Fractional-Order Total Variation Image Restoration Based on Primal-Dual Algorithm

Algorithm 1

Step 1. Choose 𝜏₀,𝜎₀>0 with 𝜏₀𝜎₀𝐿²≤1, 𝑥₀∈𝑅^𝑁, 𝑦₀∈𝑅^2𝑁, and 𝑥₀=𝑥₀.

Step 2. Update 𝑥_𝑡, 𝑦_𝑡, 𝑥_𝑡, 𝜃_𝑡, 𝜏_𝑡, 𝜎_𝑡 as follows:

𝑦_𝑡+1=argmin_{𝑦∈𝑅2𝑁}𝑦,−𝐴𝑥_𝑡𝑦+1+𝛿2𝜎_𝑡𝑦−𝑦_𝑡²₂𝑥,(A)_𝑡+1=argmin_{𝑥∈𝑅𝑁}𝜆2‖𝑥−𝑧‖²₂+𝐴𝑥,𝑦_𝑡+1+12𝜏_𝑡𝑥−𝑥_𝑡²₂𝜃,(B)_𝑡=11+2𝛾𝜏_𝑡,𝜏_𝑡+1=𝜃_𝑡𝜏_𝑡,𝜎_𝑡+1=𝜎_𝑡𝜃_𝑡,𝑥_𝑡+1=𝑥_𝑡+1+𝜃_𝑡𝑥_𝑡+1−𝑥_𝑡,