New Regularization Models for Image Denoising with a Spatially Dependent Regularization Parameter

Algorithm 2

The alternating minimization method for solving Model 2.

Input:  𝑓, 𝛼, 𝑀, and 𝑟.

Output:  𝑢^𝑘.

Initialization:  𝑔⁰₁=0 and 𝑔⁰₂=0.

  while not converged do

   Step  1. Given 𝑔₁^𝑘−1 and 𝑔₂^𝑘−1, computing 𝑢^𝑘 by solving

     min_𝑢∈𝑋𝐸₂𝑢,𝑔₁^𝑘−1,𝑔₂^𝑘−1=12𝑢−𝑓²_𝑋+∑_{1≤𝑖,𝑗≤𝑛}𝐾𝑔₁^𝑘−1∘𝑔₁^𝑘−1_𝑖,𝑗|∇𝑢|_𝑖,𝑗

            +∑_{1≤𝑖,𝑗≤𝑛}𝐾𝑔₂^𝑘−1∘𝑔₂^𝑘−1_𝑖,𝑗|𝐻𝑢|_𝑖,𝑗.       (a)

   Step  2. Given 𝑢^𝑘, computing 𝑔^𝑘₁ and 𝑔^𝑘₂ by solving

    min_{𝑔1,𝑔2∈𝑋}𝐸₂𝑢^𝑘,𝑔₁,𝑔₂=𝛼𝑔₁−𝑀𝛼⋅𝟏²_𝑋+𝑔₂−𝑀𝛼⋅𝟏²_𝑋

            +∑_{1≤𝑖,𝑗≤𝑛}𝑔₁²_𝑖,𝑗𝐾∇𝑢^𝑘_𝑖,𝑗+𝑔₂²_𝑖,𝑗𝐾𝐻𝑢^𝑘_𝑖,𝑗.  (b)

  end while