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Acta Applicandae Mathematicae 53: 59–87, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. 59 The Central Limit Theorem in Texture Component Fit Methods HELMUT SCHAEBEN Mathematical Geology and Computer Sciences in Geology, Bergakademie Freiberg University of Technology, Germany. e-mail: [email protected] DMITRY I. NIKOLAYEV Joint Institute for Nuclear Research, Laboratory for Neutron Physics, Dubna, Moscow Region, Russia 141980 (Received: 29 March 1996; in final form: 10 July 1997) Abstract. This note is concerned with implications of spherical analogues of the central limit theorem of probability in Euclidean space. In particular, it is concerned with the presumption that the analogy holds in terms of interpreting a special spherical limiting distribution, the hyperspherical Brownian distribution, as the distribution of the resultant rotation composed by a sequence of successive random rotations under similarly mild assumptions as applied in the central limit theorem for Euclidean space. This interpretation has been stressed at several instances to indicate the superiority of the spherical Brownian distribution for applications in texture component fit methods. Here it is shown, however, that this presumption is false. Thus, an explicit correspondence of the Brownian form of texture components and processes causing preferred crystallographic orientations cannot be inferred from a central limit type argument. Mathematics Subject Classifications (1991): 20H15, 60F05. Key words: texture goniometry, texture component fit methods, central limit theorems, Parthasarathytype normal distribution, spherical random walk, spherical diffusion, Brownian motion distribution. 1. Introduction The question addressed in this contribution is what might be a suitable choice of a model function f for an orientation density function defined on SO(3) or H 3 ≡ S+3 , respectively, based on philosophical and computational considerations. In Euclidean space of course f would typically take the form of a multivariate normal density. On SO(3) there are at least two natural choices for f . The first choice is the distribution of Brownian motion and the second choice is the von Mises–Fisher matrix distribution (cf. Schaeben, 1996c). The usual justification for the matrix von Mises–Fisher distribution on SO(3) or the Bingham distribution on H 3 , respectively, is that it forms a useful statistical family for inference. In particular, it forms an exponential family for the parameters VTEX(JK) PIPS No. 160034 (acapkap:mathfam) v.1.15 ACAP1262.tex; 19/08/1998; 9:16; p.1 60 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV of the distribution (cf. Beran, 1979; Hetherington, 1981; Watson, 1983, pp. 80–88). The directional von Mises–Fisher distribution on the sphere is infinitely divisible and is the distribution of stopping times for Brownian motion in Euclidean space when stopped at a unit sphere (Kent, 1995). An interesting note on Fisher’s derivation of the exp(κ cos ω) distribution is presented by Watson (1982). Brownian motion distribution on SO(3) or H 3 , respectively, has a natural probabilistic justification in terms of the sum of a large number of small independent steps. It also arises as the special case of rotational invariance of the Parthasarathytype normal (P-type normal) distribution on SO(3) derived from the characterization in terms of infinitely divisibility. Both these arguments are variants of the central limit theorem. Of course, as there is no concept of division by a positive number in SO(3) or H 3 , respectively, it is not possible to apply the central limit theorem directly. The Brownian motion distribution also provides the fundamental solution of an (isotropic) spherical diffusion equation. However, the spherical Brownian distribution is not generally the limiting distribution of the resultant rotation composed by a sequence of successive random rotations under similarly mild assumptions as applied in the Euclidean central limit theorem. This false presumption led Matthies et al. (1988) to the interpretation of the Brownian distribution as the distinguished distribution of a simple spherical analogue of the central limit theorem which in turn was stressed in texture analysis to indicate the superiority of the Brownian distribution for applications in texture component fit methods. Thus it is emphasized here that an explicit correspondence of the Brownian form of texture components and processes causing preferred crystallographic orientations cannot be inferred from an oversimplified spherical central limit theorem. In fact, any explicit correspondence of this kind has not yet been established. 2. The Inversion Problem of Texture Goniometry Mathematical texture analysis used to be primarily concerned with the resolution of the tomographic inverse problem corresponding to the fundamental projection equation of diffraction texture goniometry as follows. Let f ∈ L2 (G), G ⊂ SO(3), be a square integrable function defined on an appropriate subgroup G of the group SO(3) of orientations, i.e. proper rotations. Let x, r ∈ S 2 ⊂ R3 , then the integral operator Px : L2 (G) 7→ L2 (S 2 ) is defined as Z 1 f (g) dg = Px (r). (1) Px [f (g)](r) = (Px f )(r) = 2π {g∈G|x=gr} The functions Px are referred to as (hyper)spherical X-ray transforms of f with respect to x. Obviously, P−x (r) = Px (−r); however, generally Px (r) 6= Px (−r). For any crystal symmetry let GB ⊂ SO(3) denote the point group of proper rotations associated with the crystal symmetry class GB . Then the orientations ACAP1262.tex; 19/08/1998; 9:16; p.2 61 THE CENTRAL LIMIT THEOREM gBj g, gBj ∈ GB , are physically indistinguishable with respect to the coordinate system KA fixed to the sample, and it holds f (g) = f (gBj ◦ g), gBj ∈ GB . (2) Furthermore, for normal scattering (Friedel’s law) Equation (1) then reads Z 1 X 1 f (g) dg P̃x (r) = 2π {g∈G|x=gBj ◦gr} #G̃B (3) gBj ∈G̃B where G̃B denotes the Laue group corresponding to GB and summation is thus over all directions gB−1j x which are symmetrically equivalent with respect to G̃B . Thus, for a crystallographic orientation density function f ∈ L2 (G), its corresponding pole density function P˜h of the crystal form h = {hm | m = 1, . . . , Mh } = {gB−1j x | gBj ∈ G̃B } ⊂ S 2 with multiplicity Mh corresponding to symmetrically equivalent lattice planes {(hkl)m | m = 1, . . . , Mh } ⊂ Z3 is defined for r ∈ S 2 as P̃h (r) = (P̃h f )(r) = Mh 1 X (Phm f )(r) Mh m=1 Mh /2 1 X ({Phm + P−hm }f )(r) = Mh m=1 (4) such that Z 1 P̃h (r) ds(r) = 1 4π S 2 when 1 2π 2 Z 0 2π Z π Z π f (ϕ, θ, ω) sin2 (ω/2) dω sin θ dθ dϕ = 1 0 (5) 0 where the point q ∈ S+3 ⊂ R4 with spherical coordinates (ϕ, θ, ω/2), 0 6 ϕ < 2π, 0 6 θ 6 π, 0 6 ω 6 π represents the rotation through the angle ω about the axis with colatitude θ and longitude ϕ. It should be noted that in the standard references (Bunge, 1969, 1982; Matthies et al., 1987, 1988) the normalization of an orientation density function is with respect to Euler angles α, β, γ and reads Z Z 2π Z π Z 2π 1 f (g) dg = f (α, β, γ ) sin β dα dβ dγ = 1. 