WAVELET REPRESENTATION OF DIFFRACTION POLE FIGURES, H. Schaeben, J. Prestin, pp. 235-240

Experimental diffraction pole figure data are thought of as being discretely sampled from pole density functions P(h, r), i.e. even probability density functions defined on the cross-product S2 x S2 of two unit spheres. Several useful representations of pole density functions exist which are usually related to specific purposes: (i) series expansion into spherical harmonics, (ii) series expansion into (unimodal) radial basis functions, (iii) series expansion into piecewise constant functions. Their critical parameter may to some extent be adjusted to the total number and/or the spatial arrangement of the intensity measurements. However, they are in no way involved in the sampling process itself. After briefly reviewing the basics of wavelets and the specifics of spherical wavelets, we introduce another representation of pole figures in terms of spherical wavelets. We will show that spherical wavelets are well suited to render pole figures, and to resolve the inverse problem. Moreover, we will demonstrate that wavelets are well apt to allow for locally varying spatial resolution, thus providing a digital device to zoom into pole figure areas of special interest. Considering a measuring time of roughly 1 hour for 1000 intensity values, such a device seems to be required to increase the spatial resolution by a factor of 1000 or greater locally. Eventually, we shall present a promising prospect that wavelets provide the means to control the texture goniometer and the sampling process to gradually adapt automatically to a local refinement of the spatial resolution. REPRESENTATIONS OF POLE DENSITY FUNCTIONS A diffraction pole figure is mathematically represented as the projection of an orientation density function f : SO(3) I+ lR: basically provided by the integral operator (%f)(r) =; ~gES0(3), h=gr) f(ddv(d =P(h, r> (1) where the function P(h, r) : S2 xS2 I+ BI: for a given crystallographic direction h E S2 c lR3 may be referred to as hyperspherical X-ray transform of f with respect to h. The path of integration {g E S0(3)]h = g } r in (1) is a great-circle of the three-dimensional sphere S3 C JR4 parametrized by h and r, cf. [8]. A crystallographic pole density function is the superposition of X-ray transforms with respect to crystal-symmetrically equivalent directions h,, m = 1,. . . , Mh. There are several representations of X-ray transforms and hence pole density functions, including the representation by Copyright(c)JCPDS-International Centre for Diffraction Data 2001,Advances in X-ray Analysis,Vol.44 235