  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.
