An Algorithm for Variable Density Sampling with Block-Constrained Acquisition

Reducing acquisition time is of fundamental importance in various imaging modalities. Variable density sampling (VDS) provides an appealing framework for addressing this issue. 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 (for example, continuity of the acquisition trajectory in magnetic resonance imaging (MRI)) into account. In this paper, we propose a numerical method to perform variable density sampling with block constraints. Our main contribution is 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 carefully study 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 with standard strategies such as equiangularly distributed radial lines.