8π 2 G 0 0 0 The operator P̃h is referred to as the pole figure projection operator. The spherical function P̃h is referred to as a crystallographic pole density function. Since any G̃ contains the operation of inversion as an element of symmetry, it always induces summation for +x and −x implying P̃x (r) = P̃x (−r), (6) ACAP1262.tex; 19/08/1998; 9:16; p.3 62 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV i.e. that P̃x is an even function defined on S 2 . Therefore a pole density function may actually be thought of as a real function defined on the upper (lower) hemisphere P̃h (r): S+2 7→ R+ . Pole density functions are experimentally accessible and can be discretely sampled by diffraction (X-ray, neutron) with a texture goniometer. Since an orientation density function is generally neither even nor odd the inverse tomographic problem of texture goniometry – to determine an orientation density function f from pole density functions P̃h according to Equation (4) – does not possess a unique solution, which is known as the ‘ghost problem’ in the community of texture analysts. To resolve the problem some additional mathematical modeling assumptions are required. The orientation density function f may be thought of as being defined on SO(3). A proper rotation may be represented as point of S 3 ⊂ R4 by virtue of its Rodrigues parameters (cf. Altmann, 1986), whereby antipodal points represent the same rotation. Thus, an orientation density function may also be thought of as an even density function on S 3 or actually as a density function defined on the three-dimensional projective hyperplane H 3 ≡ S+3 ⊂ R4 . 3. Texture Component Fit Methods In the past quite a few contributions on ‘texture component fit’ methods, more precisely ‘major pole figure component fit’ methods, have been published (Savelova, 1984; 1989; 1993; 1995; Bukharova et al., 1988; Helming and Eschner, 1990; Nikolayev et al., 1992; Bucharova and Savyolova, 1993; Eschner, 1993; 1994; 1995; Helming, 1995). The common basic idea is to approximate one or a few given pole density functions of several crystal forms by the superposition of a few spherical mathematical model functions of known simple form referred to as ‘pole figure components’, each of which is also known to be the pole figure projection, (Equation (1)) of some simple density function defined on SO(3) or S+3 ⊂ R4 , respectively, i.e. of some mathematical model orientation density function referred to as ‘texture component’ (cf. Schaeben, 1996a,b). Resolution of the tomographic inverse problem corresponding to the fundamental projection equation (4) of texture goniometry by pole figure component fit methods starts with the choice of mathematical model functions K4 (ω(g0−1 g); ρ) representing a texture component in the orientation space such that Px [K4 (ω(g0−1 g); ρ)](r) = K3 (η(r, g0−1 x); ρ) (7) represents the corresponding pole figure component on the sphere. Next, the experimental pole density function is approximated by a superposition of pole density ‘component’ functions K Mh 1 XX αk K3 (rrk,m ; ρk ) (K small), P̃ˆ h (r) = Mh k=1 m=1 (8) ACAP1262.tex; 19/08/1998; 9:16; p.4 63 THE CENTRAL LIMIT THEOREM with parameters αk , rk,m , ρk , k = 1, . . . , K, to be fit such that P̃ˆ h (rl ) ≈ P̃h (rl ) = ιl , l = 1, . . . , L (9) in some sense to be specified. Then fˆ(g) = K X αk K4 (ω(ggk−1 ); ρ) (10) k=1 provides an exploratory analysis of the inverse projection problem. As an example, a texture component (in orientation space) may be thought of as of unimodal rotationally invariant von Mises–Fisher form fvMF (g; g0 , κ) = CM (κ) exp(κ cos ω), (11) with the angle ω = ω(g0 ◦g −1 ) = 12 (tr(M(g0 ◦g −1 ))−1) ∈ [0, π ] of the composed rotation g0 ◦ g −1 referred to as the orientation distance of g0 and g; then κ κ Px [f (g; g0 , κ)](r) = CM (κ)I0 (1 + cos η) exp (cos η − 1) , (12) 2 2 where Iν denotes the modified Bessel function and η = η(r, g0−1 x) = arccos(r g0−1 x). It should be noted that the distribution (12) is not of von Mises–Fisher form; its parameters κ and g0−1 x have to be fitted to the experimental pole density data. Thus both the given pole density functions and an unknown orientation density function are expanded into a ‘very finite’ and not necessarily orthogonal series of a few spherical and corresponding hyperspherical functions, respectively, each representing a major component of preferred crystallographic orientation. Since the pattern of preferred orientation observed in normal diffraction pole figures is heuristically explained by a few major pole figure (spherical) components of say L2 (S+2 ) in a first step, and then represented by the corresponding (hyperspherical) preferred orientation components of say L2 (S+3 ) in a second step, this procedure is much more an approach to model the observed pattern, generalizing the concept of ‘ideal crystal orientation (Ideallagen)’ by Grewen and Wassermann, than a method of completely analysing a texture given in terms of discretely sampled pole density functions. Just from recalling that the number of components is to some extent arbitrary, this statement should be obvious. In exactly the same way, the goodness-of-fit, i.e. the acceptance of a fit as sufficiently good, is a matter of subjective judgement and thus to some extent arbitrary; in practice it will depend on the real-world problem to be resolved. As usual, the more parameters control the shape of each individual spherical function representing a component, the more flexible and versatile it is to represent an evidently major component well, but the more cumbersome is the mathematical procedure to fit all parameters. Without going into details of this aspect of the various methods, instances of mathematically ill-posed problems are anticipated ACAP1262.tex; 19/08/1998; 9:16; p.5 64 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV that may require practical resolution by user-driven, i.e. subjective and gradually arbitrary decisions. Component fit methods may provide a concise and instructive representation of simple patterns of preferred crystallographic orientation; if the complexity of the pattern requires a large number of components for a sufficiently good fit, the method generalizes to a finite series expansion into non-orthogonal model functions, and gives eventually rise to a column-rank deficient system of linear equations (Schaeben, 1996b). The major practical problem of component fit methods remains which functions are appropriate to represent texture components, and this problem may well depend on the process of texture formation. From a mathematical point of view there are several families of valid candidate functions to be used to represent components. Initially, they may be ranked only according to the tractability of the mathematical and/or numerical problems involved in the corresponding fit procedure. With some experience they may be appraised by their performance in practical applications, e.g., spherical densities with long tails as the generalized wrapped Cauchy (referred to as ‘Lorentzian’ standard orientation distribution in the texture community) may prove less appropriate for modeling purposes. A family of functions which could be related to the physical process(es) of generation and development of texture would be a favorite candidate for obvious reasons: In this case the parameters should correspond to some features of the process and vice versa. The central limit theorem of probability apparently suggests such a relationship. 