Download The Central Limit Theorem in Texture Component Fit Methods

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. Bunge
(ed.), Proc. 10th Internat. Conf. Textures of Materials, Materials Science Forum, pp. 157–162,
31–42.
Altmann, S. L.: 1986, Rotations, Quaternions, and Double Groups, Oxford University Press, Oxford.
Anderson, T. W.: 1984, An Introduction to Multivariate Statistical Analysis, 2nd edn, Wiley, New
York.
Arnold, K. J.: 1941, On spherical probability distributions, PhD thesis, Massachusetts Institute of
Technology, MA, U.S.A.
Beran, R. J.: 1979, Exponential models for directional data, Ann. Statist. 7, 1162–1178.
Billingsley, P.: 1968, Convergence of Probability Measures, Wiley, New York.
ACAP1262.tex; 19/08/1998; 9:16; p.26
THE CENTRAL LIMIT THEOREM
85
Bingham, N. H.: 1972, Random walk on spheres, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 22,
169–192.
Breitenberger, E.: 1963, Analogues of the normal distribution on the circle and the sphere, Biometrika
50, 81–88.
Bucharova, T. I. and Savyolova, T. I.: 1993, Application of normal distribution on SO(3) and S n for
orientation distribution function approximation, Textures and Microstructures 21, 161–176.
Bukharova [identical with Bucharova], T. I., Kapcherin, A. S., Nikolayev, D. I., Papirov, I. I., Savyolova, T. I. and Shkuropatenko, V. A.: 1988, A new method of reconstructing the grain orientation
distribution function – Axial texture, Phys. Met. Metal. 65, 94–99.
Bunge, H. J.: 1969, Mathematische Methoden der Texturanalyse, Akademie-Verlag, Berlin.
Bunge, H. J. (trans. P. R. Morris): 1982, Texture Analysis in Materials Science, Butterworths, London.
Butzer, P. L. and Nessel, R. J.: 1971, Fourier Analysis and Approximation, Birkhäuser, Basel,
Stuttgart.
Dubrovin, B. A., Fomenko, A. T. and Novikov, S. P.: 1984, Modern Geometry Methods and
Applications, Springer-Verlag, New York.
Eschner, T.: 1993, Texture analysis by means of model functions, Textures and Microstructures 21,
139–146.
Eschner, T.: 1994, Generalized model function for quantitative texture analysis: in H. J. Bunge,
S. Siegesmund, W. Skrotzki and K. Weber (eds), Textures of Geological Materials, DGM
Informationsgesellschaft, Oberursel, to appear.
Eschner, T.: 1995, Quantitative Texturanalye durch Komponentenzerlegung von Beugungspolfiguren, Laborbericht PTB-7.4-95-1, Physikalisch-Technische Bundesanstalt, Braunschweig.
deHaas-Lorentz, G. L.: 1913, Die Brownsche Bewegung und einige verwandte Erscheinungen, Friedr.
Vieweg Brunswick.
Gelfand, I. M., Minlos, R. A. and Shapiro, Z. Ya.: 1963, Representations of the Rotation and Lorentz
Groups and Their Applications, Macmillan, New York.
Grenander, U.: 1963, Probabilities on Algebraic Structures, Wiley, New York.
Hartman, P. and Watson, G. S.: 1974, ‘Normal’ distribution functions on spheres and the modified
Bessel functions, Ann. Probab. 2, 593–607.
Helming, K.: 1995, Texturapproximation durch Modellkomponenten, Habilitationsschrift, TU
Clausthal.
Helming, K. and Eschner, T.: 1990, A new approach to texture analysis of multiphase materials using
a texture component model, Crystal. Res. Technol. 25, K203–K208.
Hetherington, T. J.: 1981, Analysis of directional data by exponential models, PhD Thesis, University
of California, Berkeley, U.S.A.
Heyer, H.: 1977, Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin.
Ibragimov, I. A. and Linnik, Yu. V.: 1971, Independent and Stationary Sequences of Random
Variables, Wolters-Nordhoff, Groningen.
James, A. T.: 1954, Normal multivariate analysis and the orthogonal group, Ann. Math. Statist. 25,
40–75.
Jupp, P. E. and Mardia, K. V.: 1989, A unified view of the theory of directional statistics 1979–1989,
Int. Stat. Rev. 57, 261–294.
Kent, J. T.: 1977, The infinite divisibility of the von Mises–Fisher distribution for all values of the
parameter in all dimensions, Proc. London Math. Soc. 35, 359–384.
Kent, J. T.: 1995, pers. comm.
Khatri, C. G. and Mardia, K. V.: 1977, The von Mises–Fisher matrix distribution in orientation
statistics, J. Roy. Statist. Soc., Ser. B 39, 95–106.
Korn, G. A. and Korn, T. M.: 1968, Mathematical Handbook for Scientists and Engineers.
Definitions, Theorems and Formulas for Reference and Review, McGraw-Hill, New York.
