A NOTE ON A GENERALIZED STANDARD ORIENTATION DISTRIBUTION IN PDF-COMPONENT FIT METHODS

In (Eschner, 1993) the author derived a generalized standard model orientation distribution (odf) which he suggests and elaborates for application in texture component fit methods, where the corresponding model pole distributions (pdf’s) are fitted to experimental pdfs. This note is to add some details which seem to have slipped the attention. The new generalized standard model odf is defined by Eschner (1993, p. 141, eq. 10) as f(g, ga, go, al, Otz, a3) C(a,, , oq)exp[tr(diag(a, , ot3)M(g,(’)M(g)m(go-’)m(ga))][dM(g)] (1)

1 THE VON MISES-FISHER MATRIX DISTRIBUTION ON SO(3)   In (Eschner, 1993) the author derived a generalized standard model orientation distribution (odf) which he suggests and elaborates for application in texture component fit methods, where the corresponding model pole distributions (pdf's) are fitted to experimental pdfs.This note is to add some details which seem to have slipped the attention.
The new generalized standard model odf is defined by Eschner (1993, p. 141,  eq.10) as f(g, ga, go, al, Otz, a3) C(a,, , oq)exp[tr(diag(a, , ot3)M(g,(')M(g)m(go-')m(ga))][dM(g)] (1) where tr(A) denotes the trace of a square matrix A, diag(a 1, tx 2, oq) denotes the diagonal matrix with entries tx 1, , a 3, M(o) denotes the matrix M SO(3) representing any orientation in G SO(3), and [dM(o)] denotes the uniform distribution on SO(3).The generalized standard distribution is derived by geometric reasoning, where go denotes the mode ("center of gravity") of the distribution, and gd and oq, a 2, a provide a measure of variation with respect to the mode go in terms of an (rotated) ellipsoid in SO(3).a l, a 2, a relate to the ellipsoid's main axes when it is canonically orientated (axes parallel to the axes of the sample coordinate system), while gd denotes its actual orientation (Eschner, 1993, p. 141).
Applying tr(AB)= tr(BA), eq.(1) may be rewritten as f(g, ga, go, Otl, Otz, 3) Simplifying the notation by M(gd)=A, diag(ct l, M(g) X, eq. ( 2) is rewritten as or3) D, M(go) M, and f(X; M, K)= C(a, ot 2, ot3)exp[tr(ADA'MX')][d which is the von Mises-Fisher matrix distribution Ma(F) on SO(3) (Downs, 1972; Khatri   and Mardia, 1977; Prentice, 1986) with (positive or negative) definite parameter matrix F and X e SO(3).. Let F have singular value decomposition F ADff', with A, F S0(3), and ct l< l< I, then F ADA' AF.The decomposition F KM with K ADA', M AF SO(3), is referred to as polar decomposition into the elliptical component K and the polar component M e SO(3), cf.(Halmos, 1958; Gantmacher,   1960).If D is positive definite, then the distribution has a maximum in M; hence M may be interpreted as the mode of the distribution, and K as a measure of variation with respect to M. If D,, is negative definite, then the distribution has a minimum in M; hence M may be interpreted as the "pole", or rather "center", of an equatorial distribution, and K as a measure of variation with respect to the "girdle" or "fiber" with center M, cf.Schaeben, 1990; 1992.The rank deficient case rank(D)= 2 is not particularly interesting and will not be considered in greater detail.
The family of von Mises-Fisher matrix distributions includes the uniform distribution for ct o h tz 0. In general, the distribution is not symmetrical.If ct tz ct> 0 it reduces to the rotationally invadant unimodal von Mises-Fisher matrix distribution on SO(3) fm(to(g, g0);c0 C(tz)exp(ctcosto) sindo (4)   with o(g, go) J(ggd) denoting the angle to of the rotation gg6-1, which was labeled Gaussian standard odf in texture analysis by Matthies et al., 1987, without reference   to the von Mises-Fisher matrix distribution.
