Multi-sensor PHD and CPHD filters

We study various multi-sensor PHD and CPHD filters and their implementations. 1 Problem Statement The state of the system is the collection of individual target states xk,i ∈ Rx and is denoted by the random finite set Xk = {xk,1, . . .xk,nk} where nk ≥ 0 is number of targets present at time k. We assume that the individual target dynamics are specified according to the Markovian model of the form xk+1,i = fk+1∣k(xk,i,uk) where uk is the noise. Information about the state of the system is available from sensors 1, . . . , s. These sensors make independent measurements and we denote Zk,j to be the measurement set of the j-th sensor at timestep k. Let mk,j = ∣Zk,j ∣. Let Z[k],j be the sequence of measurement sets at the j-th sensor, i.e. Z1,j , Z2,j , . . . , Zk,j . Denote by hk,j(z∣x) the likelihood function at timestep k for sensor j for an individual measurement z and target state x. Let the probability of detection be pd,j(x). Denote by ck,j(z) the clutter spatial distribution of the j-th sensor, and let Ck,j(z) be the probability generating function (p.g.f.) of the cardinality distribution of the clutter process. 2 Product CPHD Filter For deriving the approximate product multisensor CPHD filter we make the following modeling assumptions: ˆ fk∣k−1(X) and fk∣k(X ∣Zk,j) are the distributions of i.i.d. cluster processes. ˆ The sensor clutter processes are i.i.d. cluster processes. ˆ Each target generates at most one measurement per sensor. ˆ Each measurement is either associated with one target or is generated by clutter.