  Reducing acquisition time is a crucial challenge for many imaging techniques.
Compressed Sensing (CS) theory offers an appealing framework to address this
issue since it provides theoretical guarantees on the reconstruction of sparse
signals by projection on a low dimensional linear subspace. In this paper, we
focus on a setting where the imaging device allows to sense a fixed set of
measurements. We first discuss the choice of an optimal sampling subspace
(smallest subset) allowing perfect reconstruction of sparse signals. Its
standard design relies on the random drawing of independent measurements. We
discuss how to select the drawing distribution and show that a mixed strategy
involving partial deterministic sampling and independent drawings can help
breaking the so-called "coherence barrier". Unfortunately, independent random
sampling is irrelevant for many acquisition devices owing to acquisition
constraints. To overcome this limitation, the notion of Variable Density
Samplers (VDS) is introduced and defined as a stochastic process with a
prescribed limit empirical measure. It encompasses samplers based on
independent measurements or continuous curves. The latter are crucial to extend
CS results to actual applications. Our main contribution lies in two original
continuous VDS. The first one relies on random walks over the acquisition space
whereas the second one is heuristically driven and rests on the approximate
solution of a Traveling Salesman Problem. Theoretical analysis and
retrospective CS simulations in magnetic resonance imaging highlight that the
TSP-based solution provides improved reconstructed images in terms of
signal-to-noise ratio compared to standard sampling schemes (spiral, radial, 3D
iid...).
