Unknown Clutter Estimation by FMM Approach in Multitarget Tracking Algorithm

Algorithm 1

FMM Parameters Estimation by EM approach.

Input: the convergence threshold 𝜀, the estimation of clutter model parameters at time 𝑡_𝑘−1,

           𝝁^𝑖_𝑐,𝑘(0)=𝝁^𝑖_{𝑐,𝑘−1},Σ^𝑖_𝑐,𝑘(0)=Σ^𝑖_{𝑐,𝑘−1},𝑖=1,…,𝑔_{𝑐,𝑘−1}.

             Initialization step:

             (A) Initialization of Clutter Model:

sample some clutter points randomly 𝒫_𝑘={𝑝^𝑖_𝑘}^𝑔0_𝑖=1, the number of the clutter points is 𝑔₀,

            For 𝑖=𝑔_{𝑐,𝑘−1}+1,…,𝑔_{𝑐,𝑘−1}+𝑔₀, do,

         𝝁^𝑖_𝑐,𝑘(0)=𝑝_𝑘,Σ^𝑖_𝑐,𝑘(0)=𝜎²𝐼,𝑖=𝑔_{𝑐,𝑘−1}+1,…,𝑔_{𝑐,𝑘−1}+𝑔₀

         where 𝜎²=(1/10𝑑)trace((1/𝑛_𝑘)∑^𝑛𝑘_𝑗=1(𝑧^𝑗_𝑘−𝑧_𝑘)(𝑧^𝑗_𝑘−𝑧_𝑘)^𝑇)

            end for 𝑖. 𝑔_𝑐,𝑘(0)=𝑔_{𝑐,𝑘−1}+𝑔₀.

      (B) Initialization of Target-Originated Measurement Model:

To survival targets, 𝑔_𝑠,𝑘(0)=𝑔_{𝑡,𝑘−1}, for 𝑖=1,…,𝑔_𝑠,𝑘(0), do

           𝝁^𝑖_𝑠,𝑘(0)=ℎ_𝑘(𝑥^𝑖_{𝑘|𝑘−1}),  Σ^𝑖_𝑠,𝑘(0)=𝑅_𝑘,(𝑖=1,…,𝑔_𝑠,𝑘(0))

        ℱ_𝑠,𝑘⋅;𝝍_𝑠,𝑘(0)=∑^{𝑔𝑠,𝑘(0)}_𝑖=1𝜋^𝑖_𝑠,𝑘(0)𝒩(⋅;𝝁^𝑖_𝑠,𝑘(0),Σ^𝑖_𝑐,𝑘(0)). end for 𝑖.

To Spontaneous Birth Targets, 𝑔_𝑏,𝑘(0), 𝝁^𝑖_𝑏,𝑘(0),Σ^𝑖_𝑏,𝑘(0)  will be set according to prior

information, for 𝑖=1,…,𝑔_𝑏,𝑘(0), do

        ℱ_𝑏,𝑘⋅;𝝍_𝑏,𝑘(0)=∑^{𝑔𝑏,𝑘(0)}_𝑖=1𝜋^𝑖_𝑏,𝑘(0)𝒩(⋅;𝝁^𝑖_𝑏,𝑘(0),Σ^𝑖_𝑏,𝑘(0)). end for 𝑖.

To Spawned by Existent Targets, 𝑔_𝑝,𝑘(0)=𝑝_𝑘𝑔_𝑠,𝑘(0), for  𝑖=1,…,𝑔_𝑠,𝑘(0), 𝑗=1,…,𝑝_𝑘, do

    𝝁^𝑖,𝑗_𝑝,𝑘(0)=ℎ_𝑘(𝑥^𝑖_{𝑘|𝑘−1})+𝑑^𝑗_𝑝.𝑘, ℱ_𝑝,𝑘(⋅;𝝍_𝑝,𝑘(0))=∑^𝑝𝑘∑^𝑔_𝑗=1^𝑝,𝑘(0)_𝑖=1𝜋^𝑖𝑗_𝑝,𝑘(0)𝒩(⋅;𝝁^𝑖𝑗_𝑝,𝑘(0),Σ^𝑖𝑗_𝑝,𝑘(0)).

             end for 𝑗; end for 𝑖.

𝑔_𝑡,𝑘(0)=𝑔_𝑠,𝑘(0)+𝑔_𝑏,𝑘(0)+𝑔_𝑝,𝑘(0).𝑔_𝑘(0)=𝑔_𝑐,𝑘(0)+𝑔_𝑡,𝑘(0)𝜋^𝑖_𝑘(0)=1/𝑔_𝑘(0). Set 𝑡∶=0.

