Abstract
We study the normal distribution on the rotation group SO(3). If we take as the normal distribution on the rotation group the distribution defined by the central limit theorem in Parthasarathy (1964) rather than the distribution with density analogous to the normal distribution in Eucledian space, then its density will be different from the usual () exp(−(x − m) 2/2σ2) one. Nevertheless, many properties of this distribution will be analogous to the normal distribution in the Eucledian space. It is possible to obtain explicit expressions for density of normal distribution only for special cases. One of these cases is the circular normal distribution.The connection of the circular normal distribution SO(3) group with the fundamental solution of the corresponding diffusion equation is shown. It is proved that convolution of two circular normal distributions is again a distribution of the same type. Some projections of the normal distribution are obtained. These projections coincide with a wrapped normal distribution on the unit circle and with the Perrin distribution on the two‐dimensional sphere. In the general case, the normal distribution on SO(3) can be found numerically. Some algorithms for numerical computations are given. These investigations were motivated by the orientation distribution function reproduction problem described in the Appendix.
Highlights
Gaussian or normally distributed random variables, Gaussian processes and systems play important roles in the theory of probability and mathematical statistics
We will define a normal distribution on SO(3) based on Parthasarathy’s work, which satisfies the central limit theorem on SO(3)
We have found formulae for l-2 and 3, but it seems possible to reach arbitrary accuracy calculating normal distribution with three different parameters only by numerical computations
Summary
Gaussian or normally distributed random variables, Gaussian processes and systems play important roles in the theory of probability and mathematical statistics. We will define a normal distribution on SO(3) based on Parthasarathy’s work, which satisfies the central limit theorem on SO(3). Consider a normal distribution in canonical form on the SO(3) group with three parameters, fS Z Z Z Tg(l) ctinnTlmn(g)dg- exp{B,),. - Consider examples of fulfilling the conditions of the central limit theorem for the rotation group SO(3) when convergencefn(g) f(g) takes place in norms L2(SO(3)) and C(SO(3)) as n. Example Let fn(g) be the circular normal distribution (15) with parameter en2" Consider rotations around the OZ axis, or g3 (t). From Theorem under the condition that n En2 < C- const, we get that parameters of the limit normal distribution, f(g)- limn_ f,n(g), are the following: aij O, except a33 2 cr 2. Note that the function (19) was obtained in (Parthasarathy (1967)) from the central limit theorem on the circle SO(2).
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