  Compressed sensing is a theory which guarantees the exact recovery of sparse
signals from a small number of linear projections. The sampling schemes
suggested by current compressed sensing theories are often of little practical
relevance since they cannot be implemented on real acquisition systems. In this
paper, we study a new random sampling approach that consists in projecting the
signal over blocks of sensing vectors. A typical example is the case of blocks
made of horizontal lines in the 2D Fourier plane. We provide theoretical
results on the number of blocks that are required for exact sparse signal
reconstruction. This number depends on two properties named intra and
inter-support block coherence. We then show through a series of examples
including Gaussian measurements, isolated measurements or blocks in
time-frequency bases, that the main result is sharp in the sense that the
minimum amount of blocks necessary to reconstruct sparse signals cannot be
improved up to a multiplicative logarithmic factor. The proposed results
provide a good insight on the possibilities and limits of block compressed
sensing in imaging devices such as magnetic resonance imaging,
radio-interferometry or ultra-sound imaging.
