Diffusion sampling schemes: A generalized methodology with nongeometric criteria

Abstract

Purpose: The aim of this paper is to show that geometrical criteria for designing multishell $q$ -space sampling procedures do not necessarily translate into reconstruction matrices with high figures of merit commonly used in the compressed sensing theory. In addition, we show that a well-known method for visiting k-space in radial three-dimensional acquisitions, namely, the Spiral Phyllotaxis, is a competitive initialization for the optimization of our nonconvex objective function.

Theory and methods: We propose the gradient design method WISH (WeIghting SHells) which uses an objective function that accounts for weighted distances between gradients within M-tuples of consecutive shells, with $M$ ranging between 1 and the maximum number of shells $S$ . All the $M$ -tuples share the same weight ${\omega }_M$ . The objective function is optimized for a sample of these weights, using Spiral Phyllotaxis as initialization. State-of-the-art General Electrostatic Energy Minimization (GEEM) and Spherical Codes (SC) were used for comparison. For the three methods, reconstruction matrices of the attenuation signal using MAP-MRI were tested using figures of merit borrowed from the Compressed Sensing theory (namely, Restricted Isometry Property -RIP- and Coherence); we also tested the gradient design using a geometric criterion based on Voronoi cells.

Results: For RIP and Coherence, WISH got better results in at least one combination of weights, whilst the criterion based on Voronoi cells showed an unrelated pattern.

Conclusion: The versatility provided by WISH is supported by better results. Optimization in the weight parameter space is likely to provide additional improvements. For a practical design with an intermediate number of gradients, our results recommend to carry out the methodology here used to determine the appropriate gradient table.

Keywords: Voronoi; coherence; diffusion MRI; q-space sampling.