Simultaneous inpainting and denoising by directional global three-part decomposition: connecting variational and Fourier domain-based image processing

We consider the very challenging task of restoring images (i) that have a large number of missing pixels, (ii) whose existing pixels are corrupted by noise, and (iii) that ideally contain both cartoon and texture elements. The combination of these three properties makes this inverse problem a very difficult one. The solution proposed in this manuscript is based on directional global three-part decomposition (DG3PD) (Thai, Gottschlich. 2016 EURASIP. J. Image Video Process. 2016, 1–20 (doi:10.1186/s13640-015-0097-y)) with a directional total variation norm, directional G-norm and ℓ∞-norm in the curvelet domain as key ingredients of the model. Image decomposition by DG3PD enables a decoupled inpainting and denoising of the cartoon and texture components. A comparison with existing approaches for inpainting and denoising shows the advantages of the proposed method. Moreover, we regard the image restoration problem from the viewpoint of a Bayesian framework and we discuss the connections between the proposed solution by function space and related image representation by harmonic analysis and pyramid decomposition.