MultitargettrackingviaPHDfilter - u-bordeaux.frmpace/files/posterAlea... · 2009. 9. 16. ·...
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Multi target tracking via PHD filter F. Caron, P. Del Moral, P. Legrand, M. Pace {Francois.Caron,Pierre.Del-Moral,Michele.Pace}@inria.fr,[email protected] Contribution Filtering is the problem of estimating the state of a system as a set of observations becomes available in time, and it has gained a great importance in many fields of science, engineering and finance. To solve it, one begins by modeling the evolution of the system and the uncertainty in the measurements. The re- sulting models typically exhibit complex non-linearities and non- Gaussian distributions, thus precluding analytical solution. Se- quential Monte Carlo methods, also known as particle filters can be used to estimate the a posteriori distribution of the system state. They are mainly composed by two steps: the evolution of initial samples using some dynamic equation, and the resampling of par- ticles according to their likelihood (in a Bayesian sense), with re- spect to the current observation. Modern lines of research in dif- ferent fields like target tracking, computer vision, financial mathe- matics, and biology make wide use of these techniques; neverthe- less they develop and adopt each concept rather autonomously, with very few connections with each other and with the mathe- matical foundation of the field. We try to illustrate one of the pos- sible links among these disciplines, more precisely how Bayesian filtering applied to multi-target tracking can be completely and fruitfully cast into the framework of genetic algorithms. References [1] Del Moral P., Kallel L. and Rowe J. Modeling genetic algo- rithms with interacting particle systems In Revista de Matemat- ica, Teoria y aplicaciones,Vol.8,No. 2, July (2001). [2] Del Moral P. Feynman-Kac Formulae Genealogical and Inter- acting Particle Systems with Applications. Springer New York. [3] Ba-Ngu Vo, Sumeetpal Singh, Arnaud Doucet Sequential Monte Carlo methods for Multi-target Filtering with Random Finite Sets In IEEE Transaction on Aerospace and Electronic Sys- tems, Vol. X, No. XXX, June 2005 Bayesian filtering The target motion is modeled by a transition probability density: x k ∼ f k|k-1 (.|x k-1 ) (1) The target state x k generates an observation y k according to the observation model: y k ∼ g k (.|x k ) (2) The aim is to construct the posterior probability density function of the target state given the observations. The Bayes filter is a re- cursion based on two steps: prediction (3) and update (4): p k|k-1 (x k |y 1:k-1 )= Z f k|k-1 (x k |x)p k-1 (x|y 1:k-1 )dx (3) p k (x k |y 1:k )= g k (y k |x k )p k|k-1 (x k |y 1:k-1 ) R g k (y k |x)p k|k-1 (x|y 1:k-1 )dx (4) The analytical solution to the Bayes filter is generally intractable, with the exception of the linear Gaussian case whose exact solu- tion is given by Kalman filter. Extended Kalman filter (EKF) and Unscented Kalman filter (UKF) have been proposed to address non linear models. A more general solution strategy is to use se- quential Monte Carlo methods, also known as particle filters. Multi-target tracking and PHD filter Multi-target tracking involves jointly estimating the number and the states of a finite, time varying number of targets given a set of finite and time varying measurements of uncertain origins. The PHD Filter: • is a suboptimal but tractable alternative to the multi-target Bayes filter. • It propagates the first order moments of the targets random finite sets (RFS) known as intensity (or PHD) function. • Given a RFS X on X⊆ R n x with probability P the inten- sity function is a non-negative function v on X such that, for any closed subset S ⊆X : Z S v (x)dx = Z |X ∩ S |P (dX ) providing an estimate about the number of elements of a RFS in the region considered. • In the SMC-PHD filter the PHD function is approximated by the particle representation ˆ v k (x)= ∑ N k i=1 w (i) k δ x (i) k (x). • Importance sampling and resampling are applied to prop- agate the particle-approximated intensity function through the recursion v k|k =(ψ k ◦ Φ k|k-1 )v k-1|k-1 as specified in equations 8 and 9. Particle Filtering and Genetic Algorithms Genetic Algorithms In mathematical terms a genetic algorithm is a Markov chain X n =(X i n ) 1≤i≤N defined on the product space E N . The evolution of the chain is composed by a phase of selection, during