An Analysis of Block Sampling Strategies in Compressed Sensing

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 of projecting the signal over blocks of sensing vectors. A typical example is the case of blocks made of horizontal lines in the 2-D Fourier plane. We provide the theoretical results on the number of blocks that are sufficient for exact sparse signal reconstruction. This number depends on two properties named intra- and inter-support block coherence. We then show that our bounds coincide with the best so far results in a series of examples, including Gaussian measurements or isolated measurements. We also show that the result is sharp when used with specific blocks in time-frequency bases, in the sense that the minimum required amount of blocks 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.