If tz o h tz ct < 0 it reduces to the rotationally invariant equatorial von Mises- Fisher matrix distribution on SO(3) with a maximum in the fiber with center go-It should be noted that the set {g SO(3)lgx y} for any given x, y e S forms a circle on S4.Let go be the center of the circle |g SO(3)lgx=y}.Then with tz<0 f(to(g, g0);o0 (-tzygx) [dto] C(tc)exp(Kygx) [dto]  L(r/(y, gx); r) (5) with to=-ct > 0 and r/(y, gx)= arccos(ygx), which was labeled Gaussian standard odf in case of a fiber texture by Matthies et al., 1987, without reference to the von Mises- Fisher matrix distribution, and without discussion of the rrle of the sign of ct, i.e. whether the parameter matrix F is positive or negative definite.
2 THE BINGHAM DISTRIBUTION ON S 4 + Let x S+ 4 be the Rodrigues unit vector (Becker and Panchanadeeswaran, 1989;  Morawiec and Pospiech, 1989) representing the rotation X g(x) e SO(3).x e S+ is said to be distributed according to the (even) Bingham distribution B4(L, A) (Bingham,  1964; 1974) where [ds] represents the Lebesgue invariant area element on S+ , A e SO(4) is a ( 4x 4) orthogonal matrix, L is a (4 x 4) diagonal matrix with entries 11 14, and cn(L) is a normalizing constant depending on L only.It should be noted that B a(L + 214, A) and na(L, A) are indistinguishable for any e IR1, i. e. the shape parameters i, 1 4, are determined only up to an additive constant.Uniqueness can be imposed by some convention.Different sets of entries of L in eq. ( 6) give the uniform distribution (1 12 13 14), a bipolar distribution (12 + 13 < 1 + 14) with respect to the mode e a e IRa, or an equatorial distribution (l + a < 12 + 13) with respect to the center e a IRa.The intermediate case l + 14 12 + 13 corresponds to the rank deficient case rank(D, )= 2 of the von Mises-Fisher matrix distribution.If 12 13 < 14, then the Bingham distribution reduces to the rotationally invadant (rotationally symmetric) Watson distribution of polar type; if 14 < l 12 13, then x e S+ has a rotationally invariant equatorial Watson distribution.In general the l's are labeled such that 11 < 12 < 13 < 14 in the bipolar case, and a < 13 < 12 < l in the equatorial case, cf. (Prentice, 1986).
Employing the one-one correspondence between SO(3) and S+ 4 the following theorem has been deduced.X SO(3) has a von Mises-Fisher matrix distribution M3(F) if and only if x e S+ has a Bingham distribution B4(L,A) with their parameters related as tr(FX') xtALA'x when x e S+ 4 and/.t(x)X e SO(3) are equivalent representations of the same orientation (Prentice, 1986).
The one-one correspondence of the yon Mises-Fisher matrix and the hyperspherical Bingham distribution follows by a simple trigonometric identity in the special case of rotational invariance.Under these assumptions the Bingham distribution reduces to the hypersphedcal Watson distribution which reads W4(,) C4(,)exp(ossindto (16)   and with 2 cos to 1 + costo it is obviously equivalent to a rotationally invariant 2 yon Mises-Fisher matrix distribution by m do9 (17) f,M(m tO) CM(x')exp(xcosm) sinwith t ,/2, and CM(t C4(2)exp(tc).

CONCLUSIONS
The von Mises-Fisher matrix distribution was explicitly suggested for applications in texture analysis to generate model odfs which are not central in Schaeben (1990).Well known methods exist to calculate the normalization constants or to estimate the parameters of a set of data sampled from a Bingham or von Mises-Fisher matrix distribution, discussed in the context of texture analysis by Schaeben (1993).Even though the model pdfs corresponding to avon Mises-Fisher matrix distribution are not of the same type (Eschner, 1993), these methods may be helpful to know and apply to the problem of best fit of the model pdfs to experimental pdfs.Moreover, the Bingham distribution or von Mises-Fisher matrix distribution provide prerequisites for statistical tests of uniformity and rotational symmetry.
For texture analysts, the text book (Fisher et al., 1987) may prove itself an invaluable source of some more "new" types of model odfs to be introduced into texture analysis.