ACAP1262.tex; 19/08/1998; 9:16; p.27
86
HELMUT SCHAEBEN AND DMITRY I. NIKOLAYEV
Kunze, K., Heidelbach, S., Wenk, H.-R. and Adams, B. L.: 1994, Orientation imaging microscopy
of calcite rocks, in: H. J., Bunge, S., Siegesmund, W. Skrotzki and K. Weber (eds), Textures of
Geological Materials, DGM Informationsgesellschaft, Oberursel, pp. 127–146.
Mardia, K. V.: 1972, Statistics of Directional Data, Academic Press, New York.
Mardia, K. V.: 1974, Characterizations of directional distributions, in: G. P., Patil, S. Kotz and
J. K. Ord (eds), Statistical Distributions in Scientific Work, vol. 3 (D. Reidel, Dordrecht),
pp. 365–385.
Matthies, S.: 1982, Form effects in the description of the orientation distribution function (odf) of
texturized materials by model components, Phys. Stat. Sol. b 112, 705–716.
Matthies, S., Vinel, G. W. and Helming, K.: 1987, 1988, 1990, Standard Distributions in Texture
Analysis I, II, III, Akademie-Verlag, Berlin.
Matthies, S., Muller, J., and Vinel, G. W.: 1988, On the normal distribution in orientation space,
Textures and Microstructures 10, 77–96.
Nikolayev, D. I., Savyolova, T. I. and Feldmann, K.: 1992, Approximation of the orientation distribution of grains in polycrystalline samples by means of Gaussians, Textures and Microstructures
19, 9–27.
Nikolayev, D. I. and Savyolova, T. I.: 1996, Normal distributions on the rotation group SO(3),
Textures and Microstructures, in press.
Parthasarathy, K. R.: 1964, The central limit theorem for the rotation group, Theory Probab. Appl. 9,
248–257.
Parthasarathy, K. R.: 1967, Probability Measures on Metric Spaces, Academic Press, New York.
Parthasarathy, K. R.: 1995, pers. comm.
Perrin, F.: 1928, Etude mathematique du mouvement Brownien de rotation, Ann. Sci. Ecole Norm.
Sup. 45, 1–51.
Roberts, P. H. and Ursell, H. D.: 1960, Random walk on a sphere and on a Riemannian manifold,
Philos. Trans. Roy. Soc. London, Ser A 252, 317–356.
Roberts, P. H. and Winch, D. E.: 1984, On random rotations, Adv. Appl. Prob. 16, 638–655.
Savelova [identical with Savyolova], T. I.: 1984, Distribution functions of grains with respect to
orientation in polycrystals and their Gaussian approximations, Industrial Laboratory 50, 468–
474; translated from Zavodskaya Laboratoriya 50, 48–52.
Savelova [identical with Savyolova], T. I.: 1989, Computing polar figures and deriving orientation
distributions from them for canonical Gaussian distributions, Industrial Laboratory 55, 1045–
1048; translated from Zavodskaya Laboratoriya 55, 57–60.
Savyolova, T. I.: 1993, Approximation of the pole figures and the orientation distribution of
grains in polycrystalline samples by means of canonical normal distributions, Textures and
Microstructures 22, 17–28.
Savyolova, T. I.: 1995, A normal distribution for texture investigations, Textures and Microstructures,
to appear.
Saw, J. G.: 1984, Ultraspherical polynomials and statistics on the m-sphere, J. Multivariate Anal. 14,
105–113.
Schaeben, H.: 1992, ‘Normal’ orientation distributions, Textures and Microstructures 19, 197–202.
Schaeben, H.: 1996a, A unified view of methods to resolve the inverse problem of texture goniometry,
in: T. I., Savyolova, D. I. Nikolayev, H., Schaeben, and H. J. Bunge (eds), Proc. Workshop on
Mathematical Methods of Texture Analysis, Special Issue Textures and Microstructures 25, 171–
181.
Schaeben, H.: 1996b, Resolving the inverse problem of texture goniometry, submitted to Inverse
Problems.
Schaeben, H.: 1996c, Texture modeling or texture approximation with components represented by
4 , J. Appl.
the von Mises–Fisher matrix distribution on SO(3) and the Bingham distribution on S+
Crystall. 29, 516–525.
ACAP1262.tex; 19/08/1998; 9:16; p.28
THE CENTRAL LIMIT THEOREM
87
Schaeben, H.: 1997, A simple standard orientation density function: The hyperspherical de la Vallée
Poussin kernel, Phys. Stat. Sol. b 200, 367–376.
Vilenkin, N. Ia.: 1968, Special Functions and the Theory of Group Representations, Amer. Math.
Soc., Providence.
Watson, G. S.: 1982, Distributions on the circle and the sphere, in: Essay in Statistical Science,
Applied Probability Trust, J. Appl. Prob. special volume 19A, pp. 265–280.
Watson, G. S.: 1983, Statistics on Spheres, Wiley, New York.
Watson, G. S.: 1992, pers. comm.
Wehn, D.: 1962, Probability on Lie groups, Proc. Natl. Acad. Sci. 48, 791–795.
Yosida, K.: 1949, Brownian motion on the surface of the 3-sphere, Ann. Math. Statist. 20, 292–296.
ACAP1262.tex; 19/08/1998; 9:16; p.29