Frame-based image deblurring with balanced-compound regularization

This paper presents a novel balanced-compound regularization approach for solving the frame-based image deblurring. The proposed balanced-compound regularization employs two different frames as synthesis and analysis operators, and it is formulated as a minimization problem involving an ℓ₂ data-fidelity term, an ℓ₁ regularizer on sparsity of synthesis frame coefficients, an ℓ₁ regularizer on sparsity of analysis frame operator, and a penalty on distance of sparse synthesis frame coefficients to the range of the frame operator. Thus the proposed regularization consists of a synthesis-analysis compound regularizer and a balanced regularizer. Then the balanced-compound optimal problem is solved based on a variable splitting strategy and the classical alternating direction method of multiplier (ADMM). Numerical simulations show that the proposed balanced-compound approach can achieve less coefficient estimated error than the hybrid synthesis-analysis approach under comparable qualities in image deblurring problem. This improvement is due to the added balanced term. Moreover, by exploiting the related fast tight Parseval frames and the special structure of the observation matrix, the regularized Hessian matrix can perform efficiently for the frame-based image deblurring.