To reduce blur in noisy images, regularized image restoration methods have been proposed that use nonquadratic regularizers (like l1 regularization or total-variation) that suppress noise while preserving edges in the image. Most of these methods assume a circulant blur (periodic convolution with a blurring kernel) that can lead to wraparound artifacts along the boundaries of the image due to the implied periodicity of the circulant model. Using a noncirculant model could prevent these artifacts at the cost of increased computational complexity. In this paper, we propose to use a circulant blur model combined with a masking operator that prevents wraparound artifacts. The resulting model is noncirculant, so we propose an efficient algorithm using variable splitting and augmented Lagrangian (AL) strategies. Our variable splitting scheme, when combined with the AL framework and alternating minimization, leads to simple linear systems that can be solved noniteratively using fast Fourier transforms (FFTs), eliminating the need for more expensive conjugate gradient-type solvers. The proposed method can also efficiently tackle a variety of convex regularizers, including edge-preserving (e.g., total-variation) and sparsity promoting (e.g., l1-norm) regularizers. Simulation results show fast convergence of the proposed method, along with improved image quality at the boundaries where the circulant model is inaccurate.

Image restoration is a well studied problem and there are several proposed methods for deblurring and denoising. Recently, there is increasing interest in iterative schemes that employ non-quadratic regularizers, especially edge-preserving like Total Variation (TV) and sparsity promoting like l1regularization. Most methods make simplifying assumptions concerning the system model and the most common one is the use of a circulant blurring model because it facilitates using the FFT. In this work we focus on a more realistic non-circulant blurring model and apply existing algorithms for image restoration with non-quadratic regularization, tailored to work with our non-circulant model.