NORMAL DISTRIBUTION ON THE ROTATION GROUP SO ( 3 )

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 (1 / exp(-(xm)2/2cr2) 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.


INTRODUCTION
Gaussian or normally distributed random variables, Gaussian processes and systems play important roles in the theory of probability and mathematical statistics. This is due to the existence of the central limit theorem. This work concerns the normal distribution on the rotation group of three-dimensional space SO(3) and on the two-dimensional sphere S 2. There are many abstract works concerning probability measures (for example see Heyer (1971)), but there are few studies concerning probability distributions on specific groups, including SO(3). Mardia (1972) investigated the probability distribution on the abelian SO(2) group (unit circle) in detail and compiled some of the socalled "normal" distributions on the sphere S 2. In his book (Mardia (1972)) the normal distribution on SO(2) group is given with the probability density f(x) av/ E exp (x + 27rn) 2 } 20.2 (1) This distribution appears as a superposition of distributions on R with the equivalence relation xl _= x2 (mod 270. The normal distribution on SO(2) (1) satisfies the central limit theorem. Moreover the normal distribution (1) can be obtained as the solution to the diffusion equation Of(x, t) (x, t) Ot 2 Ox 2 (2) with the initial condition in which the whole mass is concentrated at x 0: f(x,O)= 6(x), and the boundary condition f(-r, t)=f(Tr, t), thus f(x) crv'-n= exp Another normal distribution on SO(2) was suggested by Mizes (see Mardia (1972)), where I0 (k) is the modified Bessel function of order 0. The normal distribution of Mizes approximates (1) and is easy to compute but does not satisfy the central limit theorem on SO(2) and is not a solution of the diffusion equation.
We will define a normal distribution on SO(3) based on Parthasarathy's work, which satisfies the central limit theorem on SO(3). Another way to define a normal distribution on SO(3) is as a solution of the corresponding diffusion equation. In this work we consider the fundamental solution of the diffusion equation on the sphere S n-1 in Rn. At n--4 a circular normal distribution on SO(3) is obtained. At n 3 the normal distribution coincides with the one obtained by Perrin (1972) and Roberts and Ursell (1960) during their study of Brownian motion and random walk on S 2 in R3. A normal distribution on S 2 in the present work is defined as projections corresponding to a normal distribution on SO(3). Some properties of these distributions are given. These properties of the normal distributions on SO(3) and S 2 are analogous to properties on Rn.
Only special cases of normal distributions on SO(3) and S 2 can be calculated analytically; the rest can be found numerically. In the present work some recipes for calculating normal distributions are given. How series convergence depending on parameters of these distributions is also considered and some estimates are given.
Appendix presents an application of these normal distributions on SO(3) and S 2 to the solution of the orientation distribution function reproduction problem in the quantitative texture analysis that motivated this investigation.
where aij is a nonnegative symmetric matrix and ai are real constants.
We remark that parameters ai (i 1,2, 3) correspond to the "center" of the normal distribution.

/=0
Make a change of variables 27rv, q-exp(-e2), then we get q-l 0 92 (V) f(t) 27r sin(t/2) v In the paper by Khatri and Mardia (1977) the Mises Fisher matrix distribution was proposed. The expression for that distribution written as a series expansion in terms of spherical harmonics and related functions (according to Watson (1983) such distributions are of some practical interest) depending on the orientation distance (in our notations) is exp{Scos t) -It(S) I(+l)(S) sin(l + 1/2)t f(t) Io(S) I(S) =o Io(S) II(S) sin(t/2) (16) here S is parameter. Correspondence between functions (15) and (16) is discussed by Matthies et al. (1988). In the case where a ij-O, except a33-2o-2, we get a degenerate normal distribution on SO (3), This function is the same as the normal distribution on the circle SO(2) or the wrapped normal distribution (Mardia (1972)).
THEOREM 3 Convolution of two circular normal distributions on the SO(3) group is again a circular normal distribution.
Proof Without losing generality it is possible to consider the case when the center of one distribution is e and another is go.
