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Reducing acquisition time is a crucial challenge for many imaging techniques.Compressed Sensing (CS) theory offers an appealing framework to address thisissue since it provides theoretical guarantees on the reconstruction of sparsesignals by projection on a low dimensional linear subspace. In this paper, wefocus on a setting where the imaging device allows to sense a fixed set ofmeasurements. We first discuss the choice of an optimal sampling subspace(smallest subset) allowing perfect reconstruction of sparse signals. Itsstandard design relies on the random drawing of independent measurements. Wediscuss how to select the drawing distribution and show that a mixed strategyinvolving partial deterministic sampling and independent drawings can helpbreaking the so-called "coherence barrier". Unfortunately, independent randomsampling is irrelevant for many acquisition devices owing to acquisitionconstraints. To overcome this limitation, the notion of Variable DensitySamplers (VDS) is introduced and defined as a stochastic process with aprescribed limit empirical measure. It encompasses samplers based onindependent measurements or continuous curves. The latter are crucial to extendCS results to actual applications. Our main contribution lies in two originalcontinuous VDS. The first one relies on random walks over the acquisition spacewhereas the second one is heuristically driven and rests on the approximatesolution of a Traveling Salesman Problem. Theoretical analysis andretrospective CS simulations in magnetic resonance imaging highlight that theTSP-based solution provides improved reconstructed images in terms ofsignal-to-noise ratio compared to standard sampling schemes (spiral, radial, 3Diid...).

Variable density sampling with continuous trajectories. Application to MRI