4. Spherical Brownian Motion Distribution, Parthasarathy-Type Normal Distribution on SO(3), Spherical Diffusion and Gauss–Weierstrass Kernel 4.1. SPHERICAL RANDOM WALKS The Brownian motion distribution on S 3 or S+3 ≡ H 3 , respectively, has a natural probabilistic justification in terms of the sequence of a large number of small independent steps. This argument is a variant of the central limit theorem. Of course as there is no concept of division by a positive number in S 3 nor H 3 , respectively, it is not possible to apply the central limit theorem directly. An isotropic random walk on the sphere is considered in which the steps are geodesic arcs of random length (whose distribution may vary from step to step) and random direction (with the same distribution at every step). The distribution of the displacement (with respect to a given starting point) after n steps is determined in terms of the distribution of the n step-lengths by means of a n-fold operation involving the directional distribution (cf. Bingham, 1972). In physical terms the problem reads as follows (cf. Roberts and Winch, 1984). The initial orientation of a rigid body is given. That body receives a sequence of n right-handed rotations, each rotation being through an angle (whose distribution ACAP1262.tex; 19/08/1998; 9:16; p.6 65 THE CENTRAL LIMIT THEOREM may vary from step to step) about an axis of rotation (with the same distribution at every step). The current rotational configuration of the body is defined from the standard configuration (reference state fixed in space) by angles such that the corresponding rotation would carry the standard configuration of the body directly into the current configuration. The sequence of random rotations will thus be represented by motions on the sphere. What is the probability distribution for the orientation of the body after these n steps? If the step-length distribution is not concentrated on ±1, then (i) the distribution of the position of the random walk on the sphere after n steps tends to the uniform distribution on the sphere as n → ∞; (ii) the distribution of the cosine of the displacement after n steps of the random walk converges to the corresponding distribution on [−1, 1], i.e. the distribution of cos 6 (x, e) where e denotes a fixed radius vector with end-point on S k and x a random radius vector whose end-point is distributed uniformly on S k (cf. Bingham, 1972). The distribution of the cosine of the displacement of the special random walk of N steps, each of deterministic step length tN −1/2 , converges to the Brownian motion distribution (cf. Bingham, 1972). The circular Brownian distribution B1 (ϕ) is the wrapped normal distribution (cf. Mardia, 1972; Hartman and Watson, 1974). The (rotationally invariant) spherical Brownian distribution is defined for −π 6 ϕ 6 π, 0 6 θ 6 π as (cf. Hartman and Watson, 1974; Watson, 1983) ∞ 1 X (2l + 1) exp[−l(l + 1)κ]Pl (cos θ) B2 (ϕ, θ; κ) = 4π l=0 (13) with the usual Legendre polynomials Pl for S 2 ⊂ R3 ; the (rotationally invariant) hyperspherical Brownian distribution is defined for −π 6 ϕ 6 π, 0 6 θ 6 π, 0 6 ψ 6 π with the Legendre polynomials Pl,3 for S 3 ⊂ R4 as (cf. Hartman and Watson, 1974; Watson, 1983) B3 (ϕ, θ, ψ; κ) = = ∞ 1 X (l + 1)2 exp[−l(l + 2)] Pl,3 (cos ψ) 2π 2 l=0 ∞ 1 X (l + 1) exp[−l(l + 2)] Cl(1) (cos ψ) 2π 2 l=0 ∞ 1 X = (l + 1) exp[−l(l + 2)] Ul (cos ψ) 2π 2 l=0 ∞ sin[(l + 1)ψ] 1 X (l + 1) exp[−l(l + 2)] = . 2 2π l=0 sin ψ (14) The last equation is an application of the identity for Gegenbauer polynomials Cl(1) (cos ψ) = Ul (cos ψ), where Ul (cos ψ) denotes the Chebyshev polynomial of the second kind. ACAP1262.tex; 19/08/1998; 9:16; p.7 66 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV Following an argument introduced by Arnold (1941) (cf. Watson, 1983) the corresponding even function B̃3 ∈ L2 (S+3 ) on the projective hyperplane H 3 ≡ S+3 ⊂ R4 is represented by 1 B̃3 (ϕ, θ, ψ; κ) = [B3 (ϕ, θ, ψ; κ) + B3 (ϕ, θ, π − ψ; κ)] 2 ∞ 1 X = (l + 1)2 exp[−l(l + 2)κ]Pl,3 (cos ψ) 2π 2 l=0(2) ∞ 1 X = (2l + 1)2 exp[−4l(l + 1)κ]P2l,3 (cos ψ) 2π 2 l=0 = ∞ 1 X (2l + 1)2 exp[−4l(l + 1)κ]P2l,3 (cos ψ) 2π 2 l=0 = ∞ 1 X sin[(2l + 1)ψ] . (2l + 1) exp[−4l(l + 1)κ] 2 2π l=0 sin ψ (15) Since the point with hyperspherical coordinates (ϕ, θ, ψ) ∈ S+3 ⊂ R4 with 0 6 ψ 6 π/2 represents the rotation about the axis (ϕ, θ) ∈ S 2 ⊂ R3 by the angle of rotation ω = 2ψ, 0 6 ω 6 π , the probability of any rotational state (ϕ, θ, ω) is prob{(ϕ, θ, ω/2) ∈ ds3 } = h̃3 (ϕ, θ, ω/2) sin2 (ω/2)dω sin θdθdϕ, which reduces in case of rotationally invariance to prob{ω ∈ dω} = h̃3 (ω/2) sin2 (ω/2)dω. Summarily, if h̃3 (ϕ, θ, ψ) is any even probability density function on S 3 , the corresponding orientation density function is h̃3 (ϕ, ϑ, ω/2). With the even order Chebyshev polynomials of second kind Dn (ω) = sin((2n + 1)ω/2) sin(ω/2) (16) a rotationally invariant orientation density function f (g, g0 ; κ) is referred to as Brownian if ∞ 1 X (2l + 1) exp[−l(l + 1)κ] Dl (ω) fB (g, g0 ; κ) = 2π 2 l=0 = B̃3 (ω; κ), (17) ω ∈ [0, π ], κ ∈ (0, 1], κ0 = 0+, where ω ∈ [0, π ] is the angle of rotation of the composed rotation g0 ◦ g −1 with an arbitrary g0 ; in this case ω is referred to as orientation distance of the orientation g from g0 . ACAP1262.tex; 19/08/1998; 9:16; p.8 67 THE CENTRAL LIMIT THEOREM 4.2. PARTHASARATHY- TYPE NORMAL DISTRIBUTION ON SO (3) AND A CENTRAL LIMIT THEOREM FOR SO (3) Let gi (t), 0 6 t 6 ∞, i = 1, 2, 3 denote parametric subgroups of SO(3) with tangent vectors at the identity e gi (t) − e . t →0 t ei = lim (18) Let g → Tg(l) denote the complete set of irreducible representations of SO(3) (cf. Gelfand et al., 1963); then Tg(l) may be considered as the matrix functions on SO(3) satisfying Tg(l) = Tg(l) , Tg(l) 1 2 1 g2 where l = 0, 1, . . . are the weighs of the representation, and where the corresponding matrix has the order 2l + 1. Next, an infinitesimal operator of a representation g → Tg(l) , g ∈ SO(3), is defined by A(l) i = lim t →0 Tg(l) −E i (t ) t , (19) where E is the identity matrix of the same order as Tg(l) . i (t ) A measure µ is said to be infinitely divisible if for every positive integer n a ?n distribution µn exists such that µ?n n = µ, where µn denotes n-fold convolution of the distribution µn . A measure µ is said to have idempotent factors if µ ? µ = µ. Following Parthasarathy (1964; 1967) the P-type normal distribution on the rotation group SO(3) is the distribution which is infinitely divisible, without idempotent factors, and possesses for every l the representation X Z 3 X 3 3 X (l) (l) (l) (l) Tg dµ(g) = exp aij Ai Aj + ai Ai , l = 0, 1, . . . , (20) SO(3) i=1 j =1 i=1 where aij is a nonnegative symmetric matrix and ai are real constants. Considering a sequence of probability measures µn → δ(g) as n → ∞, or E of G = limn→∞ nµn (G − U ) = 0, where U is any neighborhood of the identity R SO(3) and denoting the ‘mean value’ of the measure µn as gn = SO(3) gdµn (g), then it has been shown (Nikolayev and Savyolova, 1996) that if n(1 − det(gn )) is bounded as n → ∞, i.e. lim n(e − gn ) = Q = (qij ), (21) n→∞ then any limit µ?n n is normal with parameters aij , ai defined by expressions ai = 1/2(qj k − qkj ), i, j, k = 1, 2, 3, i 6= j 6= k, j > k, aii = 1/2(qii − qjj − qkk ), i 6= j 6= k, aij = −1/2(qij + qj i ). (22) ACAP1262.tex; 19/08/1998; 9:16; p.9 68 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV The proof follows from (Wehn, 1962; Parthasarathy, 1964; 1967). If the probability measure µ(g) is absolutely continuous with density f (g) : dµ(g) = f (g)dg where dg denotes the invariant measure on SO(3) the P-type normal distribution dµ(g) is explicitly derived from its characterization (20) with parameters provided by (22) by an inversion of its Fourier transform. In that case the density is represented by its series expansion into generalized spherical harmonics f (g) = ∞ X l l X X Clmn Tlmn (g), (23) l=0 m=−l n=−l with the basis functions {Tlmn (g)} that correspond to the unitary representation of the group SO(3). The series (23) converges continuously and proportionally to the nonnegative function f (g) (Grenander, 1963). Applying a canonical basis the matrix of the irreducible representation with weight l has the form Tg(l) = (Tlmn (α, β, γ )), l = 0, 1, . . . ; m, n = −l, −l + 1, . . . , 0, . . . , l − 1, l, where g = g(α, β, γ ) is an arbitrary rotation given by its Euler angles {α, β, γ }, 0 6 α, γ < 2π, 0 6 β 6 π . The basis functions Tlmn (g) are referred to as generalized spherical harmonics of order l Tlmn (α, β, γ ) = exp(im α)Plmn (cos β) exp(im γ ), Plmn (x) = (−1)l−m i n−m ((l − m)!