Diffusion sampling schemes: A generalized methodology with nongeometric criteria

PurposeThe aim of this paper is to show that geometrical criteria for designing multishell q$$ q $$-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 MethodsWe 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$$ M $$ ranging between 1 and the maximum number of shells S$$ S $$. All the M$$ M $$-tuples share the same weight omega M$$ {\omega}_M $$. 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. ResultsFor 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. ConclusionThe 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.