which individuals are selected according to certain criteria (fitness) and by a phase of state exploration. X n =(X i n ) 1≤i≤N selection ------→ b X n =( b X i n ) 1≤i≤N mutation ------→ X n+1 =(X i n+1 ) 1≤i≤N • For the selection phase a positive function (fitness) is given and each new individual is selected proportionally to his fit- ness. ψ (ˆ x (i) n )= ∑ N i=1 G(X i n ) ∑ N j =1 G(X j n ) δ X i n • During the mutation phase the state space is explored ac- cording to the Markov transition M (x (i) n+1 , ˆ x (i) n ) x (i) n+1 ∼ M (x (i) n+1 , ˆ x (i) n ) Filtering In filtering problems, the correction/prediction phases of the optimal filter: Law(X n |Y 0 ...Y n-1 )= η n correction -------→ Law(X n |Y 0 ...Y n )=Ψ G n (η n )= b η n prediction --------→ η n+1 = b η n M n+1 are estimated using a Markov process defined on the product space E N : ξ (N ) n = ξ (N,i) n 1≤i≤N selection --------→ b ξ (N ) n = b ξ (N,i) n 1≤i≤N mutation -------→ ξ (N ) n+1 This genetic algorithm is a particle filter as the occupation measures of the predicted and updated populations (or selected and mutated in terms of genetic algorithms) converge almost surely to the optimal predictor and filter. η N n := 1 N N X i=1 δ ξ (N,i) n , b η N n := 1 N N X i=1 δ b ξ (N,i) n et lim N →∞ η N n = η n , lim N →∞ b η N n = b η n Feynman Kac Measures Let’s consider the probability distribution flow (η n ) n≥0 defined by equations: η n =Φ n (η n-1 ) := Ψ G n-1 (η n-1 )M n (5) The measures η n and the corresponding updated measures b η n := Ψ G n (η n ) can be expressed as weighted path integrals known as Feynman-Kac formulae. These functional representations are defined on test functions f n by the following equations: η n (f n )= γ n (f n )/γ n (1) et b η n (f n )= b γ n (f n )/b γ n (1) (6) With γ n and b γ n being unnormalized measures defined by: γ n (f n )= E[f n (X n ) Q 0≤k<n G k (X k )] et b γ n (f n )= γ n (f n G n ) (7) PHD filter The PHD prediction and update operators Φ k|k-1 and Ψ k used to approximate the first moment of the posterior distribution are respec- tively defined as (Φ k|k-1 α)(x)= R [P S,k|k-1 (x)f k|k-1 (x|ξ )+ b k|k-1 (x|ξ )]α(ξ )λ(dξ )+ γ k (x) (8) (Ψ k α)(x)= h (1 - P D (x)) + ∑ z ∈Z k P D (x)g k (z |x) κ k (z )+hP D (x)g k (z |x),αi i α(x) (9) Equipe ALEA: Advanced Learning Evolutionary Algorithms Joint team with the IMB (CNRS and university Bordeaux 1 et 2). INRIA Bordeaux - Sud Ouest. http://alea.bordeaux.inria.fr/ Simulations Simulation 1: Particle filter approximating the posterior distribution Law(X n |Y 0 ...Y n ) of a target moving on a square. Three sensors report only angular measurements, the observation noise is Gaussian, centered with variance σ 2 =0.1rad. A random walk is used as a priori model for target evolution. 1000 particles are used. -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Distance X (mt) Distance Y (mt) Particle approximated posterior distribution (Time = 2 sec) -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Distance X (mt) Distance Y (mt) Particle approximated posterior distribution (Time = 12 sec) -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Distance X (mt) Distance Y (mt) Particle approximated posterior distribution (Time = 24 sec) -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Distance X (mt) Distance Y (mt) Particle approximated posterior distribution (Time = 39 sec) Fig.1: Time = 2 sec. Fig.2: Time = 12 sec. Fig.3: Time = 24 sec. Fig.4: Time = 39 sec. Simulation 2: PHD Filter estimating the posterior intensity function of multiple targets moving over a 1D surveillance zone with false alarms and detection uncertainty. 0 20 40 60 80 100 -100 -80 -60 -40 -20 0 20 40 60 80 100 Observation and clutter Time Sourveillance Zone 0 20 40 60 80 100 -100 -80 -60 -40 -20 0 20 40 60 80 100 Observation and clutter Time Sourveillance Zone 0 20 40 60 80 100 0 10 20 30 40 50 0 2 4 Fig.5: Targets (green), measurements (blue) and clutter (black) Fig.6: SMC-PHD filter output. Fig.7: 3D plot of the posterior intensity. Simulation 3: PHD Filter applied to a realistic 3D scenario, radar-like measurements containing false alarms are used to estimate the position of multiple targets (airplanes and boats) maneuvering in the surveillance zone. Fig.8: Maneuvering targets (blue), measurements (red) Fig.9: SMC-PHD Filter estimated target positions. Fig.10: PHD particles surrounding true target and clutter (black). trajectories during filtering.