(l + n)!)1/2 , 2l (l − m)!((l + m)!(l − n)!)1/2 d(l−n) ((1 − x)l−m (1 + x))l+m dx (l−n) with x = cos β and the Jacobi polynomials Plmn (x) (cf. Vilenkin, 1968). Let the parameter subgroups gi (t) of (18) be 1 0 0 cos t 0 − sin t 0 , g1 (t) = 0 cos t − sin t , g2 (t) = 0 1 0 sin t cos t sin t 0 cos t (1 − x)− n−m 2 (1 + x)− n+m 2 cos t − sin t 0 g3 (t) = sin t cos t 0 . 0 0 1 Their corresponding tangent matrices at the identity according to (18) are 0 0 0 0 0 −1 e2 = 0 0 0 , e1 = 0 0 −1 , 0 1 0 1 0 0 ACAP1262.tex; 19/08/1998; 9:16; p.10 69 THE CENTRAL LIMIT THEOREM 0 −1 0 e3 = 1 0 0 . 0 0 0 (l) Then the matrices A(l) i (19) corresponding to the representations Tgi (t ) are given (cf. Gelfand et al., 1963) by l A(l) i = (Ai )mn , l = 0, 1, . . . ; m, n = −l, −l + 1, . . . , 0, . . . , l − 1, l, i (Al1 )mn = − (ωn δm+1,n + ωm δm−1,n ) 2 (Al1 )mn = (Al1 )nm , 1 (Al2 )mn = (ωn δm+1,n − ωm δm−1,n ), 2 (Al2 )mn = −(Al2 )nm , (Al3 )mn = −im δmn , √ where ωm = (l + m)(l − m + 1) Choosing the Lie algebra basis, the normal distribution on SO(3) can be reduced to canonical form with three different parameters (Nikolayev and Savyolova, 1996). Let e1 , e2 , e3 denote the Lie algebra basis of the SO(3). Moreover, let aij = V −1 3V , where 3 is a diagonal matrix of third order with elements λi , and where V is an orthogonal matrix. If e10 , e20 , e30 constitute a proper orthogonal basis corresponding to eigenvalues λ1 , λ2 , λ3 , then 3 X aij ei ej = i,j =1 3 X λi ei0 ei0 (24) i=1 Because of the theorem by Dubrovin et al. (1984), and the orthogonality of matrix V , it holds [e10 , e20 ] = e30 , [e20 , e30 ] = e10 , [e30 , e10 ] = e20 , and ei0 = V ei . Thus e10 , e20 , e30 is a Lie algebra basis of SO(3). Moreover, the orthogonal matrix V = (vij ) corresponds to some rotation g0 , and to that transformation corresponds a transformation of the group g → g0−1 g Tg → Tg −1 g . It should be noted that the parameters 0 ai , i = 1, 2, 3, correspond to the ‘center’ of the normal distribution. Next, consider a normal distribution on SO(3) in canonical form with three parameters Z Tg(l) SO(3) ∞ X l l X X Clmn Tlmn (g) dg = exp{Bl }, (25) l=0 m=−l n=−l where Tg(l) are matrix elements of the representation Tg , and Bl = are five-diagonal matrices P3 i=1 2 aii (A(l) i ) ACAP1262.tex; 19/08/1998; 9:16; p.11 70 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV Bl = (1) bl 0 bl(2) (1) (2) 0 bl−1 0 bl−1 (2) (1) bl 0 bl−2 0 . . .. .. .. . (1) 0 b 0 b1(2) 1 (2) (1) b1 0 b0 0 b1(2) (2) (1) b1 0 b1 0 . . .. .. .. . (1) 0 b 0 bl(2) (l−2) (2) (1) bl−1 0 bl−1 0 (2) (1) bl 0 bl 0 0 , (26) with elements 1 (1) = − ((2m + 1)l − m2 )(a11 + a22 ) − (l − m)2 a33 , bl−m 2 m = 0, 1, . . . , l 1p (2) = − (m + 1)(m + 2)(2l − m)(2l − m − 1)|a11 − a22 |, bl−m 4 m = 1, . . . , l (27) and with all other elements zero. In the case a11 = a22 = ν 2 , a33 = %2 , the canonical P-type normal distribution with two parameters f (g) = ∞ l X X (2l + 1) exp{−l(l + 1)ν 2 } exp{m2 (ν 2 − %2 ) − l=0 m=−l − im(α + γ )}Plmm (cos β) (28) is obtained. In particular, when ν 2 = %2 = ε 2 the rotationally invariant P-type normal distribution on SO(3) ∞ X sin[(2l + 1)t/2] (2l + 1) exp{−l(l + 1)ε 2 } f (t) = sin(t/2) l=0 (29) is obtained by applying the formula for the characters of a representation l X sin[(2l + 1)t/2] Tlmm (g), = sin(t/2) m=−l ACAP1262.tex; 19/08/1998; 9:16; p.12 71 THE CENTRAL LIMIT THEOREM where cos(t/2) = cos(β/2) cos((α + γ )/2) = 12 (tr(g) − 1) (cf. Vilenkin, 1968). The same distribution has been obtained by a random walk argument in (Roberts and Winch, 1984). In (Savyolova, 1984) the approximation 2 t/2 −t , exp f (t) ∼ C(ε) sin(t/2) 4ε 2 2 1 π ε ε erfc + 2 , ε < 0.5, t ∼ 0 C(ε) = 3 exp ε 4 2 ε was presented. Compared with the approximation of the von Mises–Fisher matrix distribution (Khatri and Mardia, 1977) the similarity of the rotationally invariant P-type normal distribution on SO(3) with the von Mises–Fisher directional distribution is recovered. The close relationship of the rotationally invariant normal distribution on SO(3) with the ϑ2 -Jacobi function (cf. Korn and Korn, 1968) ϑ2 (v) = 2 ∞ X 2 q (l+1/2) cos(2l + 1)π v l=0 is emphasized; with the change t = 2π v, q = exp(−ε 2 ) of variables, f is rewritten as f (t) = − ∂ q −1/4 ϑ2 (v). 2π sin(t/2) ∂v In the case where aij = 0, except a33 = 2σ 2 , the rotationally invariant P-type normal on SO(3) degenerates to ∞ X (β + 2π l)2 1 (30) exp − f (g) = f (β) = √ 2σ 2 σ 2π l=−∞ which is identical to the wrapped normal distribution on the circle (Mardia, 1972, p. 55) and may be referred to as the P-type normal distribution on the circle SO(2). Its close relationship to the ϑ3 function is emphasized by Mardia (1972, p. 55) for numerical evaluation. In (Nikolayev and Savyolova, 1996) it is shown that the convolution of two rotationally invariant P-type normal distributions on SO(3) is again a rotationally invariant P-type normal distribution. Without loss of generality it is possible to consider the case when the centers of the distributions are e and g0 , respectively. The convolution on the rotation group is defined as Z f (g1 , ε12 )f (g1 )−1 g0 g, ε22 dg1 , f (g, ε12 ) ? f (g0 g, ε22 ) = SO(3) f (g0 g, ε22 ), f (g, ε12 ) where are rotationally invariant functions of the form (29), and g0 g denotes the composition by succession of two rotations. Expanding the ACAP1262.tex; 19/08/1998; 9:16; p.13 72 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV rotationally invariant P-type normal distribution in a series of generalized spherical harmonics the convolution is rewritten as X Z ∞ l X 2 2 2 mm Cl (ε1 ) Tl (g1 ) × f (g, ε1 ) ? f (g0 g, ε2 ) = SO(3) × l=1 X ∞ m=−l Cj (ε22 ) j =1 l X Tjnn −1 (g1 ) g0 g, ε22 dg1 , n=−l with Ck (ε 2 ) = (2l + 1) exp(−l(l + 1)ε 2 ). Using the addition theorem Tlmn (g1 g2 ) = l X Tlmk (g1 )Tlkn (g2 ) k=−l and the orthogonality of the generalized spherical harmonics it holds f (g, ε12 ) ? f (g0 g, ε22 ) = ∞ X Cl (ε12 + ε22 ) l=1 l X Tlmm (g0 g) dg1 m=−l = f (g0 g, ε12 + ε22 ) using Cl (ε12 )Cl (ε22 ) = (2l + 1)Cl (ε12 + ε22 ). As an example, the n-fold convolution of the von Mises–Fisher directional distribution fvMF (g; g0 , nκ0 ) converges to the rotationally invariant P-type normal distribution on SO(3) (cf. Matthies et al., 1988). 