IEEE粒子滤波PPT

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1、http:/www.ee.unimelb.edu.au/staff/bv/N.Wiener(1894-1964)A.N.Kolmogorov(1903-1987)R.E.Kalman(1930-)state-vectorstate dynamicstate spaceobservation spacexkxk-1zk-1zk fk|k-1(xk|xk-1)Markov Transition DensityMeasurement Likelihoodgk(zk|xk)Objectivemeasurement history(z1,zk)posterior pdf of the statepk(x

2、k|z1:k)System Modelstate-vectorstate dynamicstate spaceobservation spacexkxk-1zk-1zkBayes filterpk-1(xk-1|z1:k-1)pk|k-1(xk|z1:k-1)pk(xk|z1:k)predictiondata-update pk-1(xk-1|z1:k-1)dxk-1 fk|k-1(xk|xk-1)gk(zk|xk)pk|k-1(xk|z1:k-1)fk|k-1(xk|xk-1)gk(zk|xk)gk(zk|xk)pk-1(xk-1|z1:k-1)dxkstate-vectorstate dy

3、namicstate spaceobservation spacexkxk-1zk-1zk fk|k-1(xk|xk-1)gk(zk|xk)pk-1(.|z1:k-1)pk|k-1(.|z1:k-1)pk(.|z1:k)predictiondata-updateBayes filterN(.;mk-1,Pk-1)N(.;mk|k-1,Pk|k-1)N(.;mk,Pk)Kalman filteri=1Nwk|k-1,xk|k-1i=1N(i)(i)wk,xk i=1 N(i)(i)wk-1,xk-1(i)(i)Particle filterstate-vectorstate dynamicsta

4、te spaceobservation spacexkxk-1zk-1zk fk|k-1(xk|xk-1)gk(zk|xk)Not detectedDetectedor Number of false observationsunknown random False+Observation=Not detectedDetectedFalsestate-vectorstate dynamicstate spaceobservation spacexkxk-1zk-1zkobservation produced by objectsstate dynamicstate spaceobservati

5、on space5 objects3 objectsXk-1XkObjective:Jointly estimate the number&states of objectsNumerous applications:defence,surveillance,robotics,biomed,Challenges:Random number of objects and measurementsDetection uncertainty,clutter,association uncertaintyTrueMulti-object stateEstimatedMulti-object state

6、|2XX2 objects2 objects()min|0perm XXX00 11 00 11 TrueMulti-object stateEstimatedMulti-object State2 objectsno objectTrueMulti-object stateEstimatedMulti-object State2 objects1 object 00 11 00 11 00 ()min|0perm XXXstatesmulti-object statemulti-object observation X X observations X Z pk-1(Xk-1|Z1:k-1)

7、pk(Xk|Z1:k)pk|k-1(Xk|Z1:k-1)predictiondata-update sample u uniform0,1if u r,sample x p(.),end;EESample n Poiss(r),for i=1:n,sample xi p(.),end;ESample n c(.),for i=1:n,sample xi p(.),end;pk-1(Xk-1|Z1:k-1)pk(Xk|Z1:k)pk|k-1(Xk|Z1:k-1)predictiondata-update?statesmulti-object statemulti-object observati

8、on X X observations X ZMulti-object Bayes filterBelief“density”of f:F(E)0,)b(T)=T f(X)dXBelief“distribution”of b(T)=P(T),T EEProbability density of p:F(E)0,)P(T)=T p(X)m(dX)Probability distribution of P(T)=P(T ),T F(E)F(E)Collection of finite subsets of E State space Mahlers Finite Set Statistics(19

9、94)Choquet(1968)TTConventional integralSet integralPoint Process Theory(1950-1960s)VSD(2005)Computationally expensive!single-object Bayes filter multi-object Bayes filter state of system:random vectorfirst-moment filter(e.g.a-b-g filter)state of system:random setfirst-moment filter(“PHD”filter)Singl

10、e-object Multi-object pk-1(Xk-1|Z1:k-1)pk(Xk|Z1:k)pk|k-1(Xk|Z1:k-1)predictiondata-update Multi-object Bayes filterx0state spacev PHD(intensity function)of an RFS S v(x)dx=expected number of objects in SSv(x0)=density of expected number of objects at x0state space vk vk-1 PHD filter Mahler 03 vk-1(xk

11、-1|Z1:k-1)vk(xk|Z1:k)vk|k-1(xk|Z1:k-1)PHD predictionPHD update Multi-object Bayes filter pk-1(Xk-1|Z1:k-1)pk(Xk|Z1:k)pk|k-1(Xk|Z1:k-1)predictionupdate Avoids data association!vk|k-1(xk|Z1:k-1)=fk|k-1(xk,xk-1)vk-1(xk-1|Z1:k-1)dxk-1+gk(xk)intensity from previoustime-step term for spontaneousobject bir

