  Compressed Sensing (CS) is an appealing framework for applications such as
Magnetic Resonance Imaging (MRI). However, up-to-date, the sensing schemes
suggested by CS theories are made of random isolated measurements, which are
usually incompatible with the physics of acquisition. To reflect the physical
constraints of the imaging device, we introduce the notion of blocks of
measurements: the sensing scheme is not a set of isolated measurements anymore,
but a set of groups of measurements which may represent any arbitrary shape
(parallel or radial lines for instance). Structured acquisition with blocks of
measurements are easy to implement, and provide good reconstruction results in
practice. However, very few results exist on the theoretical guarantees of CS
reconstructions in this setting. In this paper, we derive new CS results for
structured acquisitions and signals satisfying a prior structured sparsity. The
obtained results provide a recovery probability of sparse vectors that
explicitly depends on their support. Our results are thus support-dependent and
offer the possibility for flexible assumptions on the sparsity structure.
Moreover, the results are drawing-dependent, since we highlight an explicit
dependency between the probability of reconstructing a sparse vector and the
way of choosing the blocks of measurements. Numerical simulations show that the
proposed theory is faithful to experimental observations.