4.3. SPHERICAL DIFFUSION In approximation theory the Brownian distribution is referred to as the Gauss– Weierstrass kernel (cf. Butzer and Nessel, 1971). It is the fundamental solution of a diffusion problem; its corresponding singular convolution integral applied to a given function provides the solution of a diffusion problem with initial conditions given by that function. Considering the special case of isotropy of the general equation of spherical diffusion for S n−1 ⊂ Rn (cf. Watson, 1983, p. 100) it reduces to 1 ∂f (β, t) ∂ n−2 ∂f (β, t) sin = , n−2 ∂β R ∂t sin β ∂β 1 (31) with β ∈ [0, π ], R > 0, t > 0, n > 2. Its fundamental solution fn−1 (β, t) ≡ Fn−1 (u, t) with u = cos β possesses the following properties (cf. Nikolayev and ACAP1262.tex; 19/08/1998; 9:16; p.14 73 THE CENTRAL LIMIT THEOREM Savyolova, 1996) Z π fn−1 (β, t) sinn−2 βdβ = 1, sn−1 0 fn−1 (β, 0) = δ(β) 2π n/2 = , s n−1 0(n/2) sn−1 sinn−2 β (32) and is given by (cf. Nikolayev and Savyolova, 1996) ∞ 1 Xl+p p fn−1 (β, t) ≡ Fn−1 (u, t) = exp{−l(l + 2p)Rt}Cl (u), sn−1 l=0 p (33) p with Gegenbauer polynomials Cl (u) where p = (n − 2)/2 > 0. For the special cases of n = 2, 3, 4 refering to the circle, sphere and hypersphere, respectively, Equation (33) will be explicitly rewritten. For n = 2 it reads ∞ X 1 2 exp{−l Rt} cos lβ (34) 1+2 F1 (u, t) = 2π l=1 which is the wrapped normal on the circle (Mardia, 1972, pp. 55, 90; Watson, 1983, p. 98). It should be noted that the function (34) was obtained from the central limit theorem for the circle SO(2) (Parthasarathy, 1964) as well as from the central limit theorem for SO(3) for the degenerate case (30) above. For n = 3 the function (33) reads ∞ 1 X (2l + 1) exp{−l(l + 1)Rt}Pl (u) F2 (u, t) = 4π l=0 (35) which is referred to as the Brownian motion distribution on the sphere (Roberts and Ursell, 1960; Mardia, 1972, p. 228). It may also be addressed as generalized wrapped normal for the sphere, or the spherical P-type normal distribution. For n = 4 the solution (33) specializes to ∞ 1 X sin(l + 1)β (l + 1) exp{−l(l + 2)Rt} F3 (u, t) = 2π 2 l=0 sin β (36) which is the hyperspherical Brownian motion distribution (Bingham, 1972; Roberts and Winch, 1984). To adjust (36) for the transition from SU(2) to SO(3) the summation is restricted to the even terms (Arnold, 1941; Watson, 1983, p. 107) F̃3 (u, t) = ∞ 1 X sin[(2l + 1)β̃/2] , (2l + 1) exp{−l(l + 1)4Rt} 2 2π l=0 sin(β̃/2) (37) ACAP1262.tex; 19/08/1998; 9:16; p.15 74 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV with β̃ = 2β which is the antipodally symmetric hyperspherical Brownian or the P-type normal distribution from the the central limit theorem for SO(3). It has been shown that the fundamental solution (33) is nonnegative and infinitely differentiable at t > 0 (Nikolayev and Savyolova, 1996). 4.4. THE BROWNIAN DISTRIBUTION IN TEXTURE ANALYSIS Applying the pole figure projection operator (1) to the even Brownian distribution on S 3 yields Px [B̃3 (ω; κ)](r) = ∞ X (2l + 1) exp[−l(l + 1)κ]Pl (cos η) = B2 (η; κ) (38) l=0 with ω = ω(g0−1 g), η = η(r, g0−1 x) = arccos(r g0−1 x). The result is due to Savyolova (1984; 1989); see also (Bucharova and Savyolova, 1993). Thus, the Brownian distribution is preserved by the projection operator and provides a central (rotationally invariant) model orientation density function and pole density function, respectively. The convolution of two spherical Brownian distributions is again a Brownian distribution, and the spherical Brownian distribution is thus infinitely divisible and possesses the convolution semi-group property. To summarize, the spherical Brownian distribution provides an appropriate candidate to model texture components. The remaining major question is whether it provides a model which is superior to other candidates in terms of its characteristic properties, in particular in terms of interpretations related to the texture generating process(es). 5. Central Limit Theorems and Their Physical Interpretation 5.1. ON THE MISUSE OF A ‘ SIMPLE ’ CENTRAL LIMIT ARGUMENT IN TEXTURE MODELING The spherical Brownian distribution is the special case of distributions satisfying the ‘central limit theorem for the rotation group’ (Parthasarathy, 1964) when they are unimodal and rotationally invariant. But can it be interpreted as the limiting distribution of a sequence of independent random rotations in the sense of the distribution of the rotation resulting from successive composition, gn∗ = ni=1 gi , satisfying a Lindeberg-type constraint. In the affirmative case a complete analogy to the central limit theorem in Euclidean space would have been established. However, it should be obvious now from the preceeding section that the central limit theorem in Euclidean space (cf. Appendix) does not have a simple (hyper)spherical analogue for S n−1 ⊂ Rn , n > 2, nor for the groups SO(n) as, e.g., the elements of SO(3) can be represented as points of the upper hypersphere S+3 ⊂ R4 , i.e. as points of the three-dimensional hyperplane H 3 ≡ S+3 . ACAP1262.tex; 19/08/1998; 9:16; p.16 75 THE CENTRAL LIMIT THEOREM To be more explicit, taking the assumptions of the classic central limit theorem in Euclidean space for S n ⊂ Rn , n > 2, and replacing summation of random variables by a spherical random walk the limiting distribution is the uniform distribution (cf. Roberts and Winch, 1984, p. 643). Taking additional mathematical assumptions concerning the distributions of Xn and/or the limiting distribution may lead to bell shaped limiting distributions. In terms of Brownian motion these additional assumptions amount to the implantation of some attraction in the north pole of the hypersphere representing the identity element of the group SO(n) (Watson, 1992), and imply unimodality of the limiting distribution (cf. Parthasarathy, 1964, pp. 255–256; Savelova, 1984, p. 471). Theorems of this type may be referred to as (generalized) central limit theorem, and the limiting distribution as (generalized) normal or Gaussian, respectively (cf. Savelova, 1984, p. 473; Nikolayev, Savyolova and Feldmann, 1992, pp. 9, 22–23). However, the significance of the additional assumptions may not be immediately recognized; in particular, it escaped the attention of many texture analysts (cf. Matthies, Muller and Vinel, 1988, p. 85; Helming, 1995, p. 35; Eschner, 1995, p. 32) that they are too restrictive compared to the Euclidean case to allow an analogous interpretation. There is just no simple spherical analogue of the central limit theorem in Euclidean space, the key to the difference between the Euclidean and the spherical case being the compactness of the sphere. In the same way the flourishing uncritical use of the term ‘normal’ or ‘Gaussian’ distribution on SO(n), S n respectively, in the texture community is misleading. The normal distribution in Euclidean space does not have a unique generalization for arbitrary spaces; different characterizations will generally lead to different distributions which may all deserve the label normal or Gaussian, respectively, in some sense (cf. Hartmann and Watson, 1974; Watson, 1983; Schaeben, 1992). Parthasarathy (1964, p. 248) in his paper uses the term Gaussian distribution in quotation marks at the first instance. In fact, Gauss measures in the sense of Parthasarathy are those infinitely divisible measures which do not admit Poisson factors. Bernstein started from the characterization of the classical normal distribution as being the only probability measure invariant under rotations. Then, general Gauss measures in the sense of Bernstein differ from Gauss measures in the sense of Parthasarathy by the fact that the former admit idempotent factors (Heyer, 1977). 