Thresholding and compressed sensing in combination with both wavelet and shearlet transforms have been very successful in inpainting tasks. Recent results have demonstrated that shearlets outperform wavelets in the problem of image inpainting. In this paper, we provide a general framework for universal shearlet systems in high dimensions. This theoretical framework is used to analyze the recovery of missing data via l^1 minimization in an abstract model situation. In addition, we set up a particular model inspired by seismic data and a box mask to model missing data. Finally, the results of numerical experiments comparing various inpainting methods are presented.

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