¾
q6
`
Ú
6B
%
E
(
×
454 C×
¡
?
°
`
Ú
¾
4544
°
¡
`
Ú
¾

`
Ú
CB
% 
H
(
°
`
Ú
¾
5445
°
¡
`
Ú
¾
? ×
454 Cק¡
(
×
45456ק¡
Ï,
Õź
%
â
_×
?
 ±
1
£
°
`
Ú
¾
£
m
``
¡
³
(12)
where we denote by
1
£
°
the number of
µ
equal to
¦
and
â
stands for Dirichlet distribution.

`
Ú
¾
has to be sampled according to the density
(
* ? B
% 
C
°
`
Ú
¾
q6×
`
Ú
¾
i
¡
£
m
(

£
? B
% 

°
`
Ú
¾
qC×
`
Ú
¾

(13)
The values of
°
`
Ú
¾
can imply that one object is associated
with zero or several measurements that is why we decom-
pose the preceding product in two products:
Ð
£

µ
V
` `
µ
í
ì
S
õ
ñò
V
m
£
(
*
£
? B
% 
@
6F
µ
V
4545F
µ
í
6×
`
Ú
¾
Ð
£

µ
ì
S
õ
ñCò
V
m
£
(
*
£
? B
%
/6×
`
Ú
¾
 
(14)
æ
Let
¦
be an integer in the first product. We can write
(
*
£
? B
%

6F
µ
V
454 6F
µ
í
6×
`
Ú
¾
E
MO[Zì
V
S
` `
Zì
í
S§T ï
í
SW
M/O
ï
í
S5T
ý
Ò

S*UNV
W
MO[Zì
V
S
` `
Zì
í
SóT
ý
Ò

SUNV
W
(15)
We are not able to sample directly from the density
M/OªZì
V
S
` `
Zì_í
SóT ï
í
S
W
MO
ï
í
S
T
ý
Ò

S*UNV
W
MO[Z
ì
V
S
` `
Z
ì
í
S
T
ý
Ò

SUNV
W
, for the same reasons as those
exposed in section 2 to justify the use of the particle fil-
ter (intractability of the integrals). We propose to build the
particle set
Ú
¾
Ã
ê
Ú
¾
ë
Ú
¾
m
``
k
whose weights
ë
Ú
¾
measure the likelihood of the observations affected by
°
`
Ú
¾
to object
£
. More precisely, we let:
J
K
L
ê
Ú
¾
=
#
`
£
ë
Ú
¾
MO[Zì
V
S
` `
Zìí
S
T ï
í
S
mð
R
ñò
V
W5aR
SUNV
b
c
R5d
V
M/OªZ
ì
V
S
` `
Z
ì
í
S
T ï
í
S
mð
R
ñCò
V
W/a
R
S*UNV
(16)
The density
Ú
¾
j
k
m
ë
Ú
¾
z
ð
R
ñò
V
converges weakly
to the density
(

£
? F
µ
V
4545F
µ
í
6B
%
4
. Not being able to
sample from this last density,
£
`
Ú
¾
is drawn as a realiza-
tion from
Ú
¾
.
D
Now let
¦
be an integer in the second product. As we
do not have any measure to correct the predicted particles
we draw a realization from the density
jk
]m
+
@
z
P
Q
R
S
for
£
`
Ú
¾
.
ö
After a finite number of iterations, we estimate the vector
º
by the average of its last realizations:
n
º
£
,
Ù

o

ÕÅÙ
o
x
Úø
R5ú
Ú
m
Ú÷*ø¨ù
º
£
`
Ú
(17)
Finally the weights are computed according to (8) using the
estimation
n
º
£
of
º
£
. Figure 2 summarizes the whole proce-
dure.
4. Application to bearings-only problems
We first deal with the classical bearings-only prob-
lem using synthetic data. The objects are then “point-
objects” in the
Õ
F
plane. Their state vector
rep-
resents the coordinates and the velocities in the
Õ
F
plane:
¼*
F
6Ë3
ËNF
. The following multitarget
scenario has been considered: three targets follow a near-
constant-velocity model defined by (18). The discretized
state equation associated with time period
p8
is:
¾


Ü
p8
Ü
¸
Ü
Ô
r
Ü
Ü
p8
Ü
(18)
where
Ü
is the identity matrix in dimension
ß
and
is
a Gaussian zero-mean vector of covariance matrix
Ó
! #"
$&%
%
"
')(
. Let
n
be the estimation of
computed by
the MOPF with
g
A
, i.e. ,
n

j
k
m
+
=
#
. For this
application we use a bootstrap filter, i.e. , the importance
function
>
is in fact the prior law
(
*4? @4
. Each object
produces one measurement at each time period according to
(19) except during the time interval
»
*
¸3¸
!+
¸¸
½
where the first
object does not produce any measurement and the second
produces two
F
and
F
Ü
according to:
B
,.-/10,.2
¤