Transcript of MultitargettrackingviaPHDfilter - u-bordeaux.frmpace/files/posterAlea... · 2009. 9. 16. ·...
Multi target trackingviaPHDfilterF. Caron, P. Del Moral, P. Legrand, M. Pace
{Francois.Caron,Pierre.Del-Moral,Michele.Pace}@inria.fr,[email protected]
ContributionFiltering is the problem of estimating the state of a system as aset of observations becomes available in time, and it has gaineda great importance in many fields of science, engineering andfinance. To solve it, one begins by modeling the evolution ofthe system and the uncertainty in the measurements. The re-sulting models typically exhibit complex non-linearities and non-Gaussian distributions, thus precluding analytical solution. Se-quential Monte Carlo methods, also known as particle filters canbe used to estimate the a posteriori distribution of the system state.They are mainly composed by two steps: the evolution of initialsamples using some dynamic equation, and the resampling of par-ticles according to their likelihood (in a Bayesian sense), with re-spect to the current observation. Modern lines of research in dif-ferent fields like target tracking, computer vision, financial mathe-matics, and biology make wide use of these techniques; neverthe-less they develop and adopt each concept rather autonomously,with very few connections with each other and with the mathe-matical foundation of the field. We try to illustrate one of the pos-sible links among these disciplines, more precisely how Bayesianfiltering applied to multi-target tracking can be completely andfruitfully cast into the framework of genetic algorithms.
References[1] Del Moral P., Kallel L. and Rowe J. Modeling genetic algo-
rithms with interacting particle systems In Revista de Matemat-ica, Teoria y aplicaciones,Vol.8,No. 2, July (2001).
[2] Del Moral P. Feynman-Kac Formulae Genealogical and Inter-acting Particle Systems with Applications. Springer New York.
[3] Ba-Ngu Vo, Sumeetpal Singh, Arnaud Doucet SequentialMonte Carlo methods for Multi-target Filtering with RandomFinite Sets In IEEE Transaction on Aerospace and Electronic Sys-tems, Vol. X, No. XXX, June 2005
Bayesian filteringThe target motion is modeled by a transition probability density:
xk ∼ fk|k−1(.|xk−1) (1)
The target state xk generates an observation yk according to theobservation model:
yk ∼ gk(.|xk) (2)
The aim is to construct the posterior probability density functionof the target state given the observations. The Bayes filter is a re-cursion based on two steps: prediction (3) and update (4):
pk|k−1(xk|y1:k−1) =
∫fk|k−1(xk|x)pk−1(x|y1:k−1)dx (3)
pk(xk|y1:k) =gk(yk|xk)pk|k−1(xk|y1:k−1)∫gk(yk|x)pk|k−1(x|y1:k−1)dx
(4)
The analytical solution to the Bayes filter is generally intractable,with the exception of the linear Gaussian case whose exact solu-tion is given by Kalman filter. Extended Kalman filter (EKF) andUnscented Kalman filter (UKF) have been proposed to addressnon linear models. A more general solution strategy is to use se-quential Monte Carlo methods, also known as particle filters.
Multi-target tracking and PHD filterMulti-target tracking involves jointly estimating the number andthe states of a finite, time varying number of targets given a set offinite and time varying measurements of uncertain origins.
The PHD Filter:
• is a suboptimal but tractable alternative to the multi-targetBayes filter.
• It propagates the first order moments of the targets randomfinite sets (RFS) known as intensity (or PHD) function.
• Given a RFS X on X ⊆ Rnx with probability P the inten-sity function is a non-negative function v on X such that,for any closed subset S ⊆ X :∫
Sv(x)dx =
∫|X ∩ S|P(dX)
providing an estimate about the number of elements of aRFS in the region considered.
• In the SMC-PHD filter the PHD function is approximatedby the particle representation v̂k(x) =
∑Nki=1 w
(i)k δ
x(i)k
(x).
• Importance sampling and resampling are applied to prop-agate the particle-approximated intensity function throughthe recursion
vk|k = (ψk ◦ Φk|k−1)vk−1|k−1
as specified in equations 8 and 9.
Particle Filtering and Genetic AlgorithmsGenetic AlgorithmsIn mathematical terms a genetic algorithm is a Markov chain Xn = (Xi
n)1≤i≤N defined on the product space EN . The evolution of thechain is composed by a phase of selection, during which individuals are selected according to certain criteria (fitness) and by a phase ofstate exploration.
