An adaptive ADMM technique for ill-posed image deblurring with non-smooth regularization

Abstract This paper introduces a novel variational model for image deblurring that incorporates spatially adaptive, non-smooth regularization. The proposed formulation combines harmonic regularization with a sparsity-promoting L 1 / L 2 gradient-based term, enabling an effective trade-off between data fidelity and smoothness. This design facilitates adaptive diffusion—preserving sharp edges while suppressing noise and smoothing homogeneous regions. To solve the resulting optimization problem, we develop an efficient ADMM-based algorithm equipped with a Hybrid Conjugate Gradient method featuring Adaptive Preconditioning and Anderson Acceleration (HCG-APAA), a semi-smooth Newton solver, and a fast shrinkage-thresholding scheme. Extensive numerical experiments demonstrate that the proposed approach outperforms several state-of-the-art deblurring methods in both quantitative metrics and visual quality.