Õ
w
Q
F
Õ
F
w
Q
Ô
7
(19)
where
is a zero-mean Gaussian noise of covariance
ê
Ü
3
independent of
.
w
Q
and
F
w
Q
are the Cartesian coordi-
nates of the observer, which are known. The trajectories of
the three targets and of the observer are plotted in figure 3.1
and the differences between the three couples of bearings
simulated are plotted in figure 4. As soon as the difference
between two bearings issued from two different targets is
lower than the standard deviation of the observation noise,
the two measures cannot be distinguished, which makes this
scenario very difficult. This difficulty is increased by the
detection gap for the first object. One particular run of the
particle filter with
ü
¸¸3¸
particles is presented in figure 3.2.
The plot of the three estimated trajectories shows that the
data association is overcome. There is no trajectories re-
versal and the estimations are quite satisfactory. Figure 5
shows the results of the estimation of the three components
of
º
and figure 3.3 represents the average of each compo-
nent
º
£
over successive intervals of
,
¸3¸
time steps and over
the
ß
¸
trials. When there is an ambiguity about the origin
of the measurements (i.e., when the differences between the
bearings are lower than the standard deviation noise), the

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Proceedings Article
01 Mar 2004
TL;DR: A probabilistic approach for moving object detection from a mobile robot using a single camera in outdoor environments and the positions of moving objects are estimated using an adaptive particle and EM detection.
Abstract: Robust detection of moving objects from a mobile robot is required for safe outdoor navigation, but is not easily achievable since there are two motions involved: the motions of moving objects and the motion of the sensors used to detect the objects. We have experimented with a probabilistic approach for moving object detection from a mobile robot using a single camera in outdoor environments. The ego-motion of the camera is compensated using corresponding feature sets and outlier detection, and the positions of moving objects are estimated using an adaptive particle lter and EM al- gorithm. The algorithms are implemented and tested on three different robot platforms (robotic helicopter, Segway RMP, and Pioneer2 AT) in an outdoor environment, and the detection results are analyzed.

181 citations


Cites methods from "A particle filter to track multiple..."

  • ...Work focusing on robust multiple target tracking using probabilistic filters includes [11] which uses a particle filter to track people indoors (corridors) using a laser rangefinder, and [12] which also uses a particle filter to track multiple objects using a stationary camera....

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Journal ArticleDOI
TL;DR: This paper proposes a dual-mode two-way Bayesian inference approach which dynamically switches between an offline general model and an online dedicated model to deal with single isolated object tracking and multiple occluded object tracking integrally by forward filtering and backward smoothing.
Abstract: Multiple object tracking (MOT) is a very challenging task yet of fundamental importance for many practical applications. In this paper, we focus on the problem of tracking multiple players in sports video which is even more difficult due to the abrupt movements of players and their complex interactions. To handle the difficulties in this problem, we present a new MOT algorithm which contributes both in the observation modeling level and in the tracking strategy level. For the observation modeling, we develop a progressive observation modeling process that is able to provide strong tracking observations and greatly facilitate the tracking task. For the tracking strategy, we propose a dual-mode two-way Bayesian inference approach which dynamically switches between an offline general model and an online dedicated model to deal with single isolated object tracking and multiple occluded object tracking integrally by forward filtering and backward smoothing. Extensive experiments on different kinds of sports videos, including football, basketball, as well as hockey, demonstrate the effectiveness and efficiency of the proposed method.

125 citations


Cites background from "A particle filter to track multiple..."

  • ...For example, Hue et al. [15] used a stochastic vector to assign the observation in the particle filter process....

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References
More filters
Journal ArticleDOI
01 Apr 1993
TL;DR: An algorithm, the bootstrap filter, is proposed for implementing recursive Bayesian filters, represented as a set of random samples, which are updated and propagated by the algorithm.
Abstract: An algorithm, the bootstrap filter, is proposed for implementing recursive Bayesian filters. The required density of the state vector is represented as a set of random samples, which are updated and propagated by the algorithm. The method is not restricted by assumptions of linear- ity or Gaussian noise: it may be applied to any state transition or measurement model. A simula- tion example of the bearings only tracking problem is presented. This simulation includes schemes for improving the efficiency of the basic algorithm. For this example, the performance of the bootstrap filter is greatly superior to the standard extended Kalman filter.

8,018 citations


"A particle filter to track multiple..." refers background or methods in this paper

  • ...The original particle filter, named bootstrap filter [4], proposesto approximatethe densities „ by a finite weightedsumof 0 Dirac densitiescentredon elementsof , namedparticles....