We take into account here that C,(e2)C,(e)-(2/+ 1)C,(el 2 + If we have two arbitrary rotations g and g2, then because of the invariance of integration on the group we have the proof of the theorem. 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 -. Here e is the identity matrix corresponding to the rotation g {0, 0, 0}.

< Cconst, we get
From Theorem under the condition that n E n that parameters of the limit normal distribution, f(g)-limn_ f,n(g), are the following: aij O, except a33 2 cr 2. Therefore the limit distribution is a normal distribution of rotations around the OZ axis or the normal distribution on the circle.
Example 2 Let fn(g) be the normal distribution (14) of canonical form with two parameters. In that case g (g.), where gl gn2__ exp{_Un 2 2},n gn33 exp{-2Un2}, and all other gj.-0. From here we get that if nu --u2 no 2 2, then the limit distribution will be normal with two parameters. In particular, when u -0 -e we get the circular normal distribution (15). THO,EM 4 The convergence fk f (g) as k -cx where f(g) is the normal distribution on SO(3) is assured if and only iffor eigenvalues of matrices of Fourier coefficients of distributions fk(g) hold." where are some real constants.
From Eq. (18) For the normal distribution on the circle SO(2). Note that the function (19) was obtained in (Parthasarathy (1967)) from the central limit theorem on the circle SO(2). We obtained it from the central limit theorem on SO(3) as the degenerate case (17). From Eq. (18) at n--3 we get for the normal distribution on the sphere S 2 in R3. At n--4 from Eq. (18) where P/(cos/3) are Legendre polynomials.
We remark that the circular normal distribution on S 2 (26) was obtained by Perrin (1972) from a study of Brownian motion. We suggest the following definition of a normal distribution on S 2. DEFINITION The normal distribution S 2 is defined by (26) for f(g) from (26) at aij-0 when 0 or j 3. In particular, the circular normal distribution on $2 can be expressed by (27).
The distribution on the sphere S 2 (27) approximates the well-known Fisher distribution. This is discussed in detail by Watson (1983 Explicit expressions for the matrix and its elements are given by (12) and (13).
Since BI is a symmetric matrix, it can be represented in the form B1-(U1) -AU, where U is orthogonal, AI is a diagonal and its ele- 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. We give an algorithm calculating the matrix exp Bt and estimates of the spectrum of matrix Bt. Decomposition  We remark that the coefficients (Cl)mn with both m and n even depend only on Rt and with both rn and n odd depend only on St. Moreover, (Ct)m, 0 when rn + n is odd. If Bt is symmetric about the origin then the matrix Ct exp Bt is symmetric about the origin, too. For the matrix Ct--exp Bt the following symmetry relationships hold: (Cl)mn-(Cl)nm-(Cl)2l+l-m, 2l+l-n--(Cl)2l+l-n, 2l+l-m.
It is possible to see this by direct computation of the product of arbitrary symmetrical matrices and the following formula for a matrix exponent: D D 2 expD--E++-.I +'", where D is an arbitrary matrix.
It can be seen that the matrices R and St have a lot of zeros. By means of elementary transformations (Qt)m, of transformation of lines and columns they can be evaluated to more convenient form for computation. Matrix (Qt)mn is the following: (Ql)mn E-(El)ram -(El)nn -a t-(El)mn --(El)nm, where Et is an identity matrix of order 2l + 1, (Et) O is a square matrix of order 2l + 1, with ones on the intersection of ith line and jth column and all other elements zeros. If At is some arbitrary matrix then (Qt)mnAt can be obtained from At by means of transposition of the mth and nth lines, At(Qt)mn can be obtained from At by means of transposition of the mth and nth columns. Moreover, (Qt)mn(Qt)mn El. THEOREM 6 Computation of exp Bt where matrix Bt (12) with matrix elements (13) has five diagonals of order 21 + is equivalent to computation of exp rt and exp st where the matrices rt and st have orders + and respectively and they are symmetrical tridiagonal. Estimates and asymptotics with oo of spectrum Bt and matrix elements of exp Bt Here we will give estimates of Jacobi's matrices spectra as a function of normal distribution parameters.
Proof of the lemma is analogous to that of Lemma 1.
