  We present improved sampling complexity bounds for stable and robust sparse
recovery in compressed sensing. Our unified analysis based on l1 minimization
encompasses the case where (i) the measurements are block-structured samples in
order to reflect the structured acquisition that is often encountered in
applications; (ii) the signal has an arbitrary structured sparsity, by results
depending on its support S. Within this framework and under a random sign
assumption, the number of measurements needed by l1 minimization can be shown
to be of the same order than the one required by an oracle least-squares
estimator. Moreover, these bounds can be minimized by adapting the variable
density sampling to a given prior on the signal support and to the coherence of
the measurements. We illustrate both numerically and analytically that our
results can be successfully applied to recover Haar wavelet coefficients that
are sparse in levels from random Fourier measurements in dimension one and two,
which can be of particular interest in imaging problems. Finally, a preliminary
numerical investigation shows the potential of this theory for devising
adaptive sampling strategies in sparse polynomial approximation.