5.2. SPHERICAL PROBLEMS IMPEDING A ‘ SIMPLE ’ SPHERICAL ANALOGUE OF THE CENTRAL LIMIT THEOREM The central question is whether the P-type normal distribution on SO(3) and hence its special case of the even hyperspherical Brownian distribution can be interpreted as limiting distribution of a sequence of successively composed random rotations under the same or quite similar assumptions as involved in the central limit theorem in Euclidean spaces. ACAP1262.tex; 19/08/1998; 9:16; p.17 76 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV First doubts are caused by Watson’s observation of a curious phenomenon on SO(n). If any random rotation is succeeded (multiplied) by a rotation that is uniformly distributed, then their composition (product) is also uniformly distributed (cf. Mardia, 1972, pp. 43, 50–51; Watson, 1992). Thus, Parthasarathy’s definition of a normal distribution must rule out such idempotent measures (µ ∗ µ = µ) (Watson, 1992). Furthermore, the sequence of successive rotations must converge to the identity at a well defined rate to obtain the diffusion distribution, otherwise the uniform distribution is recovered; cf. (Mardia, 1972, p. 88; Roberts and Winch, 1984, p. 643, Equation (4.5)). Basically, a simple spherical analogy to the central limit theorem in locally compact Euclidean space fails because both the group SO(3) and the sphere S n−1 ⊂ Rn are compact. Moreover, the one-one correspondence of summation of random variables in Euclidean space and convolution of their distributions defined on Rn cannot generally be transferred to other spaces. For random rotations summation is generalized to composition by succession, with respect to SO(3) in terms of matrix multiplication and with respect to S+3 in terms of quaternion multiplication. A convolution of their distributions is well defined with respect to SO(3) but not generally for any parametrization, e.g., for S+3 or S 3 a convolution is well defined only for rotationally invariant distributions. Since SO(3) is a group, any two distributions can be combined together applying the convolution generated by the group. For any distribution on the compact group SO(3) whose support is not contained in any subgroup, its n-fold convolution will converge to the uniform distribution (Haar measure) on the group (Parthasarathy, 1995). Thus, the even hyperspherical Brownian distribution is not generally the limiting distribution of the resultant rotation composed of a sequence of successive random rotations. 5.3. SPHERICAL CONVOLUTION VS CONVOLUTION ON SO (3) A convolution of probability density functions defined on the hypersphere S 3 is well defined only provided they are rotationally invariant. It should be noted that a distribution on SO(3) corresponds uniquely to an even distribution on S 3 , i.e. to a distribution on H 3 ≡ S+3 . The relationship of the convolution of distributions on SO(3) using the convolution generated by the group and the convolution of rotationally invariant even distributions on S 3 seems to pose an open question (Kent, 1995). The composition of any two (random) rotations is always well defined (whether their probability density function is rotationally invariant or not) and most elegantly represented in terms of the Rodrigues parameters (quaternion components) using the Euler construction (Altmann, 1986, pp. 155–156). Let the first rotation be given by Q1 = Q1 (ρ1 , q1 ) ∈ S+3 where ρ1 = cos(ω1 /2), 0 6 ω1 6 π (39) ACAP1262.tex; 19/08/1998; 9:16; p.18 77 THE CENTRAL LIMIT THEOREM with the angle ω1 of the first rotation, and q1 = sin(ω1 /2)n1 (40) with the axis n1 ∈ S 3 R3 of the first rotation, and the second rotation be analogously given by Q2 = Q2 (ρ2 , q2 ). Then the composition Q(ρ, q) = Q2 ◦ Q1 of the two rotations is given by ρ = ρ1 ρ2 − q1 q2 (41) q = ρ1 q2 + ρ2 q1 − q1 × q2 . (42) and To avoid any difficulties it should be noted that as usual in texture analysis, the formulae (41), (42) refer to passive rotations. They seem to provide the canonical geometric interpretation of the composition of two rotations of SO(3) for S+3 . Obviously any two rotations of SO(3) can be composed, the cross product term of the representation (41), (42) indicating that composition of rotations is not generally commutative. Excluding the trivial case, it is commutative if either (i) they have a common axis of rotation, or if (ii) they are infinitesimal, i.e. if their angle of rotation is infinitesimal (cf. Altmann, 1986). In this case ρq = ρ1 q1 + ρ2 q2 . (43) Thus, the composition of rotations can be represented in terms of multiplication of quaternions. However, the convolution of rotationally invariant spherical distributions correponds to a spherical addition of axially symmetric random points on S n−1 ⊂ Rn , n > 2 as defined by Kent (1977, pp. 376–378). This spherical addition is generally a different operation from composition by succession, any subtle connection is yet missing (Kent, 1995). 6. Common Properties of the Brownian Motion Distribution 6.1. INVARIANCE WITH RESPECT TO THE POLE FIGURE PROJECTION OPERATOR The Brownian distribution, Equation (17), is not the only distribution the type of which is preserved by the pole figure projection operator. It is also true for the ‘Lorentz-type’ standard orientation density ∞ X sin[(2l + 1)ω/2] (2l + 1)κ 2l fC (g; g0 , κ) = sin ω/2 l=0 (1 + κ 2 )2 + 4κ 2 cos2 (ω/2) [(1 + κ 2 )2 − 4κ 2 cos2 (ω/2)]2 2 = L4 (ω; κ ), ω ∈ [0, π ], κ ∈ [0, 1), κ0 = 1−, = (1 − κ 2 ) (44) ACAP1262.tex; 19/08/1998; 9:16; p.19 78 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV as was essentially shown in (Matthies et al., 1987) however without recognizing the permanence and without appreciating that the corresponding pole density function is of ‘Lorentz-type’ (cf. Schaeben, 1996b). The function L4 is referred to as the spherical Abel–Poisson kernel (which is closely related to harmonic functions and Laplace’s equation) in approximation theory (Butzer and Nessel, 1971) and ‘Lorentz-type’ distribution for SO(3) by Matthies et al. (1987). Above all, it may be addressed as a multivariate generalization of the circular wrapped Cauchy distribution (Mardia, 1972), see also (Saw, 1984). Matthies’ label as of Lorentzian type may be all together misleading as Lorentz was essentially interested in investigations of Brownian motion phenomena (cf. deHaas-Lorentz, 1913). In one dimensional theory for a function defined on the circle, its corresponding singular integral is related to a Laplace differential equation and provides a solution of Dirichlet’s problem for the unit disc; i.e. it is a harmonic function. Furthermore, it is also true for the de la Vallée Poussin kernel (cf. Butzer and Nessel, 1971) 3 2k + 1 cos2k (ω/2), ω ∈ [0, π ], k ∈ N, k0 = ∞ (45) Vk (ω) = B , 2 2 for which ) B( 32 , 2k+1 2 Px [Vk (ω)](r) = 2π Z cos [−π,π] 2k t dt cos2k (η/2) = νk (η) 2 (46) holds (Schaeben, 1997); here B denotes the Beta function. Thus, the de la Vallée Poussin kernel is invariant under the projection operator and provides a central (rotationally invariant) model orientation density function and pole density function, respectively. 