Xn = (Xin)1≤i≤N
selection−−−−−−→ X̂n = (X̂i
n)1≤i≤Nmutation−−−−−−→ Xn+1 = (Xi
n+1)1≤i≤N
• For the selection phase a positive function (fitness) is givenand each new individual is selected proportionally to his fit-ness.
ψ(x̂(i)n ) =
∑Ni=1
G(Xin)∑N
j=1 G(Xjn)δXi
n
• During the mutation phase the state space is explored ac-cording to the Markov transition M(x
(i)n+1, x̂
(i)n )
x(i)n+1 ∼M(x
(i)n+1, x̂
(i)n )
FilteringIn filtering problems, the correction/prediction phases of the optimal filter:
Law(Xn|Y0 . . . Yn−1) = ηncorrection−−−−−−−→ Law(Xn|Y0 . . . Yn) = ΨGn (ηn) = η̂n
prediction−−−−−−−−→ ηn+1 = η̂nMn+1
are estimated using a Markov process defined on the product space EN :
ξ(N)n =
(ξ(N,i)n
)1≤i≤N
selection−−−−−−−−→ ξ̂
(N)n =
(ξ̂(N,i)n
)1≤i≤N
mutation−−−−−−−→ ξ
(N)n+1
This genetic algorithm is a particle filter as the occupation measures of the predicted and updated populations (or selected and mutatedin terms of genetic algorithms) converge almost surely to the optimal predictor and filter.
ηNn :=1
N
N∑i=1
δξ(N,i)n
, η̂Nn :=1
N
N∑i=1
δξ̂(N,i)n
et limN→∞
ηNn = ηn , limN→∞
η̂Nn = η̂n
Feynman Kac MeasuresLet’s consider the probability distribution flow (ηn)n≥0 defined by equations:
ηn = Φn (ηn−1) := ΨGn−1 (ηn−1)Mn (5)
The measures ηn and the corresponding updated measures η̂n := ΨGn (ηn) can be expressed as weighted path integrals known asFeynman-Kac formulae. These functional representations are defined on test functions fn by the following equations:
ηn(fn) = γn(fn)/γn(1) et η̂n(fn) = γ̂n(fn)/γ̂n(1) (6)
With γn and γ̂n being unnormalized measures defined by:
γn(fn) = E[fn(Xn)∏
0≤k<nGk(Xk)] et γ̂n(fn) = γn(fnGn) (7)
PHD filterThe PHD prediction and update operators Φk|k−1 and Ψk used to approximate the first moment of the posterior distribution are respec-tively defined as
(Φk|k−1α)(x) =∫
[PS,k|k−1(x)fk|k−1(x|ξ) + bk|k−1(x|ξ)]α(ξ)λ(dξ) + γk(x) (8)
(Ψkα)(x) =[(1− PD(x)) +
∑z∈Zk
PD(x)gk(z|x)κk(z)+〈PD(x)gk(z|x),α〉
]α(x) (9)
Equipe ALEA: Advanced Learning Evolutionary AlgorithmsJoint team with the IMB (CNRS and university Bordeaux 1 et 2). INRIA Bordeaux - Sud Ouest. http://alea.bordeaux.inria.fr/
SimulationsSimulation 1: Particle filter approximating the posterior distribution Law(Xn|Y0 . . . Yn) of a target moving on a square. Threesensors report only angular measurements, the observation noise is Gaussian, centered with variance σ2 = 0.1rad. A random walk isused as a priori model for target evolution. 1000 particles are used.
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Particle approximated posterior distribution (Time = 39 sec)
Fig.1: Time = 2 sec. Fig.2: Time = 12 sec. Fig.3: Time = 24 sec. Fig.4: Time = 39 sec.
Simulation 2: PHD Filter estimating the posterior intensity function of multiple targets moving over a 1D surveillance zone withfalse alarms and detection uncertainty.
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Fig.5: Targets (green), measurements (blue) and clutter (black) Fig.6: SMC-PHD filter output. Fig.7: 3D plot of the posterior intensity.
Simulation 3: PHD Filter applied to a realistic 3D scenario, radar-like measurements containing false alarms are used to estimate theposition of multiple targets (airplanes and boats) maneuvering in the surveillance zone.
Fig.8: Maneuvering targets (blue), measurements (red) Fig.9: SMC-PHD Filter estimated target positions. Fig.10: PHD particles surrounding true target
and clutter (black). trajectories during filtering.
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