An Adaptive Total Generalized Variation Model with Augmented Lagrangian Method for Image Denoising

Algorithm 1

Algorithm TGV 𝟐 ID-ADMM:  Second-order TGV-regularized image denoising with ADMM.

Input:  𝐮₀,𝑐.

(1) Initialize 𝐮⁰,𝐩⁰,𝐰⁰,𝐲⁰,𝐳⁰,𝝁⁰,𝝃⁰,𝜼⁰. Set 𝑘=0 and 𝛽₁,𝛽₂,𝛽₃>0.

(2)  while  stopping criterion is not satisfied,  do

(3)  Compute  𝐮^𝑘+1 according to (22);

(4)  Compute  𝐩^𝑘+1 according to (24) and (25);

(5)  Compute  𝐲^𝑘+1 according to (27);

(6)  Compute  𝐳^𝑘+1 according to (29);

(5)  if  (32) holds,  then

(6)  𝜆^𝑘+1=0 and 𝐰^𝑘+1=𝐮^𝑘+1+(𝝁^𝑘/𝛽₁);

(7)  else

(8)  Update 𝜆^𝑘+1 and 𝐰^𝑘+1 according to (34) and (31);

(9)  end  if

(10)   Update  𝝁^𝑘+1,𝝃^𝑘+1, and  𝜼^𝑘+1 according to (20);

(11)  𝑘=𝑘+1;

(12)  end  while

(13)  return   𝐮^𝑘+1.