We remark that matrix Bt is singular when two parameters out of three (al, a22, a33) are equal to zero. In all other cases Bl is nonsingular. Matrix rt is singular when Bl is singular. Matrix Sl is singular only in the case all --a22 --a33 --0. It is interesting that in the nonsingular case det(Bt) det(rt)det(st).
COROLLARY 2 As far as the spectral norm of matrix is equal to absolute value of the maximal eigenvalue, then GII exp nzl[ exp Wt.
From here we can get the following estimates for elements of matrix C: ICnl exp WI.
It is known that the square of a matrix's eucledian norm is equal to the sum of singular values, or Ct exp{Bt).
Remark All estimates become exact in the circular case.
Remark 2 Here we have considered band matrices, with only three diagonal nonzero. However, all results, almost without any changes, can be transferred to band matrices with any amount of diagonals with only the constraint that they must have zero elements with odd sum of indices.
Estimates in the circular case In computations using the circular normal distribution on SO(3) and S 2, (15) and (27), a question arises about the number of series terms required to compute these functions to a given accuracy. Expressions for the remainders of (15) and (27) are given below as functions of the parameter e in two metrics.
In the end of this section we present analytic expressions that approximate circular normal distribution on the group and sphere. Bunge (1982) suggested the following normal distribution on SO (3): where C is a constant that can be defined from (36), cost -(Tr(g) 1) and g is a rotation matrix. Matthies et al. (1988) showed that (35) and (15) approximate each other for small values of the parameter e.

CONCLUSION
We have considered the definition of the normal distributions on the nonabelian SO(3) group, its projection to the sphere S 2, methods of computation, and some properties and application to a texture analysis problem. These results can be generalized to groups SO(n) and spheres S n-1 n> 4.
It was shown that the circular normal distribution (15) satisfies Parthasarathy's central limit theorem, is a solution of the diffusion equation and that the convolution of such distributions is again a distribution of the same type.
However, many questions require further study. A normal but not circular distribution was studied inadequately. So far as this distribution can be obtained only numerically, it would be useful to have simple approximation formulas for computations. A theorem about the convolution of two normal distributions with arbitrary parameters is not obtained and it is not clear if it has a place in that case. There is no central limit theorem for the sphere S 2 for the Perrin distribution.

APPENDIX: APPLICATION OF NORMAL DISTRIBUTION TO THE SOLUTION OF A TEXTURE ANALYSIS PROBLEM
Preliminary notes Texture is a collection of monocrystalline orientations that compose the polycrystalline sample. It has an essential influence on behavior and properties of a polycrystal. This is the reason why a full quantitative description of texture and estimations of properties of samples with different texture are important.
Consider a polycrystalline sample. A coordinate system KA is connected with the sample; a coordinate system K is connected with a crystallite in that sample. Orientation of a crystallite relative to the polycrystal can be described with the rotation g K KA. A quantitatively more detailed description of texture is possible through an orientation distribution function f(g). It is important to note that "orientation distribution function" is a common but not precise term. Under an orientation distribution function one assumes density of orientation distribution or orientation density function. Let V volume of sample and d V(g) be a volume of monocrystals in the sample with orientation g. DEFINITION Orientation distribution function is the function f(g), g E SO(3), that satisfies condition where g-{c,/3, ,}, 0 _< c, ,), < 27r, 0 <_/3 _< 7r are Euler angles of rotation and dg-dc sin/3d/3d, is an invariant measure on SO(3).
Volume of sample is supposed to be equal to the joint volume of all monocrystals and only the orientation of crystallites rather than their spatial position is taken into account. It follows from the definition that f(g) is nonnegative and, moreover, f(g)dg-1.
Because of the symmetry of crystallite for rotations g and gjg where g. E GB is a crystallite symmetry group, the following relation must be fulfilled: and if there is sample symmetry, where gA, E G.4 is a sample symmetry group. If there is some function f(g), it can be symmetrized as follows to obey properties (38) and (39): j=l k=l Experiemntal information about the orientation distribution function can be obtained if all sample crystallites volumes and orientations can be measured, but such information can only be obtained with the destruction of the sample and is very expensive. The number of crystallites in a sample is tens of thousands (in large grain samples) to tens of millions (in small grain samples). Another source of information about the orientation distribution function is pole figures, denoted as Ph;(Y) which are projections of the orientation distribution function.
