An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems

Area of Science:

• Computational Mathematics
• Image Processing
• Optimization Theory

Background:

• Ill-posed linear inverse problems (IPLIP) are common in scientific imaging.
• Standard optimization tools struggle with the high dimensionality and nonsmoothness of IPLIP regularizers.
• Basis pursuit denoising is a key challenge within IPLIP, especially for image recovery.

Purpose of the Study:

• To introduce a novel, fast, and efficient algorithm for solving a specific class of constrained IPLIP.
• To address the limitations of existing optimization methods for large-scale, nonsmooth regularization problems.
• To provide a robust solution for various imaging applications, including deconvolution and compressive sensing.

Main Methods:

• The proposed algorithm is an instance of the alternating direction method of multipliers (ADMM).
• It belongs to the family of augmented Lagrangian methods, incorporating convex regularization and constraints.
• Sufficient convergence conditions for ADMM are proven to be satisfied by the new algorithm.

Main Results:

• The algorithm demonstrates high efficiency and speed for image recovery tasks.
• It effectively handles diverse imaging IPLIP, such as deconvolution and MRI reconstruction.
• Experimental results on benchmark problems confirm its competitive performance against state-of-the-art methods.

Conclusions:

• The developed algorithm offers a powerful new tool for solving constrained IPLIP in image recovery.
• Its foundation in ADMM and augmented Lagrangian methods ensures convergence and efficiency.
• The method is versatile, applicable to various regularization techniques like total-variation and frame-based methods.