Improved Crosstalk Reduction on Multiview 3D Display by Using BILS Algorithm

Algorithm 1

Crosstalk reduction algorithm.

Input: The upper triangular matrix 𝐑∈ℝ^ℎ×ℎ with positive diagonal entries, the vector

 𝐛=𝐛_𝑗∈ℤ^ℎ, the lower bound vector 𝐥=𝐥_𝑗∈ℤ^ℎ, the upper bound vector

 u = 𝐮_𝐣∈ℤ^ℎ, the initial hyper-ellipsoid bound 𝛽.

Output: The solution 𝐳̂∈ℤ^ℎ to the BILS Problem (18).

 Step  1. (Initialization) Set 𝑘∶=ℎ, 𝑇_𝑘∶=0,

   Compute 𝑠_𝑘∶=𝑏_𝑘/𝑟_𝑘𝑘, 𝑧_𝑘∶=𝑠_𝑘,

    𝑧_𝑘∶=max(𝑧_𝑘,𝑙_𝑘), 𝑧_𝑘∶=min(𝑧_𝑘,𝑢_𝑘),

    𝜌_𝑘∶=(𝑠_𝑘−𝑧_𝑘)𝑟_𝑘𝑘, Δ_𝑘∶=−sign(𝜌_𝑘).

 Step  2. Compute 𝑇∶=𝑇_𝑘+𝜌²_𝑘,

    if  𝑇<𝛽 and 𝑘≠1, then

    Go to Step  3

    else

    Go to Step  4

    end if

 Step  3. Set 𝑇_𝑘−1∶=𝑇, 𝑘∶=𝑘−1

   Compute 𝑠_𝑘∶=𝑏_𝑘−∑^ℎ_𝑚=𝑘+1𝑟_𝑘𝑚𝑧_𝑚/𝑟_𝑘𝑘,

    𝑧_𝑘∶=𝑠_𝑘, 𝑧_𝑘∶=max(𝑧_𝑘,𝑙_𝑘), 𝑧_𝑘∶=min(𝑧_𝑘,𝑢_𝑘),

    𝜌_𝑘∶=(𝑠_𝑘−𝑧_𝑘)𝑟_𝑘𝑘, Δ_𝑘∶=−sign(𝜌_𝑘).

   Go to Step  2

 Step  4. If  𝑇<𝛽, then

    Set 𝛽∶=𝑇, 𝐳̂∶=𝐳, 𝑘∶=𝑘+1,

    else if  𝑘=ℎ, then

    Terminate

    else

     𝑘∶=𝑘+1

     if  𝜌_𝑘=0, then

      𝐾∶=𝑘

      repeat

       if  𝑘=ℎ, then

         Terminate

       end if

       𝑘∶=𝑘+1

      until  𝜌_𝑘≠0

      if  𝑘≠ℎ, then

       𝑘∶=𝐾

      end if

     end if

    end if

 Step  5. If  𝑧_𝑘+Δ_𝑘>𝑢_𝑘, then

    Set Δ_𝑘∶=−1

    Compute 𝑧_𝑘∶=𝑧_𝑘+Δ_𝑘

    else if  𝑧_𝑘+Δ_𝑘<𝑙_𝑘, then

    Set Δ_𝑘∶=1

    Compute 𝑧_𝑘∶=𝑧_𝑘+Δ_𝑘

    else

    Compute 𝑧_𝑘∶=𝑧_𝑘+Δ_𝑘, Δ_𝑘∶=−Δ_𝑘−sign(Δ_𝑘)

    end if

   Compute 𝜌_𝑘∶=𝑠_𝑘−𝑧_𝑘𝑟_𝑘𝑘

   Go to Step  2.