A Quantitative Analysis on Two RFS-Based Filtering Methods for Multicell Tracking

Algorithm 2

Read image 𝐲_𝐤

Given the posterior density 𝑟^(𝑖)_𝑘−1,𝑤^(𝑖,𝑗)_𝑘−1,𝐱^(𝑖,𝑗)_𝑘−1^{𝐿(𝑖)𝑘−1}_𝑗=1^𝑀𝑘−1_𝑖=1, the birth density

𝑟^(𝑖)_Γ,𝑘,𝑤_Γ^(𝑖,𝑗),𝐱_Γ^{(𝑖,𝑗)𝐿(𝑖)Γ,𝑘}_𝑗=1^𝑀Γ,𝑘_𝑖=1, 𝑞_𝑘^(𝑖)(⋅∣𝐱_𝑘−1,𝐲_𝑘) and 𝑏_𝑘^(𝑖)(⋅∣𝐲_𝑘) which are the densities for

the particles corresponding to existing and birth cells respectively, and the cell template 𝐭_𝑘.

Step  1. Prediction for birth cells

   𝑖=0.

  for 𝑗=1,…,𝑀_Γ,𝑘

    𝑖∶=𝑖+1, 𝑟^(𝑖)_{𝑘∣𝑘−1}=𝑟^(𝑗)_Γ,𝑘.

    for   𝑙=1,…,𝐿^𝑗_Γ,𝑘

         𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1}∼𝑏_𝑘^(𝑗)(⋅∣𝐲_𝑘), 𝑤^(𝑖,𝑙)_{𝑘∣𝑘−1}=𝑝_Γ,𝑘𝐱^(𝑗,𝑙)_Γ,𝑘𝑏_𝑘^(𝑗)𝐱^(𝑗,𝑙)_Γ,𝑘∣𝐲_𝑘.

    end

     𝑤^(𝑖,𝑙)_{𝑘∣𝑘−1}=𝑤^(𝑖,𝑙)_{𝑘∣𝑘−1}∑^{𝐿(𝑗)Γ,𝑘}_𝑙=1𝑤^(𝑖,𝑙)_{𝑘∣𝑘−1}, for 𝑙=1,…,𝐿^𝑗_Γ,𝑘.

      end

Step  2. Prediction for existing cells

 for 𝑗=1,…,𝑀_𝑘−1

    𝑖∶=𝑖+1, 𝑟^(𝑖)_{𝑘∣𝑘−1}=𝑟∑^𝐿(𝑗)_𝑘−1^{(𝑗)𝑘−1}_𝑙=1𝑤^(𝑗,𝑙)_𝑘−1𝑝_𝑆,𝑘𝑥^(𝑗,𝑙)_𝑘−1.

    for   𝑙=1,…,𝐿^𝑗_𝑘−1

              𝐱^(𝑖,𝑙)_{𝑘∣𝑘−1}∼𝑞_𝑘^(𝑗)(⋅∣𝐱^(𝑗,𝑙)_𝑘−1,𝐲_𝑘),

          𝑤^(𝑖,𝑙)_{𝑘∣𝑘−1}=𝑤^(𝑗,𝑙)_𝑘−1𝑓_{𝑘∣𝑘−1}(𝐱^(𝑗,𝑙)_{𝑘∣𝑘−1}∣𝐱^(𝑗,𝑙)_𝑘−1)𝑝_𝑆,𝑘(𝐱^(𝑗,𝑙)_𝑘−1)𝑞_𝑘^(𝑗)𝐱^(𝑗,𝑙)_{𝑘∣𝑘−1}∣𝐱^(𝑗,𝑙)_𝑘−1,𝐲_𝑘.

    end

     𝑤^(𝑖,𝑙)_{𝑘∣𝑘−1}=𝑤^(𝑖,𝑙)_{𝑘∣𝑘−1}∑^{𝐿(𝑗)𝑘−1}_𝑙=1𝑤^(𝑖,𝑙)_{𝑘∣𝑘−1}, for 𝑙=1,…,𝐿^𝑗_𝑘−1.

 end

 𝑀_{𝑘∣𝑘−1}=𝑖.

Step  3. Update module

 for  𝑗=1,…,𝑀_{𝑘∣𝑘−1}

     ℓ_𝑘^(𝑗)=∑^{𝐿(𝑗)𝑘∣𝑘−1}_𝑙=1𝑤^(𝑗,𝑙)_{𝑘∣𝑘−1}𝑔_𝑘𝐲_𝑘∣𝐱^(𝑗,𝑙)_{𝑘∣𝑘−1}, 𝑟_𝑘^(𝑗)=𝑟^(𝑗)_{𝑘∣𝑘−1}ℓ_𝑘^(𝑗)1−𝑟^(𝑗)_{𝑘∣𝑘−1}+𝑟^(𝑗)_{𝑘∣𝑘−1}ℓ_𝑘^(𝑗).

     for   𝑙=1,…,𝐿^𝑗_{𝑘∣𝑘−1}

    𝐱_𝑘^(𝑗,𝑙)=𝐱^(𝑗,𝑙)_{𝑘∣𝑘−1}, 𝑤_𝑘^(𝑗,𝑙)=𝑤^(𝑗,𝑙)_{𝑘∣𝑘−1}𝑔_𝑘𝐲_𝑘∣𝐱^(𝑗,𝑙)_{𝑘∣𝑘−1}ℓ_𝑘^(𝑗)

  end

end

Step  4. Resampling, pruning and merging module.

Output  𝑟_𝑘^(𝑗),𝑤_𝑘^(𝑖,𝑗),𝑥_𝑘^{(𝑖,𝑗)𝐿𝑘}_𝑗=1^𝑀𝑘_𝑖=1.