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  • ...TheRecursi veBayesianfilter, alsonamedOptimalFilter, resolvesexactlythisproblem in two stepsat eachtime 8 [4]....

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  • ...The readerwill find moredetailson thedifferentfilters in [4], [5] or [2] andon adapti ve resamplingin [6] and[2]....

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  • ...Numerousversionshave beenusedin various contexts: the bootstrapfilter for target tracking in [4], the Condensationalgorithmin imageanalysis[5] are two examplesamongothers....

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Journal ArticleDOI
TL;DR: The Condensation algorithm uses “factored sampling”, previously applied to the interpretation of static images, in which the probability distribution of possible interpretations is represented by a randomly generated set.
Abstract: The problem of tracking curves in dense visual clutter is challenging. Kalman filtering is inadequate because it is based on Gaussian densities which, being unimo dal, cannot represent simultaneous alternative hypotheses. The Condensation algorithm uses “factored sampling”, previously applied to the interpretation of static images, in which the probability distribution of possible interpretations is represented by a randomly generated set. Condensation uses learned dynamical models, together with visual observations, to propagate the random set over time. The result is highly robust tracking of agile motion. Notwithstanding the use of stochastic methods, the algorithm runs in near real-time.

5,804 citations

Journal ArticleDOI
TL;DR: A new theoretical result is presented: the joint probabilistic data association (JPDA) algorithm, in which joint posterior association probabilities are computed for multiple targets (or multiple discrete interfering sources) in Poisson clutter.
Abstract: The problem of associating data with targets in a cluttered multi-target environment is discussed and applied to passive sonar tracking. The probabilistic data association (PDA) method, which is based on computing the posterior probability of each candidate measurement found in a validation gate, assumes that only one real target is present and all other measurements are Poisson-distributed clutter. In this paper, a new theoretical result is presented: the joint probabilistic data association (JPDA) algorithm, in which joint posterior association probabilities are computed for multiple targets (or multiple discrete interfering sources) in Poisson clutter. The algorithm is applied to a passive sonar tracking problem with multiple sensors and targets, in which a target is not fully observable from a single sensor. Targets are modeled with four geographic states, two or more acoustic states, and realistic (i.e., low) probabilities of detection at each sample time. A simulation result is presented for two heavily interfering targets illustrating the dramatic tracking improvements obtained by estimating the targets' states using joint association probabilities.

1,421 citations


"A particle filter to track multiple..." refers background in this paper

  • ...obj[3] obj[2]...

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  • ...As long as the association is considered in a deterministic way, the hypothesis associations must be exhaustively enumerated, which leads to a NPhard problem (as in JPDAF and MHT algorithms [3] for instance)....

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Journal ArticleDOI
TL;DR: In this paper, Gibbs sampling is used to evaluate the posterior distribution and Bayes estimators by Gibbs sampling, relying on the missing data structure of the mixture model. And the data augmentation method is shown to converge geometrically, since a duality principle transfers properties from the discrete missing data chain to the parameters.
Abstract: SUMMARY A formal Bayesian analysis of a mixture model usually leads to intractable calculations, since the posterior distribution takes into account all the partitions of the sample. We present approximation methods which evaluate the posterior distribution and Bayes estimators by Gibbs sampling, relying on the missing data structure of the mixture model. The data augmentation method is shown to converge geometrically, since a duality principle transfers properties from the discrete missing data chain to the parameters. The fully conditional Gibbs alternative is shown to be ergodic and geometric convergence is established in the normal case. We also consider non-informative approximations associated with improper priors, assuming that the sample corresponds exactly to a k-component mixture.

895 citations


"A particle filter to track multiple..." refers methods in this paper

  • ...To estimate them we propose to use a Gibbs sampler whose principles are briefly recalled (see [1] or [9] for more details)....

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  • ...x-coordinate in meters obj[1]...

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  • ...-1000 0 1000 2000 3000 4000 5000 6000 7000 8000 obj[1]...

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Journal Article
TL;DR: This report presents an overview of sequential simulationbased methods for Bayesian filtering of nonlinear and non-Gaussian dynamic models and proposes some original developments.
Abstract: In this report, we present an overview of sequential simulationbased methods for Bayesian filtering of nonlinear and non-Gaussian dynamic models. It includes in a general framework numerous methods proposed independently in various areas of science and proposes some original developments.

747 citations

Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "A particle filter to track multiple objects" ?

The authors consider stochastic dynamic systems nonlinearly and uncompletely observed. The authors propose an extension of the classical particle filter where the stochastic vector of assignation is estimated by a Gibbs sampler. The merit of the method is assessed in bearings-only context and the authors present one application in image-based tracking.