Pseudometrically Constrained Centroidal Voronoi Tessellations

The central slice of the reconstructed volume by the method of Wong and Roos (Left) and the proposed method (Right). the average of the percent relative error per pixel (with the brain region) of the method of Wong and Roos and of the proposed method with respect to the ground truth are respectively 30.7% and 21.3%. The reduction in percent of relative error is almost 10%.

A: 888 deterministic generators taken from the analytically exact spiral scheme and their Voronoi regions on the upper hemisphere. 888 final (B) real and (C) virtual generators from the proposed method. D: 1776 antipodally symmetric points on the sphere obtained from (B) and (C). [Ref. 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. Magnetic Resonance in Medicine 2013. DOI: 10.1002/mrm.24715 ]

Uniformity or lack thereof of the sampling points (on the unit sphere) in k-space (e.g., 3D radial MRI technique) or q-space (e.g., diffusion MRI or diffusion tensor imaging) has long been known to play an important role in the quality of the reconstructed image or of diffusion- or tensor-derived quantities. Uniformity of the sampling points on the unit sphere comes in at least two flavors; the most common flavor is in the sense as described in the Thomson problem and the other flavor imposes antipodal symmetry on the point set.

My hypothesis is that the second flavor, which is relatively unknown to 3D radial MRI practitioners, is better than the first flavor for 3D radial MRI acquisitions. Even though the hypothesis is simple, the technical tools needed to perform the test of this hypothesis were not available and had to be designed and built. The journey was long but it has been very rewarding intellectually---learning some very beautiful geometric and computational concepts and techniques while establishing several interesting results.

Besides the paper, you may find other supplementary materials included in the links here helpful. I have presented key results to different audiences with different emphases. The first one here was presented to undergraduate math students at my alma mater. The most recent talk was presented at the CIMB seminar.

POINT SETS AVAILABLE FOR ACADEMIC RESEARCH

Some point sets have been computed based on the proposed method and the size of these point sets ranges from 6,7,8,...,999, 1000,1500, 2000, 2500, ..., 19500, 20000, 22000, 24000, ..., 58000, 60000 to some very specific numbers, 3991, 5866, 8101, 33664, 67328.

If you would like to use these point sets for your (non-commercial) academic research, please send your request by email, which can be found here.

Here is an example in Mathematica (or see PDF version of the same file) for creating the graphics shown in the above figure. The main point for sharing this Mathematica file is to show how simple it is to use the point sets and the associated properties of each point such as the vertices of its Voronoi region, area and circumference of the Voronoi region, which have been stored conveniently in Java and can retrieved easily from Mathematica or Matlab.