A new method of constructing a grid in the space of 3D rotations and its applications to texture analysis

Abstract

In computational work, data sets must often be represented on the surface of a sphere or inside a ball, requiring uniform grids. We construct a new volume-preserving projection between a cube and the set of unit quaternions. The projection consists of two steps: an equal-volume mapping from the cube to the unit ball, followed by an inverse generalized Lambert projection to either of the two unit quaternion hemispheres. The new projection provides a one-to-one mapping between a grid in the cube and elements of the special orthogonal group SO(3), i.e., 3D rotations. We provide connections to other rotation representation schemes, including the Rodrigues–Frank vector and the homochoric parameterizations, and illustrate the new mapping through example applications relevant to texture analysis.