6.2. INFINITE DIVISIBILITY The Brownian or P-type normal distribution, Equation (17), is not the only infinitely divisible spherical distribution. It has been proven that the von Mises–Fisher directional distributions fvMF (ω; λ) = CM (λ) exp(λ cos ω), λ>0 (47) are infinitely divisible (Kent, 1977). It should be noted that the convolution of two von Mises–Fisher distributions is not a von Mises–Fisher distribution (Kent, 1995). The rotationally invariant von Mises–Fisher distribution on S 3 is the special case of the von Mises–Fisher matrix distribution on SO(3) when it is central, i.e. when it depends only on the orientation distance; it is equivalent to the (even) Bingham distribution on S+3 when it is unimodal and rotationally invariant fW (ω; κ) = CW (κ) exp(κ cos2 (ω/2)), κ >0 (48) which is referred to as Watson distribution. ACAP1262.tex; 19/08/1998; 9:16; p.20 79 THE CENTRAL LIMIT THEOREM The von Mises–Fisher matrix distribution as well as the Bingham distribution may be characterized as conditional multivariate normal distributions (Mardia, 1974). The convolution of two generalized wrapped Cauchy distributions is again a generalized wrapped Cauchy distribution, and the generalized wrapped Cauchy distribution is thus infinitely divisible, too. 6.3. TRANSFORMATION STRUCTURES Initially, there are several ways to construct transformation structures between SO(3) and S 3 ⊂ R4 on one hand and between SO(3) and S 2 ⊂ R3 on the other hand. There is a canonical 2 to 1 map of S 3 to SO(3) transforming rotations into axis; thus, it is essentially a correspondence between SO(3) and S+3 ≡ H 3 ⊂ R4 , where H 3 denotes the three dimensional projective hyperplane; cf. Jupp and Mardia (1989, p. 285). Therefore, any distribution on SO(3) corresponds uniquely to an even distribution on S 3 or, equivalently, to a distribution on S+3 ≡ H 3 . Roberts and Winch (1984) considered various random walks on the rotation group SO(3) in terms of random walks on S 3 ⊂ R4 . The even portion of a spherical analogue of the normal in Euclidean spaces is a corresponding hemispherical analogue, cf. (Watson, 1983, p. 107; Roberts and Winch, 1984, p. 639). Since experimental pole density functions from normal diffraction are even, only hemispherical analogues apply. Let KA , KB be two different coordinate systems. The map X(g) ∈ SO(3) with Xr = x corresponds to a transformation of coordinates of a fixed unit vector, which is given by u with respect to KA and by v with respect to KB , according to a proper rotation g: KA 7→ KB by ω about n. Furthermore X(g(α, β, γ )) = cos α cos β cos γ − sin α sin γ sin α cos β cos γ + cos α sin γ − sin β cos γ − cos α cos β sin γ − sin α cos γ − sin α cos β sin γ + cos α cos γ sin β sin γ cos α sin β sin α sin β cos β (49) where α, β, γ denote the Euler angles of g = g(α, β, γ ): KA 7→ KB by three successive rotations (i) about the axis ZA through the angle α, (ii) about the axis YA0 through the angle β, and (iii) about the axis ZA00 about the angle γ (cf. Roe, 1965; Matthies et al., 1987). Using these conventions of modern physics the angles α, β may be interpreted as spherical coordinates of the axis ZB with respect to KA . Jupp and Mardia (1989, p. 276) summarized: Roberts and Winch (1984) considered various random walks on the rotation group SO(3). In particular, they considered the cases (i) . . . , and (ii) that the angle of rotation is arbitrary but the axis of rotation makes a given angle 2 with a fixed axis. If 2 = π/2 in (ii), then the set of such rotations can be identified (via the ACAP1262.tex; 19/08/1998; 9:16; p.21 80 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV image of the north pole) with S 2 ⊂ R3 , and the distributions obtained on SO(3) correspond to those from random walks on the sphere. To be more explicit, the set of such rotations can be pictured as motion of the north pole over the reference sphere provided 2 = π/2. Thus, a distribution on SO(3) is transferred to a rotationally invariant distribution on S 2 by integration over all possible rotational states about the z axis, cf. (Roberts and Winch, 1984, p. 649, Equations (5.22), (5.23), and (5.24)). On the other hand, let g = g(α, β, γ ) ∈ SO(3), and let r ∈ S 2 R3 with r = (cos α sin β, sin α sin β, cos β)t . Then gr = e3 for all γ ∈ [0, 2π ) with e3 = (0, 0, 1)t . Thus, the projection Z 1 {f (g(α, β, γ )) sin β dβ dα} dγ 2π [0,2π) Z 1 f (g) dg = Pe3 (r) (50) = 2π {g∈G|e3 =gr} is a special case of the general projection (1). Integrating Equation (50) with respect to α reveals the equivalence of the two transformations, one considered by Roberts and Winch (1984), and the other one being at the basis of texture analysis; hence the identity of the resulting central distributions is not surprising. 7. Conclusions The truly (Matthies et al., 1987) normal distribution is not as unique as it may appear at first sight. To put it into any relationship to some elusive ‘simple’ spherical central limit theorem is grossly erroneous. Results of spherical random walk problems, and spherical generalizations of the Brownian diffusion distribution may be interpreted as generalized variants of the central limit theorem of probability in the sense discussed above, cf. (Perrin, 1928; Yosida, 1949; Roberts, and Ursell, 1960; Breitenberger, 1963; Parthasarathy, 1964; Bingham, 1972; Hartman and Watson, 1974; Heyer, 1977; Roberts and Winch, 1984; Jupp and Mardia, 1989). For S 3 ⊂ R4 , any ‘physical’ meaning of Parthasarathy’s assumptions, that is the significance of Parthasarathy’s normal distribution as well as its association with and interpretation in terms of a simple central limit theorem for a sequence of successive random rotations was not affirmed. Since the spherical Brownian distribution is not generally the limiting distribution of the composition of successive random rotations, an explicit correspondence of process(es) causing preferred crystallographic orientations and the Brownian form of texture components cannot be inferred from a simple central limit theorem argument. Therefore, there is no reason to abandon the idea that well pronounced patterns of preferred orientation preserve and reveal a distinguished history of their formation, and particularly for geological applications of texture analysis hope remains that preferred orientation may be interpreted in terms of a few distinguished deformation processes. ACAP1262.tex; 19/08/1998; 9:16; p.22 81 THE CENTRAL LIMIT THEOREM Any explicit correspondence of the form of texture components and processes causing preferred orientations has not yet been established, and their suggestions (Matthies, 1982; Matthies et al., 1988) still remains of speculative character. The common basis of all model orientation distributions so far successfully applied in texture component fit methods is provided either by the von Mises–Fisher matrix distribution for SO(3) or equivalently the Bingham distribution on H 3 ≡ S+3 ⊂ R4 or the hyperspherical Brownian distribution favored by Savyolova and co-workers. While the von Mises–Fisher type of distribution is not preserved by the pole figure projection operator, the Brownian type of distribution is preserved. Nevertheless, it should be clarified with this communication that a theoretical justification of their application is missing in either case. Since they can always be matched very closely by appropriate choices of their concentration parameters, then for texture analysis it should matter little which is used; for the time being, the distribution may be used which is most convenient. Preference of the form of the model function representing texture components may be decided on the basis of the solutions of the following problems. Statistical analysis of individual orientation measurements in orientation imaging microscopy (Adams, 1994; Kunze et al., 1994) testing the hypothesis of agreement of the sample distribution with a chosen model (population) distribution may indicate which distribution is superior in terms of goodness-of-fit. It is anticipated that the most suitable choice will depend on the texture forming process. Modeling physical processes by differential equations, e.g., Euler equations for the rotation of solid bodies, may lead to special solutions related to one or the other form of suitable model functions. Acknowledgements