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 three‐dimensional radial MRI

Abstract

The purpose of this work is to investigate the hypothesis that uniform sampling measurements that are endowed with antipodal symmetry play an important role in image quality when the raw data and image data are related through the Fourier relationship. Currently, it is extremely challenging to generate large and uniform antipodally symmetric point sets suitable for three-dimensional radial MRI. A novel approach is proposed to solve this long-standing problem in a unique and optimal way. The proposed method is based on constrained centroidal Voronoi tessellations of the upper hemisphere with a novel pseudometric. The time complexity of the proposed tessellations was shown to be effectively linear, i.e., on the order of the number of sampling measurements. For small sample size, the proposed method was comparable with the state-of-the-art method (a direct iterative minimization of the electrostatic potential energy of a collection of electrons antipodal-symmetrically distributed on the unit sphere) in terms of the sampling 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 artifacts, when compared with those of the commonly used methods. This work proposed a unique and optimal approach to solving a long-standing problem in generating uniform sampling points for three-dimensional radial MRI.