Pseudometrically Constrained Centroidal Voronoi Tessellations: Generating uniform antipodally symmetric points on the unit sphere with a novel acceleration strategy and its applications to Diffusion and 3D radial MRI

Purpose: The purpose of this work is to investigate the hypothesis that uniform sampling measurements that are endowed with antipodal symmetry play an important role when the raw data and image data are related through the Fourier relationship as in q-space diffusion MRI and 3D radial MRI. Currently, it is extremely challenging to generate large uniform antipodally symmetric point sets suitable for 3D radial MRI. A novel approach is proposed to solve this important and long-standing problem. Methods: The proposed method is based upon constrained centroidal Voronoi tessellations of the upper hemisphere with a novel pseudometric. Geometrically intuitive approach to tessellating the upper hemisphere is also proposed. Results: The average time complexity of the proposed centroidal tessellations was shown to be effectively on the order of the product of the number of iterations and the number of generators. For small sample size, the proposed method was comparable to the state-of-the-art iterative method in terms of the uniformity. For large sample size, in which the state-of-the-art method is infeasible, the reconstructed images from the proposed method has less streak and ringing artifact as compared to those of the commonly used methods. Conclusion: This work solved a long-standing problem on generating uniform sampling points for 3D radial MRI.Comment: 33 pages, 5 figure