An alternating direction method for frame-based image deblurring with balanced regularization

In this paper, we propose an efficient algorithm for solving a balanced approach in frame-based image deblurring. The balanced approach is usually formulated as a minimization problem involving an ℓ₂ data-fidelity term, an ℓ₁ regularizer on sparsity of frame coefficients, and a penalty on distance of sparse frame coefficients to the canonical frame coefficients. The balanced approach bridges synthesis-based and analysis-based approaches. Our algorithm is based on a variable splitting strategy and the classical alternating direction method (ADM). This paper shows how the proposed algorithm can be applied to solve the balanced approach efficiently. More precisely, a regularized version of the Hessian matrix of the ℓ₂ data-fidelity term is involved, and by exploiting fast tight frame and circular structure of the observation matrix, the matrix can perform efficiently for image deblurring application. Convergence of the proposed algorithm is guaranteed by the existing ADM theory. Numerical simulations illustrate the efficiency of our proposed algorithm in frame-based image deblurring.