资源描述
,Click to edit Master title style,*,Click to edit Master title style,*,Random Finite Sets in Stochastic Filtering,Ba-Ngu Vo,EEE Department University of MelbourneAustralia,http:/,IEEE Victorian Chapter July 28,2009,Stochastic Filtering History,N.Wiener(,1894-1964),A.N.Kolmogorov(1903-1987),R.E.Kalman(1930-),1940s:Wiener filter,Pioneering work by Wiener,Kolmogorov,1950s:,Kalman,filter,Work by Bode&Shannon,Zadeh,&,Ragazzini,Levinson,Swerling,Stratonovich,etc.,1970s:Aerospace applications,Sorenson&,Alspach,Singer,Bar-Shalom,Reid,etc.,1960s:,Publication of the,Kalman,filter,Kalman-Bucy,filter,Schmidts 1,st,implementation Apollo program,LMS algorithm by,Widrow,&Hoff,Particle Filter(1990s-),Computational tools for non-linear non-Gaussian filtering,Gordon,Salmond,&Smith,Doucet,Random Finite Set(1990s-),Unified framework for multi-object filtering&control,Probability Hypothesis Density(PHD)filters,Bernoulli filter,Pioneering work by Mahler,Stochastic Filtering:The Present,The,Bayes,(nonlinear)Filter,Practical Challenges,Multi-Object Filtering,Random Finite Set,PHD/CPHD Filters&Applications,Conclusions,Outline,The Bayes(nonlinear)Filter,state-vector,state dynamic,state space,observation space,x,k,x,k-,1,z,k-,1,z,k,f,k|k-,1,(,x,k,|x,k-,1,),Markov Transition Density,Measurement Likelihood,g,k,(,z,k,|x,k,),Objective,measurement history,(,z,1,z,k,),posterior pdf of the state,p,k,(,x,k,|z,1:,k,),System Model,state-vector,state dynamic,state space,observation space,x,k,x,k-,1,z,k-,1,z,k,Bayes filter,p,k-,1,(,x,k-,1,|z,1:,k-,1,),p,k|k-,1,(,x,k,|z,1:,k-,1,),p,k,(,x,k,|z,1:,k,),prediction,data-update,p,k-,1,(,x,k-,1,|z,1:,k-,1,),d,x,k-,1,f,k|k-,1,(,x,k,|x,k-,1,),g,k,(,z,k,|x,k,),p,k|k-,1,(,x,k,|z,1:,k-,1,),The Bayes(nonlinear)Filter,f,k|k-,1,(,x,k,|x,k-,1,),g,k,(,z,k,|x,k,),g,k,(,z,k,|x,k,),p,k-,1,(,x,k-,1,|z,1:,k-,1,),d,x,k,state-vector,state dynamic,state space,observation space,x,k,x,k-,1,z,k-,1,z,k,The Bayes(nonlinear)Filter,f,k|k-,1,(,x,k,|x,k-,1,),g,k,(,z,k,|x,k,),p,k-,1,(,.,|z,1:,k-,1,),p,k|k-,1,(,.,|z,1:,k-,1,),p,k,(,.,|z,1:,k,),prediction,data-update,Bayes filter,N,(,.,;,m,k-,1,P,k-,1,),N,(,.,;,m,k|k-,1,P,k|k-,1,),N,(,.,;,m,k,P,k,),Kalman filter,i,=1,N,w,k|k-,1,x,k|k-,1,i,=1,N,(,i,),(,i,),w,k,x,k,i,=1,N,(,i,),(,i,),w,k-,1,x,k-,1,(,i,),(,i,),Particle filter,state-vector,state dynamic,state space,observation space,x,k,x,k-,1,z,k-,1,z,k,Practical Challenges,f,k|k-,1,(,x,k,|x,k-,1,),g,k,(,z,k,|x,k,),So far,we,assumed exactly 1 observation,at each time,Holds only for a small number of applications,Practical measuring device:,may fail to detect true observation(,detection uncertainty,)&,picks up false observations(,clutter,),Practical Challenges,Not detected,Detection uncertainty:,Detected,False observations(clutter),or,Number of false observations,unknown,random,False,Practical Challenges,No information on which is the observation of the state,Number of observations is a random variable.,+,Observation,=,Not detected,Detected,False,state-vector,state dynamic,state space,observation space,x,k,x,k-,1,z,k-,1,z,k,Practical Challenges,Summary of practical challenges:,Number of observations is random&time varying,True observation may not be present,Do not know which observations are false/true,Ordering of observations not relevant,observation,produced by objects,state dynamic,state space,observation space,5 objects,3 objects,X,k-,1,X,k,Objective:,Jointly estimate,the,number,&,states,of objects,Numerous applications,:defence,surveillance,robotics,biomed,Challenges:,Random number of objects and measurements,Detection uncertainty,clutter,association uncertainty,Multi-Object Filtering,Estimate is correct but estimation error,?,True,Multi-object,state,Estimated,Multi-object,state,How can we mathematically,represent,the,multi-object state?,2 objects,Usual practice:stack individual states into a large,vector,!,2 objects,Remedy:use,Fundamental inconsistency:,Multi-Object Filtering,True,Multi-object,state,Estimated,Multi-object,State,2 objects,no object,True,Multi-object,state,Estimated,Multi-object,State,2 objects,1 object,What are the estimation errors?,Multi-Object Filtering,Miss-distance:error,between,estimate,and,true,state,measures how close an estimate is to the true value,well-understood for single target:Euclidean distance,MSE,etc,fundamental,in estimation/filtering&control,Vector representation,doesnt admit multi-object miss-distance,Finite set representation,admits multi-object miss-distance,e.g.,Haussdorf,Wasserstein,OSPA,Schuhmacher,et.al.08,In fact the“distance”,is a distance for sets not vectors,Multi-Object Filtering,Multi-Object Filtering,states,multi-object state,multi-object observation,X,observations,X,Z,p,k-,1,(,X,k-,1,|,Z,1:,k-,1,),p,k,(,X,k,|,Z,1:,k,),p,k|k-,1,(,X,k,|,Z,1:,k-,1,),prediction,data-update,Reconceptualise as a finite,set-valued,filtering problem,Multi-object,state,&,observation,represented by,finite sets,Bayesian framework treats,state/observation,as,random variables,Bayesian multi-object filtering,re
展开阅读全文