A Unified Framework for Sparse Reconstruction via Preconditioning and Nonconvex Regularization

Compressed Sensing (CS) is an effective technique to recover sparse signals with fewer samples than what is required by the classical Shannon Nyquist sampling theorem. The sensing matrix, sparsifying transform, and sparse recovery algorithm are three key factors for accurate reconstruction in CS. Traditional CS uses a convex $l_{1}$-norm sparse regularizer which may lead to biased estimates and is suboptimal in promoting sparsity. Another challenge is the design of incoherent sensing matrices which is crucial for accurate sparse recovery. In this paper, we propose a novel CS framework combining a preconditioned sensing matrix and nonconvex regularization for improved sparse signal recovery. First, we formulate an optimization problem to find an incoherent sensing matrix via a preconditioner. It allows for a direct computation of the optimal preconditioner and preconditioned sensing matrix, simultaneously. Secondly, we consider a generalized CS model for signal recovery based on the incoherent sensing matrix and a nonconvex $\ell _{1/2}$-norm regularizer. We then derive an Alternating Direction Method of Multipliers (ADMM) algorithm to solve this nonconvex optimization problem. The proposed model is applied to sparse-view Computed Tomography (CT) reconstruction with highly-undersampled and noisy data. Qualitative and quantitative results show significantly better image reconstruction using the preconditioned sensing matrix and $\ell _{1/2}$ regularizer, compared to methods without preconditioning and using the $\ell _{1}$ regularizer.