Fast iteratively reweighted least squares algorithms for analysis-based sparse reconstruction.

Area of Science:

• Signal Processing
• Image Reconstruction
• Applied Mathematics

Background:

• Sparsity reconstruction is crucial for efficient data representation and signal recovery.
• Existing methods often struggle with computational cost and handling diverse sparsity structures.
• Total Variation (TV) regularization is effective but computationally intensive.

Purpose of the Study:

• To introduce a novel, efficient algorithm for analysis-based sparsity reconstruction.
• To address generalized sparsity problems using structured sparsity and TV regularization.
• To accelerate the iterative reweighted least squares (IRLS) framework for practical applications.

Main Methods:

• The algorithm employs a structured sparsity regularization with an orthogonal basis and total variation (TV) regularization.
• It is based on the iterative reweighted least squares (IRLS) framework, accelerated by a preconditioned conjugate gradient method.
• The method leverages the diagonally dominant nature of the Hessian matrix in many applications.

Main Results:

• The proposed algorithm demonstrates linear convergence, comparable to traditional IRLS methods.
• A specifically devised preconditioner significantly reduces computational cost, enabling large-scale problem solutions.
• The method effectively handles standard sparsity, group sparsity, overlapping group sparsity, and TV-based problems.

Conclusions:

• The novel algorithm offers superior performance in accuracy and computational efficiency for sparsity reconstruction.
• It provides a versatile and accelerated solution for various sparsity-related challenges.
• Successful application in compressed sensing magnetic resonance imaging validates its practical utility.