                   Repeat:

Expectation-step: calculate the conditional expectation of missing-data 𝐸_𝑘={𝑒¹_𝑘,…,𝑒^𝑛𝑘_𝑘}

           For 𝑖=1,…,𝑔_𝑘, 𝑗=1,…,𝑛_𝑘, do

       𝐸_{𝝍𝑘(𝑡)}[𝑒_𝑘^𝑖𝑗;𝑍_𝑘]=𝜏_𝑘^𝑖𝑗(𝑡)=𝜋^𝑖_𝑘(𝑡)𝑓^𝑖_𝑘(𝑧^𝑗_𝑘;𝜽^𝑖_𝑘(𝑡))/∑^{𝑔𝑘(𝑡)}_𝑖=1𝜋^𝑖_𝑘(𝑡)𝑓^𝑖_𝑘(𝑧^𝑗_𝑘;𝜽^𝑖_𝑘(𝑡)). end for 𝑗; end for 𝑖.

calculate the conditional expectation of complete-data log likelihood given 𝒫_𝑘 and 𝝍_𝑘(𝑡)

   𝑄(𝝍_𝑘;𝝍_𝑘(𝑡))=𝐸_{𝝍𝑘(𝑡)}log𝒞_𝑘(𝑌_𝑘;𝝍_𝑘)𝑍_𝑘=∑^𝑔𝑘∑^𝑛_𝑖=1^𝑘_𝑗=1𝜏_𝑘^𝑖𝑗(𝑡)log𝜋^𝑖_𝑘(𝑡)+log𝑓^𝑖_𝑘(𝑧^𝑗_𝑘;𝜽^𝑖_𝑘(𝑡))

Maximization-step: require the global maximization of 𝑄(𝝍_𝑘;𝝍_𝑘(𝑡)) with respect to 𝝍_𝑘 over the

     parameter space to give the updated estimate 𝝍_𝑘(𝑡)=argmax_𝝍𝑘𝑄(𝝍_𝑘;𝝍_𝑘(𝑡)).

            for 𝑖=1,…,𝑔_𝑘, 𝑗=1,…,𝑛_𝑘, do

𝜋^𝑖_𝑘1(𝑡)=𝑛_𝑘^𝑛𝑘_𝑗∑₌₁𝜏_𝑘^𝑖𝑗(𝑡),𝝁^𝑖_𝑘∑(𝑡)=^{𝑛𝑘𝑗=1}𝜏_𝑘^𝑖𝑗(𝑡)𝑧^𝑗_𝑘∑^𝑛𝑘_𝑗=1𝜏_𝑘^𝑖𝑗(𝑡),𝑖=1,…,𝑔_𝑘(𝑡),^𝑖∑_𝑐,𝑘∑(𝑡)=^{𝑛𝑘𝑗=1}𝜏_𝑘^𝑖𝑗(𝑡)(𝑧^𝑗_𝑘−𝝁^𝑖_𝑐,𝑘(𝑡))(𝑧^𝑗_𝑘−𝝁^𝑖_𝑐,𝑘(𝑡))^𝑇∑^𝑛𝑘_𝑗=1𝜏_𝑘^𝑖𝑗(𝑡)

                end for 𝑗; end for 𝑖.

Component management step: manage the components of the set 𝝍_𝑘(𝑡)={𝜋^𝑖_𝑘(𝑡),𝝁^𝑖_𝑘(𝑡),Σ^𝑖_𝑘(𝑡)}^𝑔𝑘_𝑖=1

        according to the merging and pruning strategy described in IV.C.

 Update 𝑔_𝑘(𝑡+1) and let 𝝍_𝑘(𝑡+1)={𝜋^𝑖_𝑘(𝑡+1),𝝁^𝑖_𝑘(𝑡+1),Σ^𝑖_𝑘(𝑡+1)}^{𝑔𝑘(𝑡+1)}_𝑖=1. denote the managed

          component number and parameter set. Set 𝑡∶=𝑡+1.

        Until   ℒ_𝑘(𝑌_𝑘;𝝍_𝑘(𝑡+1))−ℒ_𝑘(𝑌_𝑘;𝝍_𝑘(𝑡))<𝜀ℒ_𝑘(𝑌_𝑘;𝝍_𝑘(𝑡)).

        Output: the set of estimated parameters 𝝍_𝑘=𝝍_𝑘(𝑡+1)