12、thsfk|k-1(xk,xk-1)=ek|k-1(xk-1)fk|k-1(xk|xk-1)+bk|k-1(xk|xk-1)Markovtransitionintensityprobabilityof objectsurvivalterm for objectsspawned byexisting objectsMarkov transition densitypredictedintensityNk|k-1=vk|k-1(x|Z1:k-1)dxpredicted expected number of objects(Fk|k-1a)(xk)fk|k-1(xk,x)a(x)dx+gk(xk)v

13、k|k-1 Fk|k-1vk-1 vk(xk|Z1:k)zZkDk(z)+kk(z)pD,k(xk)gk(z|xk)+1 pD,k(xk)vk|k-1(xk|Z1:k-1)Dk(z)=pD,k(x)gk(z|x)vk|k-1(x|Z1:k-1)dx Nk=vk(x|Z1:k)dxBayes-updated intensitypredicted intensity(from previous time)intensity offalse alarmssensor likelihood functionprobabilityof detectionexpected number of object

14、smeasurementvk Ykvk|k-1(Yka)(x)=zZk+kk(z)yk,z(x)+1 pD,k(x)a(x)vk-1(.|Z1:k-1)vk(.|Z1:k)vk|k-1(.|Z1:k-1)1|Fkk kY wk-1,xk-1j=1Jk-1(j)(j)j=1Jk|k-1(j)(j)wk|k-1,xk|k-1 wk,xk j=1 Jk(j)(j)wk-1,mk-1,Pk-1j=1Jk-1(j)(j)(j)wk|k-1,mk|k-1,Pk|k-1j=1Jk|k-1(j)(j)(j)wk,mk,Pk j=1 Jk(j)(j)(j)Data courtesy of Czyz et.al.

15、Data courtesy of K.Smith IDIAP Research Institute.CPHD filter Mahler 06,07,Gaussian Mixture CPHD filter VVC 06,07 vk-1(xk-1|Z1:k-1)vk(xk|Z1:k)vk|k-1(xk|Z1:k-1)intensity predictionintensity update ck-1(n|Z1:k-1)ck(n|Z1:k)ck|k-1(n|Z1:k-1)cardinality predictioncardinality update Courtesy of Lockheed Ma

16、rtinCourtesy of Lockheed MartinOSPA distance(satisfies all metric axioms)=per target cardinality&state error0102030405060708090100050010001500TimeCardinality TrueEstimate1020304050607080901000102030TimeOSPA(km)1020304050607080901000102030TimeOSPA Loc(km)1020304050607080901000102030TimeOSPA Card(km)R

17、obot poseMapMeasurementsControlsMeasurement likelihoodSet integralTransition densityRFS-SLAM predictionRFS-SLAM updateSet integralRFS-SLAM Mullane et.al.08PHD of the posterior map RFSExperiment:Nanyang Technological University Campus Low clutter:All 3 algorithms can close the loopHigher clutter:Only

18、 PHD-SLAM can close the loopGround truth plotted in greenThank You!For more info please see http:/randomsets.ee.unimelb.edu.auSee also:http:/www.ee.unimelb.edu.au/staff/bv/publications.htmlBooksD.Daley and D.Vere-Jones,An Introduction to the Theory of Point Processes,Springer-Verlag,1988.D.Stoyan,D.

19、Kendall,J.Mecke,Stochastic Geometry and its Applications,John Wiley&Sons,1995I.Goodman,R.Mahler,and H.Nguyen,Mathematics of Data Fusion.Kluwer Academic Publishers,1997.R.Mahler,Statistical Multisource-Multitarget Information Fusion,ArtechHouse,2007.M.Mallick,V.Krisnamurthy,B.-N.Vo(eds),Advanced Topi

20、cs and Applications in Integrated Tracking,Classification,and Sensor Management,IEEE-Wiley(under review)PapersR.Mahler,“Multi-target Bayes filtering via first-order multi-target moments,”IEEE Trans.AES,vol.39,no.4,pp.11521178,2003.B.-N.Vo,S.Singh,and A.Doucet,“Sequential Monte Carlo methods for mult

21、i-target filtering with random finite sets,”IEEE Trans.AES,vol.41,no.4,pp.12241245,2005.B.-N.Vo,and W.K.Ma,“The Gaussian mixture PHD filter,”IEEE Trans.Signal Processing,IEEE Trans.Signal Processing,Vol.54,No.11,pp.4091-4104,2006.R.Mahler,“PHD filter of higher order in target number,”IEEE Trans.Aero

22、space&Electronic Systems,vol.43,no.4,pp.15231543,2007B.T.Vo,B.-N.Vo,and A.Cantoni,Analytic implementations of the Cardinalized Probability Hypothesis Density Filter,IEEE Trans.Signal Processing,Vol.55,No.7,Part 2,pp.3553-3567,2007.B.-T.Vo,B.-N Vo,and A.Cantoni,The Cardinality Balanced Multi-target Multi-Bernoulli filter and its implementations,IEEE Trans.Signal Processing,vol.57,no.2,pp.409423,2009.J.Mullane,B.-N.Vo,M.Adams and S.Wijesoma,A Random Set Formulation for Bayesian SLAM,International Conference on Intelligent Robots and Systems,Nice,France,2008.