These functions are measured in diffraction experiments (X-ray or neutron); for a detailed description see Bunge (1982). If y is a unit vector that describes some direction in a sample coordinate system and hi is a unit vector that describes some direction in a crystallite coordinate system, d V(hil[ y) is a volume of crystallites whose crystallographic direction hi is parallel to the direction y of the sample, hi and y can be described by cartesian coordinates (hil, hi2, hi3) and (yl, y2, Y3) or by spherical coordinates (0, q)) and (X,r/) with 0 _< , r/< 2r, 0 _< 0, X < r, respectively. DEFbaTION A pole figure is a function Phi(Y), Y E $2 that satisfies condition [4], dV(hi y) v Phi(Y)dy, here dy sin X dx d is a measure on the sphere 8 2. The volume of the sample here is also assumed to be equal to the joint volume of all monocrystals. Only the orientation or the crystallites rather than their spatial positions are taken into account. It follows from the definition that Phi(Y) is a nonnegative and Phi(Y)dy-1.
Because of the symmetry of the crystallites for rotations g and gjg where gj E G is a crystallite symmetry group, the following relation must be fulfilled: and if there is sample symmetry, where gAk E GA is the sample symmetry group. If there is some function Phi(Y), to obey properties (40) and (41)  Phi(Y) --/3(hi, Y,g)f(g). The next formula presents the explicit appearance of the operator P(hi, y, g) (Matthies (1979) /=1 m=-l n=-l An addition theorem is known (see Vilenkin (1968)) for the spherical function, Yln (hi) Z Ylm (y) zlmn (g)" From (42) with expansion (43) using the addition theorem it is possible to obtain a formula that connects the expansion coefficients of the pole figures with the orientation distribution function coefficients: 47r Phi(Y)---Z Z 21+1 l=1 m=-I n=-I + (-1) czm,, yzm(y) yff (hi). (44) This sum depends only on terms with even indices 1. If we write f(g) in the form f (g) f (g) +f(g), here jT(g)corresponds to the part of the expansion (43) with even indices l, f(g) corresponds to the part of the expansion (43) with odd indices l. The operator/5(hi, y, g) acts in such a way that (hi, y, g)(g) O.
In other words, it has nonzero kernel. This is a consequence of the symmetry of the experiment with respect to directions y and -y and also hi and -hi. From (44) it follows that there is no information in the pole figures about the odd part of the orientation distribution function j(g). That fundamental fact does not depend on the method used to compute the orientation distribution function but depends only on the information the pole figures are identical. That means that the problem of reproducing the orientation distribution function from measured pole figures is ill-posed (there exist no unique solution).
Therefore, only f(g) can be produced from the experimentally measured pole figures.
The most commonly used method to solve the problem is Bunge-Roe method. In 1965, almost  In the Bunge-Roe method, coefficients with odd are assumed to be zero. This is the reason why a computed orientation distribution function has negative values and false maxima (ghost phenomena). This was noticed by many scientists but was interpreted as computational errors (errors because of calculation of coefficients C/''n, errors of summation and truncation of the series). Matthies (1979) showed that the ghost phenomena and negative values are in fact due to the incompleteness of the information contained in the pole figures. At present several improved methods to find an orientation distribution function are known.
Our idea for the solution of that problem is the following. It happens that the circular normal distribution on the SO(3) group and the sphere S 2 are connected to each other by the integral relation (42). We suggest approximating the experimentally measured pole figures by a linear combination of normal distributions on S 2 by means of fitting variance like parameters and weights. This will be a nonlinear problem about unknown parameters. The orientation distribution function is taken as a linear combination of normal distributions with parameters and weights found. As the odd part of the orientation distribution function we assume odd part of the sum obtained in this way. Examples of such an approach can be found in Bukharova et al. (1988), Savyolova (1993), Nikolayev et al. (1992). It must be mentioned that also several other methods how to approximate the odd part have been suggested. For a survey see, e.g., Wagner and Dahms (1991).