Tensor metrics and charged containers for 3D Q-space sample distribution.

This paper extends Jones' popular electrostatic repulsion based algorithm for distribution of single-shell Q-space samples in two fundamental ways. The first alleviates the single-shell requirement enabling full Q-space sampling. Such an extension is not immediately obvious since it requires distributing samples evenly in 3 dimensions. The extension is as elegant as it is simple: Add a container volume of the desired shape having a constant charge density and a total charge equal to the negative of the sum of the moving point charges. Results for spherical and cubic charge containers are given. The second extension concerns the way distances between sample point are measured. The Q-space samples represent orientation, rather than direction and it would seem appropriate to use a metric that reflects this fact, e.g. a tensor metric. To this end we present a means to employ a generalized metric in the optimization. Minimizing the energy will result in a 3-dimensional distribution of point charges that is uniform in the terms of the specified metric. The radically different distributions generated using different metrics pinpoints a fundamental question: Is there an inherent optimal metric for Q-space sampling? Our work provides a versatile tool to explore the role of different metrics and we believe it will be an important contribution to further the continuing debate and research on the matter.