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

Algorithm 3

The alternating minimization method for solving (41).

Input:  𝑓, 𝑔₁^𝑘−1, 𝑔₂^𝑘−1, 𝑟, 𝛽, and 𝐿.

Output:  𝑢^𝑘, 𝑝^𝑘, 𝑞^𝑘, 𝜆^𝑘₁, and 𝜆^𝑘₂.

Initialization:  𝑝^𝑘,0=𝑝^𝑘−1, 𝑞^𝑘,0=𝑞^𝑘−1, 𝜆₁^𝑘,0=𝜆₁^𝑘−1, and 𝜆₂^𝑘,0=𝜆₂^𝑘−1.

  while not converged and 𝑙<𝐿  do

   Step  1. Given 𝑝^{𝑘,𝑙−1}, 𝑞^{𝑘,𝑙−1}, 𝜆₁^{𝑘,𝑙−1}, and 𝜆₂^{𝑘,𝑙−1}, computing 𝑢^𝑘,𝑙 by

      𝑢^𝑘,𝑙=ℱ⁻¹ℱ𝑓−div𝜆₁^{𝑘,𝑙−1}+𝛽𝑝^{𝑘,𝑙−1}+𝐻^∗𝜆₂^{𝑘,𝑙−1}+𝛽𝑞^{𝑘,𝑙−1}1−𝛽ℱ(Δ)+𝛽ℱ(𝐻^∗𝐻),        (a)

   where ℱ and ℱ⁻¹ denote the discrete Fourier transform and the inverse discrete Fourier

   transform, respectively, and Fourier transforms of operators Δ and 𝐻^∗𝐻 are regarded as the

   transforms of their corresponding convolution kernels.

   Step  2. Given 𝑢^𝑘,𝑙, 𝜆₁^{𝑘,𝑙−1}, and 𝜆₂^{𝑘,𝑙−1}, update 𝑝^𝑘,𝑙 and 𝑞^𝑘,𝑙 by the two-dimensional shrinkage

      𝑝^𝑘,𝑙_𝑖,𝑗=max∇𝑢^𝑘,𝑙−𝜆₁^{𝑘,𝑙−1}𝛽_𝑖,𝑗−𝐾𝑔₁^𝑘−1∘𝑔₁^𝑘−1_𝑖,𝑗𝛽,0∇𝑢^𝑘,𝑙−𝜆₁^{𝑘,𝑙−1}/𝛽_𝑖,𝑗∇𝑢^𝑘,𝑙−𝜆₁^{𝑘,𝑙−1}/𝛽_𝑖,𝑗,      (b)

      𝑞^𝑘,𝑙_𝑖,𝑗=max𝐻𝑢^𝑘,𝑙−𝜆₂^{𝑘,𝑙−1}𝛽_𝑖,𝑗−𝐾𝑔₂^𝑘−1∘𝑔₂^𝑘−1_𝑖,𝑗𝛽,0𝐻𝑢^𝑘,𝑙−𝜆₂^{𝑘,𝑙−1}/𝛽_𝑖,𝑗𝐻𝑢^𝑘,𝑙−𝜆₂^{𝑘,𝑙−1}/𝛽_𝑖,𝑗,    (c)

   respectively, where 0⋅(0/0)=0 is assumed.

   Step  3. Given 𝑢^𝑘,𝑙, 𝑝^𝑘,𝑙, and 𝑞^𝑘,𝑙, update 𝜆₁^𝑘,𝑙 and 𝜆₂^𝑘,𝑙 by

         𝜆₁^𝑘,𝑙=𝜆₁^{𝑘,𝑙−1}+𝛽𝑝^𝑘,𝑙−∇𝑢^𝑘,𝑙,               (d)

         𝜆₂^𝑘,𝑙=𝜆₂^{𝑘,𝑙−1}+𝛽𝑞^𝑘,𝑙−𝐻𝑢^𝑘,𝑙,               (e)

   respectively.

end while