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Several strategies have been applied for the recovery of the missing parts in an image, with recovery performance depending significantly on the image type and the geometry of missing data. To provide a deeper understanding of such image restoration problem, King et al. recently introduced a rigorous multiscale analysis framework and proved that a shearlet based inpainting approach outperforms methods based on more conventional multiscale representations when missing data are line singularities. In this paper, we extend and improve the analysis of the inpainting problem to the more realistic and more challenging setting of images containing curvilinear singularities. We derive inpainting performance guarantees showing that exact image recovery is achieved if the size of the missing singularity is smaller than the size of the structure elements of appropriate functional representations of the image. Our proof relies critically on the microlocal and sparsity properties of the shearlet representation. [ABSTRACT FROM AUTHOR]