Single-image super-resolution with total generalised variation and Shearlet regularisations

In this study, the authors proposed a novel regularisation model for resolution enhancement of clean or noisy single image based on the total generalised variation (TGV) and Shearlet transform. The proposed model has two main contributions. Firstly, different from models with total variation regularisation, which assume that images consist of piecewise-constant areas, the author´s TGV-based model is aware of higher-order smoothness, thus eliminates the staircase-like artefacts. Secondly, various image features including edges and fine details can be preserved by their model. This is nature since the Shearlets mathematically provide an optimally sparse approximation for the class of piecewise-smooth functions with rich geometric information. Moreover, to solve the proposed model, an efficient numerical scheme is explicitly developed based on the Nesterov´s algorithm. A series of numerical experiments validate the effectiveness of the proposed method.