A fixed-point augmented Lagrangian method for total variation minimization problems

• We presented an efficient fixed-point augmented Lagrangian method (FPALM) for total variation minimization problems.
• We proposed fast alternative algorithms to solve the minimization problems.
• The proof for the convergence of the FPALM was provided under some mild assumptions.
• The wide applicability of the proposed algorithms were demonstrated on several well-known image denoing models.

In this paper, we propose a fixed-point augmented Lagrangian method (FPALM) for general convex problems arising in image processing. We can easily obtain the alternating minimization algorithm (AMA) referred to [1] from the proposed FPALM. The proof for the convergence of the FPALM is provided under some mild assumptions. We present two kinds of first-order augmented Lagrangian schemes and show their connections to first-order primal–dual algorithms [2]. Furthermore, we apply an acceleration rule to both the FPALM and AMA to achieve better convergence rates. Numerical examples on different image denosing models including the ROF model, the vectorial TVmodel, high order models and the TV-L1L1 model are provided to demonstrate the efficiency of the proposed algorithms.