This joint work was initiated at the Dubna (Russia) workshop on mathematical methods of texture analysis in March 1995, and completed during D.N’s. stay at the Institut für Metallkunde und Metallphysik, TU Clausthal, Germany, granted by Deutscher Akademischer Austausch Dienst (DAAD), Bonn, Germany. The authors would like to thank Prof. N. H. Bingham, University of London, England, Prof. J. T. Kent, University of Leeds, England, Prof. K. R. Parthasarathy, Indian Statistical Institute, Delhi Center, India, and Prof. G. S. Watson, Princeton University, NJ, U.S.A., for their comments and suggestions to improve this communication. Appendix THE CENTRAL LIMIT THEOREM IN EUCLIDEAN SPACE The central limit theorem in Euclidean space states the following. Let (Xn )n∈N be an independent sequence of real, square integrable random variables with V (Xn ) > P 0, and let Sn = nj=1 (Xj − E(Xj ))/σ (X1 + · · · + Xn ). Then the sequence of ACAP1262.tex; 19/08/1998; 9:16; p.23 82 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV measures (PSn )n∈N converges towards the standardized normal distribution ν0,1 , if the sequence (Xn ) satisfies the Lindeberg constraint n Z 1 X lim Ln (ε) = lim 2 (x − ηj )2 PXj (dx) = 0 n→∞ n→∞ sn j =1 |x−ηj |>εsn for each ε > 0 (51) with sn = σ ((X1 + · · · + Xn ) and ηn = E(Xn ). Without going into details, this condition may be interpreted that no variate dominates the others since n Z 1 X (x − ηj )2 PXj (dx), (52) P max |Xk − ηk | > εsn 6 2 2 16k 6n ε sn j =1 |x−ηj |>εsn where the last sum converges towards zero for arbitrary constant ε > 0 according to the Lindeberg constraint. For example, if the sequence (Xn ) of random variables is independent and identically distributed, then it obviously satisfies the Lindeberg constraint because with η = ηn , σ = σn independently of n it is sn 2 = nσ 2 which implies lim sn = ∞ and Z 1 (x − η)2 µ(dx). (53) Ln (ε) = 2 σ |x−η|>εsn This example could also be stated as a simple version of the central limit theorem and be proven elementarily without referring to the Lindeberg constraint: Let (Xn )n∈N be an independent and identically distributed sequence of real, square integrable random variables with E(Xn ) = 0, V (Xn ) = σ 2 > 0, and let Sn = P √ n j =1 (Xj − E(Xj ))/ n. Then the sequence of measures (PSn )n∈N converges towards the standardized normal distribution ν0,σ 2 . This result may be interpreted in the following elementary way. Would E(Xn ) > 0, then PPnj=1 Xj would move ever further to the right according to Chebyshev’s inequality and thus convergence could not occur. Therefore, the additional assumption E(Xn ) = 0 is required. However, even then the sequence PPnj=1 Xj does not generally converge, e.g., if PXn = ν(0, 1) P then the probability density function of nj=1 Xj → 0 for n → ∞. Therefore, Pn some scaling of j =1 Xj is required to impose convergence of the sequence of P corresponding measures, i.e. c1n nj=1 Xj with a suitably chosen sequence of real numbers (cn ), n ∈ N, with 0 < cn → ∞ as n → ∞ is considered. To guarantee convergence of P 1 Pnj=1 Xj towards a nondegenerate measure, (cn ) must not cn increase too fast, e.g., P 1 Pnj=1 Xj → δ0 for n → ∞ according to Chebyshev’s Pn √ inequality. Now V ( c1n nj=1 Xj ) = ncn−2 V (Xn ) suggests to choose cn = n, which yields the result stated above. It should be noted that in the general case asymptotic negligibility lim max P {|Xnj | > ε} = 0 n→∞ 16j 6kn for each ε > 0 (54) ACAP1262.tex; 19/08/1998; 9:16; p.24 83 THE CENTRAL LIMIT THEOREM of a sequence of random variables is not sufficient for convergence in the sense of the central limit theorem. There are no other assumptions concerning the individual distributions PXn nor the limiting distribution. Therefore, the central limit theorem is of great practical importance, and often used to justify the application of a normal model when a stochastic phenomenon is the resulting effect of a large number of independent random causes non of which dominates the others. The central limit theorem of probability is easily generalized for the multivariate case (cf. Anderson, 1984). There are generalizations of the central limit theorem relaxing the assumption concerning the independence of the sequence of random variables (cf. Billingsley, 1968; Ibragimoff and Linnik, 1971). SPHERICAL ANALOGUES OF THE CENTRAL LIMIT THEOREM The interpretation of the normal distribution in Euclidean spaces with respect to the central limit theorem was summarized by Watson (1983, p. 98): The Gaussian [in Euclidean spaces, these authors] also arises by addition and scaling as envisaged in the central limit theorem. On the circle one could use addition√modulo 2π . Because there seems to be no natural scaling [as on the real line by 1/ n to reduce the variance of the sum, these authors], this gives the uniform distribution as the limit, except in certain degenerate cases. It is less obvious what to do. However, Brownian motion is much more successful. The distribution of the sum (mod 2π ) of independent and identically distributed random variables on the circle converges to the uniform distribution provided that the common distribution does not correspond to a lattice distribution on the circle (Mardia, 1972). P The distribution of ( θj /n1/2) (mod 2π ) of independent and identically distributed random variables on the circle with expectation 0 converges to the wrapped normal distribution with the parameter σ = E(θ 2 ) (Mardia, 1972); note the analogy to independent and identically distributed random variables on the real line as stated above as simple version of the central limit theorem. A Poincaré’s type limit theorem yields again the uniform distribution (Mardia, 1972). Concerning a central limit theorem for non-identical random variables Mardia (1972) refers the reader for a Lindeberg-type central limit theorem to Parthasarathy (1964, 1967). The limiting distribution for the circle in this case is the wrapped normal (Mardia, 1972). Following Parthasarathy (1995) the following can be stated. If P is any distribution on a compact group and its support is not contained in any subgroup, then the n-fold convolution of P will converge to the uniform distribution on the group. ACAP1262.tex; 19/08/1998; 9:16; p.25 84 HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV SO(3) is a compact group and hence the n-fold convolution will converge to the uniform distribution. {z· · · ∗ P} → uniform distribution, |P ∗ P ∗ n→∞ (55) n for any distribution P on SO(3). However, there is a variety of limiting distributions for the n-fold convolution of appropriate distributions Pn with decreasing spherical dispersion for increasing n, for example − if Pn is the rotationally invariant von Mises–Fisher distribution with density fvMF (g; g0 , nκ0 ) for SO(3), then P ∗ Pn ∗ · · · ∗ Pn → Brownian distribution, |n {z } n → ∞, (56) n − if Pn happens to be the nth root of the directional von Mises–Fisher distribution, then Pn ∗ Pn ∗ · · · ∗ Pn → von Mises–Fisher distribution, | {z } n → ∞, (57) n − if Pn itself is a Brownian or Cauchy distribution, respectively, then P ∗ Pn ∗ · · · ∗ Pn = Brownian distribution, |n {z } (58) n Pn ∗ Pn ∗ · · · ∗ Pn = Cauchy distribution. | {z } (59) n The general case is treated in Parthasarathy (1964, 1967). References Adams, B. L., Dingley, D. J., Kunze, K. and Wright, S. I.: 1994, Orientation imaging microscopy: New possibilities for microstructural investigations using automated bkd analysis, in: H. J. 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