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Reconstruction from fourier measurements using compactly supported shearlets

We study the reconstruction problem of a compactly supported function from its Fourier coefficients using a compactly supported shearlet system. We assume that only finitely many Fourier samples are accessible and the reconstruction can only be constructed using finitely many shearlets. Our main result shows that stable recovery of the signal is possible provided the number of measurements is up to a constant equal to the number of shearlet elements that are used for the approximation. Numerical experiments with MR data are presented which emphasize the advantages of shearlets compared to other systems such as wavelets.