  Recently, compressed sensing techniques in combination with both wavelet and
directional representation systems have been very effectively applied to the
problem of image inpainting. However, a mathematical analysis of these
techniques which reveals the underlying geometrical content is completely
missing. In this paper, we provide the first comprehensive analysis in the
continuum domain utilizing the novel concept of clustered sparsity, which
besides leading to asymptotic error bounds also makes the superior behavior of
directional representation systems over wavelets precise. First, we propose an
abstract model for problems of data recovery and derive error bounds for two
different recovery schemes, namely l_1 minimization and thresholding. Second,
we set up a particular microlocal model for an image governed by edges inspired
by seismic data as well as a particular mask to model the missing data, namely
a linear singularity masked by a horizontal strip. Applying the abstract
estimate in the case of wavelets and of shearlets we prove that -- provided the
size of the missing part is asymptotically to the size of the analyzing
functions -- asymptotically precise inpainting can be obtained for this model.
Finally, we show that shearlets can fill strictly larger gaps than wavelets in
this model.
