The computational approaches to calculate normal distributions on the rotation group

Abstract

There are several kinds of probability distribution widely used in quantitative texture analysis. One of them is the normal (Gaussian) distribution. The main application of the normal distribution is the orientation distribution function and pole-figure approximation on the rotation group SO(3) and sphere S2accordingly. The calculation of the normal distribution is a complicated computational task. There are currently several methods for calculating the normal distribution. Each of these methods has its advantages and disadvantages. The classical method of calculation by Fourier series summation is effective enough only in the case of continuous texture approximation. In the case of sharp texture approximation, the analytical approach is more suitable and effective. These two calculation methods result in a continuous function. The other method allows a discrete orientation set to be obtained, corresponding to a random sample of normal distribution similar to experimental electron backscatter diffraction data. This algorithm represents a statistical simulation by the particularized Monte Carlo method. A short review of these computational approaches to the calculation of normal distributions on the rotation group is presented.