  Reducing acquisition time is of fundamental importance in various imaging
modalities. The concept of variable density sampling provides a nice framework
to achieve this. It was justified recently from a theoretical point of view in
the compressed sensing (CS) literature. Unfortunately, the sampling schemes
suggested by current CS theories may not be relevant since they do not take the
acquisition constraints into account (for example, continuity of the
acquisition trajectory in Magnetic Resonance Imaging - MRI). In this paper, we
propose a numerical method to perform variable density sampling with block
constraints. Our main contribution is to propose a new way to draw the blocks
in order to mimic CS strategies based on isolated measurements. The basic idea
is to minimize a tailored dissimilarity measure between a probability
distribution defined on the set of isolated measurements and a probability
distribution defined on a set of blocks of measurements. This problem turns out
to be convex and solvable in high dimension. Our second contribution is to
define an efficient minimization algorithm based on Nesterov's accelerated
gradient descent in metric spaces. We study carefully the choice of the metrics
and of the prox function. We show that the optimal choice may depend on the
type of blocks under consideration. Finally, we show that we can obtain better
MRI reconstruction results using our sampling schemes than standard strategies
such as equiangularly distributed radial lines.
