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A new model for image restoration and decomposition has been presented here. The proposed model applies the non-convex non-smooth regularisation and the Hilbert-Sobolev spaces of negative degree of differentiability to capture oscillatory patterns. The existence of a pseudosolution to the proposed model is proved. Moreover, two numerical algorithms for solving the minimisation problem are provided by applying the variable splitting and the penalty techniques. Finally, extended experiments on denoising, deblurring and decompositions of both real and synthetic images demonstrate the effectiveness and efficiency of the proposed numerical schemes.