Compressed sensing with structured sparsity and structured acquisition

Abstract

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 (radial lines for instance). Structured acquisition with blocks of measurements are easy to implement, and they give good reconstruction results in practice. However, very few results exist on the theoretical guarantees of CS reconstructions in this setting. In this paper, we fill the gap between CS theory and acquisitions made in practice. To this end, the key feature to consider is the structured sparsity of the signal to reconstruct. In this paper, we derive new CS results for structured acquisitions and signal satisfying a prior structured sparsity. The obtained results are RIPless, in the sense that they do not hold for any s-sparse vector, but for sparse vectors with a given support S. Our results are thus support-dependent, and they offer the possibility for flexible assumptions on the structure of S. Moreover, our results are also 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.