Image is one of the important media that cannot be separated from science, industry, and human daily life. Recently, many researchers, from the perspective of maximum a posterior (MAP) estimation, have extensively investigated the image restoration problem which is to estimate an original clear image from an observed blurry and noisy image, so that some suitable image restoration models are established by introducing their reasonable prior knowledge and then efficiently solved.
We proposed a new regularization method using the weighted hyper-Laplacian prior with overlapping group sparsity (OGS-WHL) of image gradient and reduced the image restoration problem under Cauchy noise to an non-convex and non-smooth optimization problem. Here we used the median-filtered version of observed image as an initial solution and penalized the out-of-range pixels such that gray levels of the resultant image lie on the boundary of the specific range.
The model proposed by Sciacchitano et al. [SJIS, 8 (2015) 1894–1922] corresponds to the special case of our model with K=1 and q=1, and median-filtering is out of their consideration. Mei et al. introduced the median-filtering and achieved better results by the alternating direction method of multipliers (ADMM) algorithm, but it still yields staircase artifacts since it only considers the local sparseness of image gradient. To compensate its drawback, Ding et al. [AMC, 341 (2019) 128–147] adopted the total variation with overlapping group sparsity to further depict the structural characteristics, making the undesirable artifacts reduced.
In this study, we adopted the hyper-Laplacian prior with q<1 at pixel level to better describe the heavy-tailed nature of image gradients and the weighting scheme by balancing the contributions of neighboring pixels to adaptively use those priors. Then, we proposed an ADMM-based algorithm to solve the non-convex and non-smooth optimization problem and proved that it would globally converge to a stationary point.
Our research result was published in "Journal of Scientific Computing" under the title of "Weighted Hyper-Laplacian Prior with Overlapping Group Sparsity for Image Restoration under Cauchy Noise"(https://doi.org/10.1007/s10915-021-01461-8).
